CE415 Transportation Engineering II

CE415 Transportation Engineering II

Tom Mathew

B. Tech IV Semester

Transportation Systems Engineering

Civil Engineering Department

Indian Institute of Technology Bombay

Powai, Mumbai 400076, India

August 24, 2011

CE415 Transportation Engineering II

Contents

1 Transportation systems analysis 1

2 Introduction to Travel Demand Modeling 9

3 Data Collection 14

4 Trip Generation 22

5 Trip Distribution 27

6 Modal Split 37

7 Trip Assignment 44

8 Fundamental parameters of traffic flow 53

9 Fundamental relations of traffic flow 62

10 Traffic stream models 71

11 Traffic data collection 82

12 Microscopic traffic flow modeling 90

13 Modeling Traffic Characteristics 100

14 Macroscopic traffic flow modeling 108

15 Cell transmission models 110

16 Traffic intersections 120

17 Traffic signs 128

18 Road markings 134

Tom Mathew, IIT Bombay 1 August 24, 2011

19 Traffic rotaries 145

20 Traffic signal design-I 154

21 Traffic signal design-II 166

22 Traffic signal design-III 176

23 HCM Method of Signal Design 179

24 Coordinated signal design 184

25 Area Traffic Control 195

26 Parking 197

27 Congestion Studies 208

CE415 Transportation Engineering II 1. Transportation systems analysis

Chapter 1

Transportation systems analysis

1.1 Goal of Transportation System Analysis

In the last couple of decades transportation systems analysis (TSA) has emerged as a recognizedprofession. More and more government organizations, universities, researchers, consultants, andprivate industrial groups around the world are becoming truly multi-modal in their orientationand are as opting a systematic approach to transportation problems.

1.1.1 Characteristics

1. Multi-modal: Covering all modes or transport; air, land, and sea and both passenger andfreight.

2. Multi-sector: Encompassing the problem,s and viewpoints of government, private indus-try, and public.

3. Multi-problem: Ranging across a spectrum of issues that includes national and interna-tional policy, planning of regional system, the location and design of specific facilities,carrier management issues, regulatory, institutional and financial policies.

4. Multi-objective: National and regional economic development, urban development, envi-ronment quality, and social quality, as well as service to users and financial and economicfeasibility.

5. Multi-disciplinary: Drawing on the theories and methods of engineering, economics, op-eration research, political science, psychology, other natural and social sciences, manage-ment and law.

1.1.2 Context

1. Planning range: Urban transportation planning, producing long range plans for 5-25 yearsfor multi-modal transportation systems in urban areas as well as short range programs ofaction for less than five years.

2. Passenger transport: Regional passenger transportation, dealing with inter-city passengertransport by air, rail, and highway and possible with new modes.



Flow Prediction

Predication of other impacts

Process of analysis

Role of system analyst

Figure 1:1: Role of transportation system analyst

3. Freight transport: routing and management, choice of different modes of rail and truck.

4. International transport: Issues such as containerization, inter-modal co-ordination.

1.1.3 Goal of TSA

In spite of the diversity of problems types, institutional contexts and technical perspectives theiris an underlying unity: a body of theory and set of basic principles to be utilizes in every analysisof transportation systems. The core of this is the transportation system analysis approach. Thefocus of this is the interaction between the transportation and activity systems of region. Thisapproach is to intervene, delicately and deliberately in the complex fabric of society to usetransport effectively in coordination with other public and private actions to achieve the goalsof that society. For this the analyst must have substantial understanding of the transportationsystems and their interaction with activity systems; which requires understanding of the basictheoretical concepts and available empirical knowledge.

1.1.4 Role of TSA

The methodological challenge of transportation systems is to conduct a systematic analysis ina particular situation which is valid, practical, and relevant and which assist in clarifying theissues to debated. The core of the system analysis is the prediction of flows, which must becomplemented by the predication for other impacts. Refer Fig. 1:1 Predication is only a partof the process of analysis and technical analysis is only a part of the broader problem, and therole of the professional transportation system analysis is to model the process of bringing aboutchanges in the society through the means of transport.

1.1.5 Influence of TSA: Applications

Transportation system analysis can lead to different application specialties and they include:

1. highway engineering

2. freight transportation

3. marine transportation



4. transportation management

5. airport planning

6. port planning and development

7. transportation regulation

8. transportation economics

9. environmental impacts

1.1.6 Influence of TSA: Methodologies

Transportation system analysis can also lead to different methodological specialties and theyinclude:

1. Demand analysis, estimation and forecasting

2. transportation system performance like delays, waiting time, mobility, etc.

3. policy analysis and implementation

4. urban planning and development

5. land-use management

1.1.7 Influence of TSA: Methodologies

Finally, transportation system analysis can lead to different professional specialties and theyinclude:

1. technical analyst

2. project managers

3. community interaction

4. policy analyst

1.2 The Scope of TSA

1.2.1 Background: A changing world

The strong interrelationship and the interaction between transportation and the rest of thesociety especially in a rapidly changing world is significant to a transportation planner. Amongthem four critical dimensions of change in transportation system can be identified; which formthe background to develop a right perspective.



1. Change in the demand: When the population, income, and land-use pattern changes, thepattern of demand changes; both in the amount and spatial distribution of that demand.

2. Changes in the technology: As an example, earlier, only two alternatives (bus transit andrail transit) were considered for urban transportation. But, now new system like LRT,MRTS, etc offer a variety of alternatives.

3. Change in operational policy: Variety of policy options designed to improve the efficiency,such as incentive for car-pooling, road pricing etc.

4. Change in values of the public: Earlier all beneficiaries of a system was monolithicallyconsidered as users. Now, not one system can be beneficial to all, instead one mustidentify the target groups like rich, poor, young, work trip, leisure, etc.

1.2.2 Basic premise of a transportation system

The first step in formulation of a system analysis of transportation system is to examine thescope of analytical work. The basic premise is the explicit treatment of the total transportationsystem of region and the interrelations between the transportation and socioeconomic context.

P1 The total transportation system must be viewed as a single multi-modal system.

P2 Considerations of transportation system cannot be separated from considerations of social,economic, and political system of the region.

This follows the following steps for the analysis of transportation system:

• S1 Consider all modes of transportation

• S2 Consider all elements of transportation like persons, goods, carriers (vehicles), pathsin the network facilities in which vehicles are going, the terminal, etc.

• S3 Consider all movements of movements of passengers and goods for every O-D pair.

• S4 Consider the total tip for every flows for every O-D over all modes and facilitates.

As an example consider the the study of inter-city passenger transport in metro cities.

• Consider all modes: i.e rail, road, air, buses, private automobiles, trucks, new modes likeLRT, MRTS, etc.

• Consider all elements like direct and indirect links, vehicles that can operate, terminals,transfer points, intra-city transit like taxis, autos, urban transit.

• Consider diverse pattern of O-D of passenger and good.

• Consider service provided for access, egress, transfer points and mid-block travel etc.

Once all these components are identified, the planner can focus on elements that are of realconcern.



Transportation

System

T

A

Activity System

Flows

F

2

1

3

Figure 1:2: Relationship between T , A and F

1.2.3 Interrelationship of T&A

Transportation system is tightly interrelated with socio-economic system. Transportation affectthe growth and changes of socio-economic system, and will triggers changes in transportationsystem. The whole system of interest can be defined by these basic variables:

T The transportation system including different modes, facilities like highways, etc.

A The socio-economic activity system like work, land-use, housing, schools, etc. Activitysystem is defined as the totality of social, economic, political, and other transactionstaking place over space and time in a given region.

F The flow pattern which includes O-D, routes, volume or passenger/goods, etc.

Three kinds or relationships can be identified as shown in Fig. 1:2 and can be summaries asfollows:

• F is determined by T and A.

• Current F will cause changes over time in A through the pattern of T and through theresources consumed in providing T .

• Current F will also cause changes over time in T due to changes in A

Note that A is not a simple variable as it looks. Also note that transportation is not the soleagency causing changes in A.

1.2.4 Intervening TAF system

The mode of fulfilling the objective of intervening the system of TAF is important. The threemajor player in the TAF system are:

• User The users of the transportation system will decided when where and how to travel.

• Operator The operator of a particular facility or service operator will decide the modeof operation, routes, schedule, facilities, etc.



Technology

Network

Links

Vehicles

Operations

Policy

Travel

Other options

Activity Options

Transport Options

T A F

Systemm

Impacts

User

Operator

GovtGovernment

Phycal

Functional

Figure 1:3: Impact of TAF system

• Government Government will decided on taxes, subsidies, construction of new facilities,governing law, fares, etc.

Their intervention can be in either transportation or activity system. The transportationoptions available to impart changes in the system are:

1. Technology (eg. articulated bus, sky bus, etc.);

2. Network (eg. grid or radial);

3. Link characteristics (eg. signalized or flyover at an intersection);

4. Vehicles (eg. increase the fleet size);

5. System operating policy (eg. increase frequency or subsidy); and

6. Organizational policy (eg. private or public transit system in a city).

On the other hand, some of the activity options are:

1. Travel demand which is the aggregate result of all the individual travel decisions. Thedecision can be travel by train or bus, shortest distance route or shortest travel timeroute, when (time) and how (mode) to travel, etc.

2. Other options Most of the social, economic, and political actors in the activity systemdecide when, how, or where to conduct activities. For example, the choice of schoolis affected by the transportation facility, or the price of real estate influenced by thetransportation facilities.

The impacts of the transportation and activity options mentioned above diverse impact asillustrated in fig. 1:3



1.2.5 Prediction of flows

Any proposed change in transportation system will trigger a change in system activity whichneeds a procedure to predict the impacts. The impact depend upon the pattern of flows resultingfrom particular flows. The core of any TSA is th prediction of changes in flows which is themost significant impact of change in transportation system. Consider present transportationsystem T and activity system A. A particular change in transportation system will be definedin terms of changes in T .

T ′ = T ± ∆T

A′ = A ± ∆A

⇒ F → F ′ (1.1)

Initially, T , A, and F exist in an equilibrium, i.e., specification of transportation system T atany point in time and of activity system A implies the pattern of flows, F . The basic hypothesisunderlying this statement is that there is a market for transportation which can be separatedout from other markets. This is type 1 relationship and can be separated out from type 2 andtype 3 relationships (Fig. 1:2). Introducing two more variables, the first indicated the servicecharacteristics expressed by F like travel time, fare, comfort, etc. which is denoted as S andthe volume of flow in the network denoted as V , following relations can be stated.

1. Specification of transportation system T establishes service function fj which indicatehow the level of service varies as a function of the transportation option and the volumeof flows; i.e.

S = fj(T, V ) (1.2)

2. Specification of the activity system options, A establishes demand function, fd, whichgives the volume of flow as function of activity system and level of service; i.e.

V = fd(A, S) (1.3)

3. The flow pattern F consists of the volume V using the system and level of service S; i.e.

F = (V, S) (1.4)

for a particular T and A, the flow pattern that will actually occur can be found by the solutionof service function and demand function:

S = fj(T, V )

V = fd(A, S)

⇒ (V o, So), i.e.

(T, A) ⇒ (fj , fd)

⇒ [f(A, T )

⇒ V o, So


CE415 Transportation Engineering II 1. Introduction to Travel Demand Modeling

Fj

Fj

Fd

FdFd

Fj’

T constant A constant

T,A constant

Fj S(t)

S(t)

S(t)

S(t)

Vo Vo V1

to

toto

Figure 1:4: Graphical representation of flow prediction

The above relations are shown in Fig. 1:4.



Chapter 2

Introduction to Travel DemandModeling

2.1 Overview

This chapter provides an introduction to travel demand modeling, the most important aspect oftransportation planning. First we will discuss about what is modeling, the concept of transportdemand and supply, the concept of equilibrium, and the traditional four step demand modeling.We may also point to advance trends in demand modeling.

2.2 Transport modeling

Modeling is an important part of any large scale decision making process in any system. Thereare large number of factors that affect the performance of the system. It is not possible forthe human brain to keep track of all the player in system and their interactions and interrela-tionships. Therefore we resort to models which are some simplified, at the same time complexenough to reproduce key relationships of the reality. Modeling could be either physical, sym-bolic, or mathematical In physical model one would make physical representation of the reality.For example, model aircrafts used in wind tunnel is an example of physical models. In sym-bolic model, with the complex relations could be represented with the help of symbols. Drawingtime-space diagram of vehicle movement is a good example of symbolic models. Mathematicalmodel is the most common type when with the help of variables, parameters, and equations onecould represent highly complex relations. Newton’s equations of motion or Einstein’s equationE = mc2, can be considered as examples of mathematical model. No model is a perfect repre-sentation of the reality. The important objective is that models seek to isolate key relationships,and not to replicate the entire structure. Transport modeling is the study of the behavior ofindividuals in making decisions regarding the provision and use of transport. Therefore, unlikeother engineering models, transport modeling tools have evolved from many disciplines likeeconomics, psychology, geography, sociology, and statistics.



Volume

Demand

Equilibrium

Supply

Cost

Figure 2:1: Demand supply equilibrium

2.3 Transport demand and supply

The concept of demand and supply are fundamental to economic theory and is widely appliedin the field to transport economics. In the area of travel demand and the associated supplyof transport infrastructure, the notions of demand and supply could be applied. However, wemust be aware of the fact that the transport demand is a derived demand, and not a need initself. That is, people travel not for the sake of travel, but to practice in activities in differentlocations

The concept of equilibrium is central to the supply-demand analysis. It is a normal practiceto plot the supply and demand curve as a function of cost and the intersection is then plottedin the equilibrium point as shown in Figure 2:1 The demand for travel T is a function of cost Cis easy to conceive. The classical approach defines the supply function as giving the quantity Twhich would be produced, given a market price C. Since transport demand is a derived demand,and the benefit of transportation on the non-monetary terms(time in particular), the supplyfunction takes the form in which C is the unit cost associated with meeting a demand T. Thus,the supply function encapsulates response of the transport system to a given level of demand.In other words, supply function will answer the question what will be the level of service of thesystem, if the estimated demand is loaded to the system. The most common supply functionis the link travel time function which relates the link volume and travel time.

2.4 Travel demand modeling

Travel demand modeling aims to establish the spatial distribution of travel explicitly by meansof an appropriate system of zones. Modeling of demand thus implies a procedure for predictingwhat travel decisions people would like to make given the generalized travel cost of each alter-natives. The base decisions include the choice of destination, the choice of the mode, and thechoice of the route. Although various modeling approaches are adopted, we will discuss onlythe classical transport model popularly known as four-stage model(FSM).

The general form of the four stage model is given in Figure 2:2. The classic model ispresented as a sequence of four sub models: trip generation, trip distribution, modal split, trip



Trip assignment

Trip generation

Trip distribution

flows, trip matrixOutput link

Modal split

Base−year data data, zones

Network

DatabaseFuture

planningdata

Figure 2:2: General form of the four stage modeling

assignment. The models starts with defining the study area and dividing them into a numberof zones and considering all the transport network in the system. The database also include thecurrent (base year) levels of population, economic activity like employment, shopping space,educational, and leisure facilities of each zone. Then the trip generation model is evolved whichuses the above data to estimate the total number of trips generated and attracted by each zone.The next step is the allocation of these trips from each zone to various other destination zonesin the study area using trip distribution models. The output of the above model is a trip matrixwhich denote the trips from each zone to every other zones. In the succeeding step the tripsare allocated to different modes based on the modal attributes using the modal split models.This is essentially slicing the trip matrix for various modes generated to a mode specific tripmatrix. Finally, each trip matrix is assigned to the route network of that particular mode usingthe trip assignment models. The step will give the loading on each link of the network.

The classical model would also be viewed as answering a series of questions (decisions)namely how many trips are generated, where they are going, on what mode they are going, andfinally which route they are adopting. The current approach is to model these decisions usingdiscrete choice theory, which allows the lower level choices to be made conditional on higherchoices. For example, route choice is conditional on the mode choice. This hierarchical choicesof trip is shown in Figure 2:3 The highest level to find all the trips Ti originating from a zoneis calculated based on the data and aggregate cost term Ci***. Based on the aggregate travelcost Cij** from zone i to the destination zone j, the probability pm|ij of trips going to zone jis computed and subsequently the trips Tij** from zone i to zone j by all modes and all routesare computed. Next, the mode choice model compute the probability pm|ij of choosing modem based on the travel cost Cjm* from zone i to zone j, by mode m is determined. Similarly,the route choice gives the trips Tijmr from zone i to zone j by mode m through route r canbe computed. Finally the travel demand is loaded to the supply model, as stated earlier, will



performance

Supply model network

Data

Tijmr

Tijm∗

Tij∗∗

Ti∗∗∗

pj|i

pr|ijm Route choice Cijmr

Destination choice Cij ∗ ∗

Trip frequency (Ci∗∗∗)

Mode choice Cijm∗pm|ij

Figure 2:3: Demand supply equilibrium

produce a performance level. The purpose of the network is usually measured in travel timewhich could be converted to travel cost. Although not practiced ideally, one could feed thisback into the higher levels to achieve real equilibrium of the supply and demand.

2.5 Summary

In a nutshell, travel demand modeling aims at explaining where the trips come from and wherethey go,and what modes and which routes are used. It provides a zone wise analysis of thetrips followed by distribution of the trips, split the trips mode wise based on the choice of thetravelers and finally assigns the trips to the network. This process helps to understand theeffects of future developments in the transport networks on the trips as well as the influence ofthe choices of the public on the flows in the network.

2.6 Problems

1. Link travel time function relates travel time and

(a) link volume

(b) link cost

(c) level of service

(d) none of the above

2. What is the first stage of four-stage travel demand modeling?

(a) Trip generation

(b) Trip distribution

(c) Modal split


CE415 Transportation Engineering II 2. Data Collection

(d) Traffic assignment

2.7 Solutions

1. Link travel time function relates travel time and

(a) link volume√

(b) link cost

(c) level of service

(d) none of the above

2. What is the first stage of four-stage travel demand modeling?

(a) Trip distribution

(b) Trip generation√

(c) Modal split

(d) Traffic assignment



Chapter 3

Data Collection

3.1 Overview

The four-stage modeling, an important tool for forecasting future demand and performanceof a transportation system, was developed for evaluating large-scale infrastructure projects.Therefore, the four-stage modeling is less suitable for the management and control of existingsoftware. Since these models are applied to large systems, they require information about trav-elers of the area influenced by the system. Here the data requirement is very high, and may takeyears for the data collection, data analysis, and model development. In addition, meticulousplanning and systematic approach are needed for accurate data collection and processing. Thischapter covers three important aspects of data collection, namely, survey design, householddata collection, and data analysis. Finally, a brief discussion of other important surveys is alsopresented.

3.2 Survey design

Designing the data collection survey for the transportation projects is not easy. It requiresconsiderable experience, skill, and a sound understanding of the study area. It is also importantto know the purpose of the study and details of the modeling approaches, since data requirementis influenced by these. Further, many practical considerations like availability of time andmoney also has a strong bearing on the survey design. In this section, we will discuss the basicinformation required from a data collection, defining the study area, dividing the area intozones, and transport network characteristics.

3.2.1 Information needed

Typical information required from the data collection can be grouped into four categories,enumerated as below.

1. Socio-economic data: Information regarding the socio-economic characteristics of thestudy area. Important ones include income, vehicle ownership, family size, etc. Thisinformation is essential in building trip generation and modal split models.



8

45

76

12

9

3

Figure 3:1: zoning of a study area

2. Travel surveys: Origin-destination travel survey at households and traffic data fromcordon lines and screen lines (defined later). Former data include the number of tripsmade by each member of the household, the direction of travel, destination, the cost ofthe travel, etc. The latter include the traffic flow, speed, and travel time measurements.These data will be used primarily for the calibration of the models, especially the tripdistribution models.

3. Land use inventory: This includes data on the housing density at residential zones,establishments at commercial and industrial zones. This data is especially useful for tripgeneration models.

4. Network data: This includes data on the transport network and existing inventories.Transport network data includes road network, traffic signals, junctions etc. The serviceinventories include data on public and private transport networks. These particulars areuseful for the model calibration, especially for the assignment models.

3.2.2 Study area

Once the nature of the study is identified, the study area can be defined to encompass the areaof expected policy impact. The study area need not be confirmed by political boundaries, butbounded by the area influenced by the transportation systems. The boundary of the study areais defined by what is called as external cordon or simply the cordon line. A sample of the zoningof a study area is shown in figure 3:1 Interactions with the area outside the cordon are definedvia external stations which effectively serve as doorways to trips, into, out of, and through thestudy area. In short, study area should be defined such that majority of trips have their originand destination in the study area and should be bigger than the area-of-interest covering thetransportation project.

3.2.3 Zoning

Once the study area is defined, it is then divided into a number of small units called trafficanalysis zones (TAZ) or simply zones. The zone with in the study area are called internalzones.



Zones are modeled as if all their attributes and properties were concentrated in a singlepoint called the zonecentroid. The centroids are connected to the nearest road junction or railstation by centroid connectors. Both centroid and centroid connectors are notional and it isassumed that all people have same travel cost from the centroid to the nearest transport facilitywhich is the average for a zone. The intersection from outside world is normally representedthrough external zones. The external zones are defined by the catchment area of the majortransport links feeding to the study area. Although the list is not complete, few guidelines aregiven below for selecting zones.

1. zones should match other administrative divisions, particularly census zones.

2. zones should have homogeneous characteristics, especially in land use, population etc.

3. zone boundaries should match cordon and screen lines, but should not match major roads.

4. zones should be as smaller in size as possible so that the error in aggregation caused bythe assumption that all activities are concentrated at the zone centroids is minimum.

3.2.4 Network

Transport network consists of roads,junctions, bus stops, rails, railway station etc. Normallyroad network and rail network are represented separately. Road network is considered asdirected graph of nodes and links. Each node and links have their own properties. Road linkis normally represented with attributes like starting node, ending node, road length, free flowspeed, capacity, number of lanes or road width, type of road like divided or undivided etc.Road junctions or nodes are represented with attributes like node number, starting nodes of alllinks joining the current node, type of intersection (uncontrolled, round about, signalized, etc.).Similarly public transport network like bus transit network and rail network are represented,but with attributes relevant to them. These may include frequency of service, fare of travel,line capacity, station capacity etc. This completes the inventory of base-year transportationfacility.

3.3 Household data

To understand the behavior and factors affecting the travel, one has got the origin of travelwhen the decision for travel is made. It is where people live as family which is the household.Therefore household data is considered to be the most basic and authentic information aboutthe travel pattern of a city.

Ideally one should take the details of all the people in the study to get complete travel details.However, this is not feasible due to large requirement of time and resources needed. In additionthis will cause difficulties in handling these large data in modeling stage. Therefore, samesample households are randomly selected and survey is conducted to get the household data.Higher sample size is required fro large population size, and vice-versa. Normally minimum tenpercent samples are required for population less than 50,000. But for a population more thanone million require only one percent for the same accuracy.



3.3.1 Questionnaire design

The next step in the survey is the questionnaire design. A good design will ensure betterresponse from the respondent and will significantly improve the quality of data. Design ofquestionnaire is more of an art than a science. However few guiding principles can be laidout. The questionnaire should be simple, direct, should take minimum time, and should causeminimum burden to the respondent. Traditional household survey has three major sections;household characteristics, personal characteristics, and trip details.

Household characteristics This section includes a set of questions designed to obtainsocioeconomic information about the household. Relevant questions are:number of members inthe house, no.of employed people, number of unemployed people, age and sex of the membersin the house etc., number of two-wheelers in the house, number of cycles, number of cars in thehouse etc., house ownership and family income.

Personal characteristics This part includes questions designed to classify the householdmembers(older than 5) according to the following aspects:relation to the head of the household(e.g. wife, son), sex, age, possession of a driving license, educational level, and activity.

Trip data This part of the survey aims at detecting and characterizing all trips made bythe household members identified in the first part. A trip is normally defined as any movementgreater than 300 meters from an origin to a destination with a given purpose. Trips arecharacterized on the basis of variables such as: origin and destination, trip purpose, trip startand ending times, mode used, walking distance, public-transport line and transfer station orbus stop (if applicable).

3.3.2 Survey administration

Once the questionnaire is ready, the next step is to conduct the actual survey with the helpof enumerators. Enumerators has to be trained first by briefing them about the details of thesurvey and how to conduct the survey. They will be given random household addresses andthe questionnaire set. They have to first get permission to be surveyed from the household.They may select a typical working day for the survey and ask the members of the householdabout the details required in the questionnaire. They may take care that each member of thehousehold should answer about their own travel details, except for children below 12 years.Trip details of children below 5 years are normally ignored. Since the actual survey may takeplace any time during the day, the respondents are required to answer the question about thetravel details of the previous day.

There are many methods of the administration of the survey and some of them are discussedbelow:

1. Telephonic: The enumerator may use telephone to fix an appointment and then conductdetailed telephonic interview. This is very popular in western countries where phonepenetration is very high.

2. Mail back: The enumerator drops the questionnaire to the respondent and asks themto fill the details and mail them back with required information. Care should be takento design the questionnaire so that it is self explanatory.



3. Face-to-face In this method, the enumerator visits the home of the respondent andasks the questions and fills up the questionnaire by himself. This is not a very sociallyacceptable method in the developed countries, as these are treated as intrusion to privacy.However, in many developed countries, especially with less educated people, this is themost effective method.

3.4 Data preparation

The raw data collected in the survey need to be processed before direct application in themodel. This is necessary, because of various errors, except in the survey both in the selectionof sample houses as well as error in filling details. In this section, we will discuss three aspectsof data preparation; data correction, data expansion, and data validation.

3.4.1 Data correction

Various studies have identified few important errors that need to be corrected, and are listedbelow.

1. Household size correction It may be possible that while choosing the random samples,one may choose either larger or smaller than the average size of the population as observedin the census data and correction should be made accordingly.

2. Socio-demographic corrections It is possible that there may be differences betweenthe distribution of the variables sex, age, etc. between the survey, and the population asobserved from the census data. This correction is done after the household size correction.

3. Non-response correction It is possible that there may not be a response from manyrespondents, possible because they are on travel everyday. Corrections should be madeto accommodate this, after the previous two corrections.

4. Non-reported trip correction In many surveys people underestimate the non-mandatorytrips and the actual trips will be much higher than the reported ones. Appropriate cor-rection need to be applied for this.

3.4.2 Sample expansion

The second step in the data preparation is to amplify the survey data in order to represent thetotal population of the zone. This is done with the help of expansion factor which is defined asthe ratio of the total number of household addressed in the population to that of the surveyed.A simple expansion factor Fi for the zone i could be of the following form.

Fi =a

b − d(3.1)



where a is the total number of household in the original population list, b is the total numberof addresses selected as the original sample, and d is the number of samples where no responsewas obtained.

3.4.3 Validation of results

In order to have confidence on the data collected from a sample population, three validationtests are adopted usually. The first simply considers the consistency of the data by a field visitnormally done after data entry stage. The second validation is done by choosing a computa-tional check of the variables. For example, if age of a person is shown some high unrealisticvalues like 150 years. The last is a logical check done for the internal consistency of the data.For example, if the age of a person is less than 18 years, then he cannot have a driving license.Once these corrections are done, the data is ready to be used in modeling.

3.5 Other surveys

In addition to the household surveys, these other surveys are needed for complete modelinginvolving four stage models. Their primary use is for the calibration and validation of themodels, or act as complementary to the household survey. These include O-D surveys, roadside interviews, and cordon and screen line counts.

3.5.1 O-D survey

Sometime four small studies, or to get a feel of the O-D pattern without doing elaborate survey,work space interviews are conducted to find the origin-destination of employers in a location.Although they are biased in terms of the destination, they are random in terms of the mode oftravel.

3.5.2 Road side interviews

These provide trips not registered in a household survey, especially external-internal trips. Thisinvolves asking questions to a sample of drivers and passengers of vehicles crossing a particularlocation. Unlike household survey, the respondent will be asked with few questions like origin,destination, and trip purpose. Other information like age, sex, and income can also be added,but it should be noted that at road-side, drivers will not be willing to spend much time forsurvey.

3.5.3 Cordon and screen-line survey

These provide useful information about trips from and to external zones. For large study area,internal cordon-line can be defined and surveying can be conducted. The objective of the surveyis primarily to collect the origin and destination zones and for this many suitable methods can



be adopted. It could be either recording the license plate number at all the external cordonpoints or by post-card method.

Screen lines divide the study area into large natural zones, like either sides of a river, withfew crossing points between them. The procedure for both cordon-line and screen-line surveyare similar to road-side interview. However, these counts are primarily used for calibration andvalidation of the models.

3.6 Summary

Data collection is one of the most important steps in modeling. Only if accurate data isavailable, modeling becomes successful. Survey design is discussed in detail. Household datagives important information required for data collection. Questionnaire should be simple, lesstime consuming and should be designed such that the required information is obtained with lessburden on the respondent. Data collected should be prepared well before application. Variouscorrections should be made in data collection before they are used in modeling. Finally, othertypes of surveys are also discussed.

3.7 Problems

1. The data that is useful for developing trip generation models is

(a) Travel survey data

(b) Land-use inventory data

(c) Network data

(d) None of these

2. Which of the following is not a criterion for zoning?

(a) zones should match other administrative divisions, particularly census zones.

(b) zones should have homogeneous characteristics, especially in land use, populationetc.

(c) zone boundaries should match cordon and screen lines, but should not match majorroads.

(d) zones should have regular geometric shape.

3.8 Solutions

1. The data that is useful for developing trip generation models is

(a) Travel survey data


CE415 Transportation Engineering II 3. Trip Generation

(b) Land-use inventory data√

(c) Network data

(d) None of these

2. Which of the following is not a criterion for zoning?

(a) zones should match other administrative divisions, particularly census zones.

(b) zones should have homogeneous characteristics, especially in land use, populationetc.

(c) zone boundaries should match cordon and screen lines, but should not match majorroads.

(d) zones should have regular geometric shape√



Chapter 4

Trip Generation

4.1 Overview

Trip generation is the first stage of the classical first generation aggregate demand models. Thetrip generation aims at predicting the total number of trips generated and attracted to each zoneof the study area. In other words this stage answers the questions to “how many trips” originateat each zone, from the data on household and socioeconomic attributes. In this section basicdefinitions, factors affecting trip generation, and the two main modeling approaches; namelygrowth factor modeling and regression modeling are discussed.

4.1.1 Types of trip

Some basic definitions are appropriate before we address the classification of trips in detail.We will attempt to clarify the meaning of journey, home based trip, non home based trip, tripproduction, trip attraction and trip generation.

Journey is an out way movement from a point of origin to a point of destination, where asthe word “trip” denotes an outward and return journey. If either origin or destination of a tripis the home of the trip maker then such trips are called home based trips and the rest of thetrips are called non home based trips. Trip production is defined as all the trips of home basedor as the origin of the non home based trips. See figure 4:1

Trips can be classified by trip purpose, trip time of the day, and by person type. Trip

Work

Shop

Non−homebased trips

tripsHome

based

Work

Home

Production

Attraction

Attraction

Attraction

Production

Production

Attraction

Production

Figure 4:1: trip types



generation models are found to be accurate if separate models are used based on trip purpose.The trips can be classified based on the purpose of the journey as trips for work, trips foreducation, trips for shopping, trips for recreation and other trips. Among these the work andeducation trips are often referred as mandatory trips and the rest as discretionary trips. Allthe above trips are normally home based trips and constitute about 80 to 85 percent of trips.The rest of the trips namely non home based trips, being a small proportion are not normallytreated separately. The second way of classification is based on the time of the day when thetrips are made. The broad classification is into peak trips and off peak trips. The third wayof classification is based on the type of the individual who makes the trips. This is importantsince the travel behavior is highly influenced by the socio economic attribute of the travelerand are normally categorized based on the income level, vehicle ownership and house hold size.

4.1.2 Factors affecting trip generation

The main factors affecting personal trip production include income, vehicle ownership, household structure and family size. In addition factors like value of land, residential density andaccessibility are also considered for modeling at zonal levels. The personal trip attraction, onthe other hand, is influenced by factors such as roofed space available for industrial, commercialand other services. At the zonal level zonal employment and accessibility are also used. In tripgeneration modeling in addition to personal trips, freight trips are also of interest. Althoughthe latter comprises about 20 percent of trips, their contribution to the congestion is significant.Freight trips are influenced by number of employees, number of sales and area of commercialfirms.

4.2 Growth factor modeling

Growth factor modes tries to predict the number of trips produced or attracted by a household or zone as a linear function of explanatory variables. The models have the following basicequation:

Ti = fiti (4.1)

where Ti is the number of future trips in the zone and ti is the number of current trips in thatzone and fi is a growth factor. The growth factor fi depends on the explanatory variable suchas population (P) of the zone , average house hold income (I) , average vehicle ownership (V).The simplest form of fi is represented as follows

fi =P d

i × Idi × V d

i

P ci × Ic

i × V ci

(4.2)

where the subscript ” d” denotes the design year and the subscript ”c” denotes the currentyear.



Example

Given that a zone has 275 household with car and 275 household without car and the averagetrip generation rates for each groups is respectively 5.0 and 2.5 trips per day. Assuming thatin the future, all household will have a car, find the growth factor and future trips from thatzone, assuming that the population and income remains constant.

Solution

Current trip rate ti = 275 × 2.5 + 275 × 5.0 = 2062.5 trips / day.

Growth factor Fi =V d

i

V ci

= 550275

= 2.0

Therefore, no. of future trips Ti = Fiti = 2.0 × 2062.5 = 4125 trips / day.The above example also shows the limitation of growth factor method. If we thinkintuitively, the trip rate will remain same in the future.

Therefore the number of trips in the future will be 550 house holds × 5 trips per day =2750 trips per day .

It may be noted from the above example that the actual trips generated is much lower than thegrowth factor method. Therefore growth factor models are normally used in the prediction ofexternal trips where no other methods are available. But for internal trips , regression methodsare more suitable and will be discussed in the following section.

4.3 Regression methods

The general form of a trip generation model is

Ti = f(x1, x2, x3, ....xi, ...xk) (4.3)

Where xi’s are prediction factor or explanatory variable. The most common form of tripgeneration model is a linear function of the form

Ti = a0 + a1x1 + a2x2 + ...aixi... + akxk (4.4)

where ai ’s are the coefficient of the regression equation and can be obtained by doing regressionanalysis. The above equations are called multiple linear regression equation, and the solutionsare tedious to obtain manually. However for the purpose of illustration, an example with onevariable is given.

Example

Let the trip rate of a zone is explained by the household size done from the field survey. It wasfound that the household size are 1, 2, 3 and 4. The trip rates of the corresponding householdis as shown in the table below. Fit a linear equation relating trip rate and household size.



Household size(x)1 2 3 4

Trips 1 2 4 6per 2 4 5 7day(y) 2 3 3 4Σy 5 9 12 17

Solution The linear equation will have the form y = bx + a where y is the trip rate, and x isthe household size, a and b are the coefficients. For a best fit, b is given by

b =nΣxy − ΣxΣy

nΣx2 − (Σx)2

a = y − bx

Σx = 3 × 1 + 3 × 2 + 3 × 3 + 3 × 4 = 30

Σx2 = 3 × (12) + 3 × (22) + 3 × (32) + 3 × (42) = 90

Σy = 5 + 9 + 12 + 17 = 43

Σxy = 1 × 1 + 1 × 2 + 1 × 2

+ 2 × 2 + 2 × 4 + 2 × 3

+ 3 × 4 + 3 × 5 + 3 × 3

+ 4 × 6 + 4 × 7 + 4 × 4

= 127

y = 43/12 = 3.58

x = 30/12 = 2.5

b =nΣxy − ΣxΣy

nΣx2 − (Σx)2

=((12 × 127) − (30 × 43))

((12 × 90) − (30)2)= 1.3

a = y − bx = 3.58 − 1.3 × 2.5 = +0.33

y = 1.3x − 0.33

4.4 Summary

Trip generation forms the first step of four-stage travel modeling. It gives an idea about thetotal number of trips generated to and attracted from different zones in the study area. Growthfactor modeling and regression methods can be used to predict the trips. They are discussedin detail in this chapter.


CE415 Transportation Engineering II 4. Trip Distribution

4.5 Problems

1. The trip rate (y) and the corresponding household sizes (x) from a sample are shownin table below. Compute the trip rate if the average household size is 3.25 (Hint: useregression method).

Householdsize(x)1 2 3 4

Trips 1 3 4 5per 3 4 5 8day(y) 3 5 7 8

Solution Fit the regression equation as below.

Σx = 3 × 1 + 3 × 2 + 3 × 3 + 3 × 4 = 30

Σx2 = 3 × (12) + 3 × (22) + 3 × (32) + 3 × (42) = 90

Σy = 7 + 12 + 16 + 21 = 56

Σxy = 1 × 1 + 1 × 3 + 1 × 3

+ 2 × 3 + 2 × 4 + 2 × 5

+ 3 × 4 + 3 × 5 + 3 × 7

+ 4 × 5 + 4 × 8 + 4 × 8

= 163

y = 56/12 = 4.67

x = 30/12 = 2.5

b =nΣxy − ΣxΣy

nΣx2 − (Σx)2

=((12 × 163) − (30 × 56))

((12 × 90) − (30)2)= 1.533

a = y − bx = 4.67 − 1.533 × 2.5 = 0.837

y = 0.837 + 1.533x

When average household size =3.25, number of trips becomes,y = 0.837 + 1.533 × 3.25 = 5.819



Chapter 5

Trip Distribution

5.1 Overview

The decision to travel for a given purpose is called trip generation. These generated tripsfrom each zone is then distributed to all other zones based on the choice of destination. Thisis called trip distribution which forms the second stage of travel demand modeling. Thereare a number of methods to distribute trips among destinations; and two such methods aregrowth factor model and gravity model. Growth factor model is a method which respond onlyto relative growth rates at origins and destinations and this is suitable for short-term trendextrapolation. In gravity model, we start from assumptions about trip making behavior andthe way it is influenced by external factors. An important aspect of the use of gravity models istheir calibration, that is the task of fixing their parameters so that the base year travel patternis well represented by the model.

5.2 Definitions and notations

5.2.1 Trip matrix

The trip pattern in a study area can be represented by means of a trip matrix or origin-destination (O-D)matrix. This is a two dimensional array of cells where rows and columnsrepresent each of the zones in the study area. The notation of the trip matrix is given infigure 5:1.

The cells of each row i contain the trips originating in that zone which have as destinationsthe zones in the corresponding columns. Tij is the number of trips between origin i anddestination j. Oi is the total number of trips between originating in zone i and Dj is the totalnumber of trips attracted to zone j. The sum of the trips in a row should be equal to the totalnumber of trips emanating from that zone. The sum of the trips in a column is the numberof trips attracted to that zone. These two constraints can be represented as:

∑

j Tij = Oi∑

i Tij = Dj If reliable information is available to estimate both Oi and Dj , the model is saidto be doubly constrained. In some cases, there will be information about only one of theseconstraints, the model is called singly constrained.



5.2.2 Generalized cost

One of the factors that influences trip distribution is the relative travel cost between two zones.This cost element may be considered in terms of distance, time or money units. It is oftenconvenient to use a measure combining all the main attributes related to the dis-utility of ajourney and this is normally referred to as the generalized cost of travel. This can be representedas

cij = a1 tvij + a2 twij + a3 ttij + a4 Fij + a5 φj + δ (5.1)

where tvij is the in-vehicle travel time between i and j, twij is the walking time to and from stops,ttij is the waiting time at stops, Fij is the fare charged to travel between i and j, φj is theparking cost at the destination, and δ is a parameter representing comfort and convenience,and a1, a2, .... are the weights attached to each element of the cost function.

5.3 Growth factor methods

5.3.1 Uniform growth factor

If the only information available is about a general growth rate for the whole of the study area,then we can only assume that it will apply to each cell in the matrix, that is a uniform growthrate. The equation can be written as:

Tij = f × tij (5.2)

where f is the uniform growth factor tij is the previous total number of trips, Tij is the expandedtotal number of trips. Advantages are that they are simple to understand, and they are usefulfor short-term planning. Limitation is that the same growth factor is assumed for all zones aswell as attractions.

Zones 1 2 . . . j . . . n Productions1 T11 T12 . . . T1j . . . T1n O1

2 T21 T22 . . . T2j . . . T2n O2... . . . . . . . . . . . . . . . . . .

...Ti1 Ti2 . . . Tij . . . Tin Oi

... . . . . . . . . . . . . . . . . . ....

n Tni Tn2 . . . Tnj . . . Tnn On

Attractions D1 D2 . . . Dj . . . Dn Twhere Dj =

∑

i Tij , Oi =∑

j Tij , and T =∑

ij Tij.

Figure 5:1: Notation of an origin-destination trip matrix



5.3.2 Example

Trips originating from zone 1, 2, and 3 of a study area are 78, 92 and 82 respectively and thoseterminating at zones 1, 2, and 3 are given as 88, 96 and 78 respectively. If the growth factoris 1.3 and the base year trip matrix is as given below, find the expanded origin-constrainedgrowth trip table.

1 2 3 oi

1 20 30 28 782 36 32 24 923 22 34 26 82dj 88 96 78 252

Solution Given growth factor = 1.3, Therefore, multiplying the growth factor with each ofthe cells in the matrix gives the solution as shown below.

1 2 3 Oi

1 26 39 36.4 101.42 46.8 41.6 31.2 119.63 28.6 44.2 33.8 106.2

Dj 101.4 124.8 101.4 327.6

5.3.3 Doubly constrained growth factor model

When information is available on the growth in the number of trips originating and terminatingin each zone, we know that there will be different growth rates for trips in and out of each zoneand consequently having two sets of growth factors for each zone. This implies that there aretwo constraints for that model and such a model is called doubly constrained growth factormodel. One of the methods of solving such a model is given by Furness who introduced balancingfactors ai and bj as follows:

Tij = tij × ai × bj (5.3)

In such cases, a set of intermediate correction coefficients are calculated which are thenappropriately applied to cell entries in each row or column. After applying these corrections tosay each row, totals for each column are calculated and compared with the target values. If thedifferences are significant, correction coefficients are calculated and applied as necessary. Theprocedure is given below:

1. Set bj = 1

2. With bj solve for ai to satisfy trip generation constraint.

3. With ai solve for bj to satisfy trip attraction constraint.

4. Update matrix and check for errors.

5. Repeat steps 2 and 3 till convergence.



Here the error is calculated as: E =∑ |Oi − O1

i | +∑ |Dj − D1

j | where Oi corresponds to theactual productions from zone i and O1

i is the calculated productions from that zone. SimilarlyDj are the actual attractions from the zone j and D1

j are the calculated attractions from thatzone.

5.3.4 Advantages and limitations of growth factor model

The advantages of this method are:

1. Simple to understand.

2. Preserve observed trip pattern.

3. Useful in short term-planning.

The limitations are:

1. Depends heavily on the observed trip pattern.

2. It cannot explain unobserved trips.

3. Do not consider changes in travel cost.

4. Not suitable for policy studies like introduction of a mode.

Example

The base year trip matrix for a study area consisting of three zones is given below.

1 2 3 oi

1 20 30 28 782 36 32 24 923 22 34 26 82dj 88 96 78 252

The productions from the zone 1,2 and 3 for the horizon year is expected to grow to 98, 106,and 122 respectively. The attractions from these zones are expected to increase to 102, 118, 106respectively. Compute the trip matrix for the horizon year using doubly constrained growthfactor model using Furness method.

Solution The sum of the attractions in the horizon year, i.e.∑

Oi = 98+106+122 = 326.The sum of the productions in the horizon year, i.e.

∑

Dj = 102+118+106 = 326. They bothare found to be equal. Therefore we can proceed. The first step is to fix bj = 1, and findbalancing factor ai. ai = Oi/oi, then find Tij = ai × tij

So a1 = 98/78 = 1.26a2 = 106/92 = 1.15a3 = 122/82 = 1.49 Further T11 = t11 × a1 = 20 × 1.26 = 25.2. Similarly T12 = t12 × a2 =

36 × 1.15 = 41.4. etc. Multiplying a1 with the first row of the matrix, a2 with the second rowand so on, matrix obtained is as shown below.



1 2 3 oi

1 25.2 37.8 35.28 982 41.4 36.8 27.6 1063 32.78 50.66 38.74 122d1

j 99.38 125.26 101.62

Dj 102 118 106

Also d1j = 25.2 + 41.4 + 32.78 = 99.38

In the second step, find bj = Dj/d1j and Tij = tij × bj . For example b1 = 102/99.38 = 1.03,

b2 = 118/125.26 = 0.94 etc.,T11 = t11× b1 = 25.2×1.03 = 25.96 etc. Also O1i = 25.96+35.53+

36.69 = 98.18. The matrix is as shown below:

1 2 3 oi Oi

1 25.96 35.53 36.69 98.18 982 42.64 34.59 28.70 105.93 1063 33.76 47.62 40.29 121.67 122bj 1.03 0.94 1.04Dj 102 118 106

1 2 3 O1i Oi

1 25.96 35.53 36.69 98.18 982 42.64 34.59 28.70 105.93 1063 33.76 47.62 40.29 121.67 122dj 102.36 117.74 105.68 325.78Dj 102 118 106 326

Therefore error can be computed as ; Error =∑ |Oi − O1

i | +∑ |Dj − dj|

Error = |98.18 − 98| + |105.93 − 106| + |121.67 − 122| + |102.36 − 102| + |117.74 − 118| +|105.68 − 106| = 1.32

5.4 Gravity model

This model originally generated from an analogy with Newton’s gravitational law. Newton’sgravitational law says, F = G M1 M2 / d2 Analogous to this, Tij = C Oi Dj / cn

ij Introducingsome balancing factors, Tij = Ai Oi Bj Dj f(cij) where Ai and Bj are the balancing factors,f(cij) is the generalized function of the travel cost. This function is called deterrence functionbecause it represents the disincentive to travel as distance (time) or cost increases. Some of theversions of this function are:

f(cij) = e−βcij

f(cij) = c−nij

f(cij) = c−nij × e−βcij



The first equation is called the exponential function, second one is called power function whereas the third one is a combination of exponential and power function. The general form of thesefunctions for different values of their parameters is as shown in figure.

As in the growth factor model, here also we have singly and doubly constrained models.The expression Tij = Ai Oi Bj Dj f(cij) is the classical version of the doubly constrainedmodel. Singly constrained versions can be produced by making one set of balancing factors Ai

or Bj equal to one. Therefore we can treat singly constrained model as a special case whichcan be derived from doubly constrained models. Hence we will limit our discussion to doublyconstrained models.

As seen earlier, the model has the functional form, Tij = AiOiBjDjf(cij)

∑

i

Tij =∑

i

AiOiBjDjf(cij) (5.4)

But∑

i

Tij = Dj (5.5)

Therefore,Dj = BjDj

∑

i

AiOif(cij) (5.6)

From this we can find the balancing factor Bj as

Bj =1

∑

i AiOif(cij)(5.7)

Bj depends on Ai which can be found out by the following equation:

Ai =1

∑

j BjDjf(cij)(5.8)

We can see that both Ai and Bj are interdependent. Therefore, through some iteration proce-dure similar to that of Furness method, the problem can be solved. The procedure is discussedbelow:

1. Set Bj = 1, find Ai using equation 5.8

2. Find Bj using equation 5.7

3. Compute the error as E =∑ |Oi − O1

i |+∑ |Dj − D1

j | where Oi corresponds to the actualproductions from zone i and O1

i is the calculated productions from that zone. SimilarlyDj are the actual attractions from the zone j and D1

j are the calculated attractions fromthat zone.

4. Again set Bj = 1 and find Ai, also find Bj. Repeat these steps until the convergence isachieved.



Example

The productions from zone 1, 2 and 3 are 98, 106, 122 and attractions to zone 1,2 and 3 are102, 118, 106. The function f(cij) is defined as f(cij) = 1/c2

ij The cost matrix is as shownbelow

1.0 1.2 1.81.2 1.0 1.51.8 1.5 1.0

(5.9)

Solution The first step is given in Table 5:1 The second step is to find Bj. This can be

Table 5:1: Step1: Computation of parameter Ai

i j Bj DJ f(cij) BjDjf(cij)∑

BjDjf(cij) Ai = 1∑

BjDjf(cij)

1 1.0 102 1.0 102.001 2 1.0 118 0.69 81.42 216.28 0.00462

3 1.0 106 0.31 32.861 1.0 102 0.69 70.38

2 2 1.0 118 1.0 118 235.02 0.004253 1.0 106 0.44 46.641 1.0 102 0.31 31.62

3 2 1.0 118 0.44 51.92 189.54 0.005273 1.0 106 1.00 106

found out as Bj = 1/∑

AiOif(cij), where Ai is obtained from the previous step. The detailedcomputation is given in Table 5:2. The function f(cij) can be written in the matrix form as:

Table 5:2: Step2: Computation of parameter Bj

j i Ai Oi f(cij) AiOif(cij)∑

AiOif(cij) Bj = 1/∑

AiOif(cij)1 0.00462 98 1.0 0.4523

1 2 0.00425 106 0.694 0.3117 0.9618 1.03973 0.00527 122 0.308 0.19781 0.00462 98 0.69 0.3124

2 2 0.00425 106 1.0 0.4505 1.0458 0.95623 0.00527 122 0.44 0.28291 0.00462 98 0.31 0.1404

3 2 0.00425 106 0.44 0.1982 0.9815 1.01883 0.00527 122 1.00 0.6429

1.0 0.69 0.310.69 1.0 0.440.31 0.44 1.0

(5.10)



Table 5:3: Step3: Final Table1 2 3 Ai Oi O1

i

1 48.01 35.24 15.157 0.00462 98 98.4072 32.96 50.83 21.40 0.00425 106 105.193 21.14 31.919 69.43 0.00527 122 122.489Bj 1.0397 0.9562 1.0188Dj 102 118 106D1

j 102.11 117.989 105.987

Then Tij can be computed using the formula

Tij = AiOiBjDjf(cij) (5.11)

For eg, T11 = 102 × 1.0397 × 0.00462 × 98 × 1 = 48.01. Oi is the actual productions from thezone and O1

i is the computed ones. Similar is the case with attractions also. The results areshown in table 5:3. Oi is the actual productions from the zone and O1

i is the computed ones.Similar is the case with attractions also.

Therefore error can be computed as ; Error =∑ |Oi − O1

i | +∑ |Dj − D1

j | Error = |98 −98.407|+|106−105.19|+|122−122.489|+||102−102.11|+|118−117.989|+|106−105.987| = 2.03

5.5 Summary

The second stage of travel demand modeling is the trip distribution. Trip matrix can be usedto represent the trip pattern of a study area. Growth factor methods and gravity model areused for computing the trip matrix. Singly constrained models and doubly constrained growthfactor models are discussed. In gravity model, considering singly constrained model as a specialcase of doubly constrained model, doubly constrained model is explained in detail.

5.6 Problems

The trip productions from zones 1, 2 and 3 are 110, 122 and 114 respectively and the tripattractions to these zones are 120,108, and 118 respectively. The cost matrix is given below.The function f(cij) = 1

cij

1.0 1.2 1.81.2 1.0 1.51.8 1.5 1.0

Compute the trip matrix using doubly constrained gravity model. Provide one complete itera-tion.



Solution The first step is given in Table 5:4 The second step is to find Bj . This can be

Table 5:4: Step1: Computation of parameter Ai

i j Bj DJ f(cij) BjDjf(cij)∑

BjDjf(cij) Ai = 1∑

BjDjf(cij)

1 1.0 120 1.0 120.001 2 1.0 108 0.833 89.964 275.454 0.00363

3 1.0 118 0.555 65.491 1.0 120 0.833 99.96

2 2 1.0 108 1.0 108 286.66 0.003483 1.0 118 0.667 78.7061 1.0 120 0.555 66.60

3 2 1.0 108 0.667 72.036 256.636 0.003893 1.0 118 1.00 118

found out as Bj = 1/∑

AiOif(cij), where Ai is obtained from the previous step. The functionf(cij) can be written in the matrix form as:

1.0 0.833 0.5550.833 1.0 0.6670.555 0.667 1.0

(5.12)

Then Tij can be computed using the formula

Tij = AiOiBjDjf(cij) (5.13)

For eg, T11 = 102 × 1.0397 × 0.00462 × 98 × 1 = 48.01. Oi is the actual productions from thezone and O1

i is the computed ones. Similar is the case with attractions also. This step is givenin Table 5:6 Oi is the actual productions from the zone and O1

i is the computed ones. Similar

Table 5:5: Step2: Computation of parameter Bj

j i Ai Oi f(cij) AiOif(cij)∑

AiOif(cij) Bj = 1/∑

AiOif(cij)1 0.00363 110 1.0 0.3993

1 2 0.00348 122 0.833 0.3536 0.9994 1.0483 0.00389 114 0.555 0.24651 0.00363 110 0.833 0.3326

2 2 0.00348 122 1.0 0.4245 1.05 0.94943 0.00389 114 0.667 0.29621 0.00363 110 0555 0.2216

3 2 0.00348 122 0.667 0.2832 0.9483 1.0543 0.00389 114 1.00 0.44346


CE415 Transportation Engineering II 5. Modal Split

Table 5:6: Step 3: Final Table1 2 3 Ai Oi O1

i

1 48.01 34.10 27.56 0.00363 110 109.572 42.43 43.53 35.21 0.00348 122 121.173 29.53 30.32 55.15 0.00389 114 115Bj 1.048 0.9494 1.054Dj 120 108 118D1

j 119.876 107.95 117.92

is the case with attractions also.Therefore error can be computed as ; Error =

∑ |Oi − O1i | +

∑ |Dj − D1j | Error = |110 −

109.57|+ |122−121.17|+ |114−115|+ |120−119.876+ |108−107.95|+ |118−117.92| = 2.515



Chapter 6

Modal Split

6.1 Overview

The third stage in travel demand modeling is modal split. The trip matrix or O-D matrixobtained from the trip distribution is sliced into number of matrices representing each mode.First the significance and factors affecting mode choice problem will be discussed. Then a briefdiscussion on the classification of mode choice will be made. Two types of mode choice modelswill be discussed in detail. ie binary mode choice and multinomial mode choice. The chapterends with some discussion on future topics in mode choice problem.

6.2 Mode choice

The choice of transport mode is probably one of the most important classic models in transportplanning. This is because of the key role played by public transport in policy making. Publictransport modes make use of road space more efficiently than private transport. Also theyhave more social benefits like if more people begin to use public transport , there will be lesscongestion on the roads and the accidents will be less. Again in public transport, we can travelwith low cost. In addition, the fuel is used more efficiently. Main characteristics of publictransport is that they will have some particular schedule, frequency etc.

On the other hand, private transport is highly flexible. It provides more comfortable andconvenient travel. It has better accessibility also. The issue of mode choice, therefore, isprobably the single most important element in transport planning and policy making. It affectsthe general efficiency with which we can travel in urban areas. It is important then to developand use models which are sensitive to those travel attributes that influence individual choicesof mode.

6.3 Factors influencing the choice of mode

The factors may be listed under three groups:

1. Characteristics of the trip maker : The following features are found to be important:

(a) car availability and/or ownership;



(b) possession of a driving license;

(c) household structure (young couple, couple with children, retired people etc.);

(d) income;

(e) decisions made elsewhere, for example the need to use a car at work, take childrento school, etc;

(f) residential density.

2. Characteristics of the journey: Mode choice is strongly influenced by:

(a) The trip purpose; for example, the journey to work is normally easier to undertakeby public transport than other journeys because of its regularity and the adjustmentpossible in the long run;

(b) Time of the day when the journey is undertaken.

(c) Late trips are more difficult to accommodate by public transport.

3. Characteristics of the transport facility: There are two types of factors.One isquantitative and the other is qualitative. Quantitative factors are:

(a) relative travel time: in-vehicle, waiting and walking times by each mode;

(b) relative monetary costs (fares, fuel and direct costs);

(c) availability and cost of parking

Qualitative factors which are less easy to measure are:

(a) comfort and convenience

(b) reliability and regularity

(c) protection, security

A good mode choice should include the most important of these factors.

6.4 Types of modal split models

6.4.1 Trip-end modal split models

Traditionally, the objective of transportation planning was to forecast the growth in demand forcar trips so that investment could be planned to meet the demand. When personal characteris-tics were thought to be the most important determinants of mode choice, attempts were madeto apply modal-split models immediately after trip generation. Such a model is called trip-endmodal split model. In this way different characteristics of the person could be preserved andused to estimate modal split. The modal split models of this time related the choice of modeonly to features like income, residential density and car ownership.



The advantage is that these models could be very accurate in the short run, if publictransport is available and there is little congestion. Limitation is that they are insensitive topolicy decisions example: Improving public transport, restricting parking etc. would have noeffect on modal split according to these trip-end models.

6.4.2 Trip-interchange modal split models

This is the post-distribution model; that is modal split is applied after the distribution stage.This has the advantage that it is possible to include the characteristics of the journey andthat of the alternative modes available to undertake them. It is also possible to include policydecisions. This is beneficial for long term modeling.

6.4.3 Aggregate and disaggregate models

Mode choice could be aggregate if they are based on zonal and inter-zonal information. Theycan be called disaggregate if they are based on household or individual data.

6.5 Binary logit model

Binary logit model is the simplest form of mode choice, where the travel choice between twomodes is made. The traveler will associate some value for the utility of each mode. if the utilityof one mode is higher than the other, then that mode is chosen. But in transportation, we havedisutility also. The disutility here is the travel cost. This can be represented as

cij = a1tvij + a2t

wij + a3t

tij + a4tnij + a5Fij + a6φj + δ (6.1)

where tvij is the in-vehicle travel time between i and j, twij is the walking time to and from stops,ttij is the waiting time at stops, Fij is the fare charged to travel between i and j, φj is theparking cost, and δ is a parameter representing comfort and convenience. If the travel cost islow, then that mode has more probability of being chosen. Let there be two modes (m=1,2)then the proportion of trips by mode 1 from zone i to zone j is(P 1

ij) Let c1ij be the cost of

traveling from zone i to zonej using the mode 1, and c2ij be the cost of traveling from zonei to

zone j by mode 2,there are three cases:

1. if c2ij - c1

ij is positive, then mode 1 is chosen.

2. if c2ij - c1

ij is negative, then mode 2 is chosen.

3. if c2ij - c1

ij = 0 , then both modes have equal probability.

This relationship is normally expressed by a logit curve as shown in figure 6:1 Thereforethe proportion of trips by mode 1 is given by

P 1ij = T 1

ij/Tij =e−βc1ij

e−βc1ij + e−βc2ij(6.2)



0.5

1.0

B1

p1ij B2

c2ij − c1

ij

Figure 6:1: logit function

Table 6:1: Trip characterisitcstuij twij ttij fij φj

car 20 - 18 4bus 30 5 3 9ai 0.03 0.04 0.06 0.1 0.1

This functional form is called logit, where cij is called the generalized cost and β is the parameterfor calibration. The graph in figure 6:1 shows the proportion of trips by mode 1 (T 1

ij/Tij) againstcost difference.

Example

Let the number of trips from zone i to zone j is 5000, and two modes are available which hasthe characteristics given in Table 6:1. Compute the trips made by mode bus, and the farethat is collected from the mode bus. If the fare of the bus is reduced to 6, then find the farecollected.

Table 6:2: Binary logit model example: solutiontuij twij ttij fij φj cij pij Tij

ar 20 - 18 4 2.08 .52 2600bus 30 5 3 9 2.18 .475 2400ai .03 .04 .06 .1 .1



Solution The base case is given below.

Cost of travel by car (Equation)=ccar = 0.03 × 20 + 18 × 0.06 + 4 × 0.1 = 2.08

Cost of travel by bus (Equation)=cbus = 0.03× 30 + 0.04× 5 + 0.06× 3 + 0.1× 9 = 2.18

Probability of choosing mode car (Equation)= pcarij = e−2.08

e−2.08+e−2.18 = 0.52

Probability of choosing mode bus (Equation)= pbusij = e−2.18

e−2.08+e−2.18 = 0.475

Proportion of trips by car = T carij = 5000×0.52 = 2600

Proportion of trips by bus = T busij = 5000×0.475 = 2400

Fare collected from bus = T busij × Fij = 2400×9 = 21600

When the fare of bus gets reduced to 6,

Cost function for bus= cbus = 0.03 × 30 + 0.04 × 5 + 0.06 × 3 + 0.1 × 6 = 1.88

Probability of choosing mode bus (Equation)=pbusij = e−1.88

e−2.08+e−1.88 = 0.55

Proportion of trips by bus = T busij = 5000×0.55 = 2750

Fare collected from the bus T busij × Fij= 2750×6 = 16500

The results are tabulated in table

6.6 Multinomial logit model

The binary model can easily be extended to multiple modes. The equation for such a modelcan be written as:

P 1ij =

e−βc1ij

Σe−βcmij

(6.3)

6.6.1 Example

Let the number of trips from i to j is 5000, and three modes are available which has thecharacteristics given in Table 6:3: Compute the trips made by the three modes and the farerequired to travel by each mode.



Table 6:3: Trip characteristicstvij twalk

ij ttij Fij φij

coefficient 0.03 0.04 0.06 0.1 0.1car 20 - - 18 4bus 30 5 3 6 -train 12 10 2 4 -

Table 6:4: Multinomial logit model problem: solutiontvij twalk

ij ttij Fij φij C eC pij Tij

coeff 0.03 0.04 0.06 0.1 0.1 - - - -car 20 - - 18 4 2.8 0.06 0.1237 618.5bus 30 5 3 6 - 1.88 0.15 0.3105 1552.5train 12 10 2 4 - 1.28 0.28 0.5657 2828.5

Solution

Cost of travel by car (Equation)=ccar = 0.03 × 20 + 18 × 0.1 + 4 × 0.1 = 2.8

Cost of travel by bus (Equation)=cbus = 0.03× 30 + 0.04× 5 + 0.06× 3 + 0.1× 6 = 1.88

Cost of travel by train (Equation)=ctrain = 0.03 × 12 + 0.04 × 10 + 0.06 × 2 + 0.1 × 4 =1.28

Probability of choosing mode car (Equation)pcarij = e−2.8

e−2.8+e−1.88+e−1.28 = 0.1237

Probability of choosing mode bus (Equationpbusij = e−1.88

e−2.8+e−1.88+e−1.28 = 0.3105

Probability of choosing mode train (Equation)= ptrainij = e−1.28

e−2.8+e−1.88+e−1.28 = 0.5657

Proportion of trips by car,T carij = 5000×0.1237 = 618.5

Proportion of trips by bus, T busij = 5000×0.3105 = 1552.5

Similarly, proportion of trips by train,T trainij = 5000×0.5657 = 2828.5 We can put all this

in the form of a table as shown below 6:4:

• Fare collected from the mode bus = T busij × Fij = 1552.5×6 = 9315

• Fare collected from mode train = T trainij × Fij = 2828.5×4 = 11314


CE415 Transportation Engineering II 6. Trip Assignment

Table 6:5: Trip characteristicstvij twalk

ij ttij Fij φij

coefficient 0.05 0.04 0.07 0.2 0.2car 25 - - 22 6bus 35 8 6 8 -train 17 14 5 6 -

6.7 Summary

Modal split is the third stage of travel demand modeling. The choice of mode is influencedby various factors. Different types of modal split models are there. Binary logit model andmultinomial logit model are dealt in detail in this chapter.

6.8 Problems

1. The total number of trips from zone i to zone j is 4200. Currently all trips are made by car.Government has two alternatives- to introduce a train or a bus. The travel characteristicsand respective coefficients are given in table 6:5. Decide the best alternative in terms oftrips carried.

Solution

First, use binary logit model to find the trips when there is only car and bus. Then,again use binary logit model to find the trips when there is only car and train.Finally compare both and see which alternative carry maximum trips.

Cost of travel by car=ccar = 0.05 × 25 + 0.2 × 22 + 0.2 × 6 = 6.85

Cost of travel by bus=cbus = 0.05 × 35 + 0.04 × 8 + 0.07 × 6 + 0.2 × 8 = 4.09

Cost of travel by trai=ctrain = 0.05 × 17 + 0.04 × 14 + 0.07 × 5 + 0.2 × 6 = 2.96

Case 1: Considering introduction of bus, Probability of choosing car, pcarij = e−6.85

e−6.85+e−4.09

= 0.059

Probability of choosing bus, pbusij = e−4.09

e−6.85+e−4.09 = 0.9403

Case 2: Considering introduction of train, Probability of choosing car pcarij = e−6.85

e−6.85+e−2.96

= 0.02003

Probability of choosing train ptrainij = e−2.96

e−6.85+e−2.96 = 0.979

Trips carried by each mode

Case 1: T carij = 4200×0.0596 = 250.32 T bus

ij = 4200×0.9403 = 3949.546

Case 2: T carij = 4200×0.02 = 84.00 T train

ij = 4200×0.979 = 4115.8

Hence train will attract more trips, if it is introduced.



Chapter 7

Trip Assignment

7.1 Overview

The process of allocating given set of trip interchanges to the specified transportation system isusually referred to as trip assignment or traffic assignment. The fundamental aim of the trafficassignment process is to reproduce on the transportation system, the pattern of vehicularmovements which would be observed when the travel demand represented by the trip matrix,or matrices, to be assigned is satisfied. The major aims of traffic assignment procedures are:

1. To estimate the volume of traffic on the links of the network and obtain aggregate networkmeasures.

2. To estimate inter zonal travel cost.

3. To analyze the travel pattern of each origin to destination(O-D) pair.

4. To identify congested links and to collect traffic data useful for the design of futurejunctions.

7.2 Link cost function

As the flow increases towards the capacity of the stream, the average stream speed reducesfrom the free flow speed to the speed corresponding to the maximum flow. This can be seen inthe graph shown below.That means traffic conditions worsen and congestion starts developing. The inter zonal flowsare assigned to the minimum paths computed on the basis of free-flow link impedances (usuallytravel time). But if the link flows were at the levels dictated by the assignment, the link speedswould be lower and the link travel time would be higher than those corresponding to the freeflow conditions. So the minimum path computed prior to the trip assignment will not be theminimum after the trips ae assigned. A number of iterative procedures are done to convergethis difference. The relation between the link flow and link impedance is called the link costfunction and is given by the equation as shown below:

t = t0 [1 + α(x

k)β

] (7.1)



flow (x)

trav

el ti

me

Figure 7:1: Two Link Problem with constant travel time function

where t and x is the travel time and flow, respectively on the link, t0 is the free flow traveltime, and k is the practical capacity. The parameters α and β are specific the type of link andis to be calibrated from the field data. In the absense of any field data, following values couldthe assumed: α = 0.15, and β = 4.0.

The types of traffic assignment models are all-or-nothing assignment (AON), incrementalassignment, capacity restraint assignment, user equilibrium assignment (UE), stochastic userequilibrium assignment (SUE), system optimum assignment (SO), etc. The frequently usedmodels all-or-nothing, user equilibrium, and system optimum will be discussed in detail here.

7.3 All-or-nothing assignment

In this method the trips from any origin zone to any destination zone are loaded onto a single,minimum cost, path between them. This model is unrealistic as only one path between everyO-D pair is utilized even if there is another path with the same or nearly same travel cost. Also,traffic on links is assigned without consideration of whether or not there is adequate capacityor heavy congestion; travel time is a fixed input and does not vary depending on the congestionon a link. However, this model may be reasonable in sparse and uncongested networks wherethere are few alternative routes and they have a large difference in travel cost. This model mayalso be used to identify the desired path : the path which the drivers would like to travel in theabsence of congestion. In fact, this model’s most important practical application is that it actsas a building block for other types of assignment techniques.It has a limitation that it ignoresthe fact that link travel time is a function of link volume and when there is congestion or thatmultiple paths are used to carry traffic.

Example

To demonstrate how this assignment works, an example network is considered. This networkhas two nodes having two paths as links. Let us suppose a case where travel time is not afunction of flow as shown in other words it is constant as shown in the figure below.

Solution The travel time functions for both the links is given by:



151 2

10

x2

t2 =

x1

t1 =


t1 = 10t2 = 15and total flows from 1 to 2 is given by. q12 = 12Since the shortest path is Link 1 all flows are assigned to it making x1 =12 and x2 = 0.

7.4 User equilibrium assignment (UE)

The user equilibrium assignment is based on Wardrop’s first principle, which states that nodriver can unilaterally reduce his/her travel costs by shifting to another route. User Equilibrium(UE) conditions can be written for a given O-D pair as:

fk(ck − u) = 0 : ∀k (7.2)

ck − u >= 0 : ∀k (7.3)

where fk is the flow on path k, ck is the travel cost on path k, and u is the minimum cost.Equation 7.3 can have two states.

1. If ck − u = 0, from equation 7.2 fk ≥ 0. This means that all used paths will have sametravel time.

2. If ck − u ≥ 0, then from equation 7.2 fk = 0.

This means that all unused paths will have travel time greater than the minimum cost path.where fk is the flow on path k, ck is the travel cost on path k, and u is the minimum cost.

Assumptions in User Equilibrium Assignment

1. The user has perfect knowledge of the path cost.

2. Travel time on a given link is a function of the flow on that link only.

3. Travel time functions are positive and increasing.



The solution to the above equilibrium conditions given by the solution of an equivalent nonlinearmathematical optimization program,

Minimize Z =∑

a

∫ xa

0ta(xa)dx, (7.4)

subjected to∑

k

f rsk = qrs : ∀r, s

xa =∑

r

∑

s

∑

k

δrsa,kf

rsk : ∀a

f rsk ≥ 0 : ∀ k, r, s

xa ≥ 0 : a ∈ A

where k is the path, xa equilibrium flows in link a, ta travel time on link a, f rsk flow on path

k connecting O-D pair r-s, qrs trip rate between r and sand δrsa,k is a definitional constraint and

is given by

δr,sa,k =

{

1 if link a belongs to path k,0 otherwise

(7.5)

The equations above are simply flow conservation equations and non negativity constraints,respectively. These constraints naturally hold the point that minimizes the objective function.These equations state user equilibrium principle.The path connecting O-D pair can be dividedinto two categories : those carrying the flow and those not carrying the flow on which the traveltime is greater than (or equal to)the minimum O-D travel time. If the flow pattern satisfiesthese equations no motorist can better off by unilaterally changing routes. All other routeshave either equal or heavy travel times. The user equilibrium criteria is thus met for everyO-D pair. The UE problem is convex because the link travel time functions are monotonicallyincreasing function, and the link travel time a particular link is independent of the flow andother links of the networks. To solve such convex problem Frank Wolfe algorithm is useful.

Example

Let us suppose a case where travel time is not a function of flow as shown in other words it isconstant as shown in the figure below.

1 2

x1

10+3x1t1 =

t2 =15+2x2

x2




Solution Substituting the travel time in equation yield to

min Z(x) =∫ x1

010 + 3x1dx1 +

∫ x2

015 + 2x2dx2

= 10x1 +3x2

1

2+ 15x2 +

2x22

2

subject to

x1 + x2 = 12

Substituting x2 = 12 − x1, in the above formulation will yield the unconstrained formulationas below :

min Z(x) = 10x1 +3x2

1

2+ 15(12 − x1) +

2(12 − x1)2

2

Differentiate the above equation x1 and equate to zero, and solving for x1 and then x2 leads tothe solution x1 = 5.8, x2 = 6.2.

7.5 System Optimum Assignment (SO)

The system optimum assignment is based on Wardrop’s second principle, which states thatdrivers cooperate with one another in order to minimize total system travel time. This assign-ment can be thought of as a model in which congestion is minimized when drivers are toldwhich routes to use. Obviously, this is not a behaviorally realistic model, but it can be usefulto transport planners and engineers, trying to manage the traffic to minimize travel costs andtherefore achieve an optimum social equilibrium.

Minimize Z =∑

a

xata(xa) (7.6)

subject to

∑

k

f rsk = qrs : ∀r, s (7.7)

xa =∑

r

∑

s

∑

k

δrsa,kf

rsk : ∀a (7.8)

f rsk ≥ 0 : ∀ k, r, s (7.9)

xa ≥ 0 : a ∈ A (7.10)

xa equilibrium flows in link a, ta travel time on link a, f rsk flow on path k connecting O-D pair

r-s, qrs trip rate between r and s.



Example

To demonstrate how the assignment works, an example network is considered. This networkhas two nodes having two paths as links. Let us suppose a case where travel time is not afunction of flow or in other words it is constant as shown in the figure below.

1 2

x1

10+3x1t1 =

t2 =15+2x2

x2


Solution Substituting the travel time in equation , we get the following:

min Z(x) = x1 ∗ (10 + 3x1) + x2 ∗ (15 + 2x2) (7.11)

= 10x1 + 3x12 + 15x2 + 2x2

2 (7.12)

Substituting x2 = x1 − 12

min Z(x) == 10x1 + 3x12 + 15(12 − x1) + 2(12 − x1)

2 (7.13)

(7.14)

Differentiate the above equation to zero, and solving for x1 and then x2 leads to the solutionx1 = 5.3,x2= 6.7which gives Z(x) = 327.55

7.6 Other assignment methods

Let us discuss briefly some other assignments like incremental assignment, capacity restraintassignment, stochastic user equilibrium assignment and dynamic assignment.

7.6.1 Incremental assignment

Incremental assignment is a process in which fractions of traffic volumes are assigned in steps.Ineach step, a fixed proportion of total demand is assigned, based on all-or-nothing assignment.After each step, link travel times are recalculated based on link volumes. When there are manyincrements used, the flows may resemble an equilibrium assignment ; however, this method



does not yield an equilibrium solution. Consequently, there will be inconsistencies between linkvolumes and travel times that can lead to errors in evaluation measures. Also, incrementalassignment is influenced by the order in which volumes for O-D pairs are assigned, raising thepossibility of additional bias in results.

7.6.2 Capacity restraint assignment

Capacity restraint assignment attempts to approximate an equilibrium solution by iteratingbetween all-or-nothing traffic loadings and recalculating link travel times based on a congestionfunction that reflects link capacity. Unfortunately, this method does not converge and canflip-flop back and forth in loadings on some links.

7.6.3 Stochastic user equilibrium assignment

User equilibrium assignment procedures based on Wardrop’s principle assume that all driversperceive costs in an identical manner. A solution to assignment problem on this basis is anassignment such that no driver can reduce his journey cost by unilaterally changing route. VanVilet considered as stochastic assignment models, all those models which explicitly allows nonminimum cost routes to be selected. Virtually all such models assume that drivers perception ofcosts on any given route are not identical and that the trips between each O-D pair are dividedamong the routes with the most cheapest route attracting most trips. They have importantadvantage over other models because they load many routes between individual pairs of networknodes in a single pass through the tree building process,the assignments are more stable andless sensitive to slight variations in network definitions or link costs to be independent of flowsand are thus most appropriate for use in uncongested traffic conditions such as in off peakperiods or lightly trafficked rural areas.

7.6.4 Dynamic Assignment

Dynamic user equilibrium,expressed as an extension of Wardrop’s user equilibrium principle,may be defined as the state of equilibrium which arises when no driver can reduce his disutilityof travel by choosing a new route or departure time,where disutility includes, schedule delayin addition in to costs generally considered. Dynamic stochastic equilibrium may be similarlydefined in terms of perceived utility of travel. The existence of such equilibrium in complexnetworks has not been proven theoretical and even if they exist the question of uniquenessremains open.

7.7 Summary

Traffic assignment is the last stage of traffic demand modeling. There are different types oftraffic assignment models. All-or-nothing, User-equilibrium, and System-optimum assignmentmodels are the commonly used models. All-or-nothing model is an unrealistic model since onlyone path between every O-D pair is utilised and they can give satisfactory results only when the



network is least congested. User-equilibrium assignment is based on Wardrop’s first principleand it’s conditions are based on certain assumptions. Wardrop’s second principle is utilized bySystem-optimum method and it tries to minimise the congestion by giving prior information todrivers regarding the respective routes to be chosen. Other assignment models are also brieflyexplained.

7.8 Problems

Calculate the system travel time and link flows by doing user equilibrium assignment for thenetwork in the given figure 7:5. Verify that the flows are at user equilibrium.

1 2

12+3x1

t2 =10+5x2

x1

x2

t1 =

Figure 7:5: Two link problem with constant travel time

Solution Substituting the travel time in the respective equation yield to

min Z(x) =∫ x1

012 + 3x1dx1 +

∫ x2

010 + 5x2dx2

= 12x1 +3x2

1

2+ 10x2 +

5x22

2

subject to

x1 + x2 = 12

Substituting x2 = 12 − x1, in the above formulation will yield the unconstrained formulationas below :

min Z(x) = 12x1 +3x2

1

2+ 10(12 − x1) +

5(12 − x1)2

2min Z(x) = 4x2

1 − 58x1 + 480

Differentiate the above equation x1 and equating to zero,

dz(x)

dx= 0 − 58 + 8x1 = 0orx1 = 7.25Hencex2 = 12 − 7, 25 = 4.75


CE415 Transportation Engineering II 7. Fundamental parameters of traffic flow

Travel times are

t1 = 12 + 3 × 7.25 = 33.75t2 = 10 + 5 × 4.75 = 33.75i.e.t1 = t2

It follows that the travel times are at user equilibrium.



Chapter 8

Fundamental parameters of traffic flow

8.1 Overview

Traffic engineering pertains to the analysis of the behavior of traffic and to design the facilitiesfor a smooth, safe and economical operation of traffic. Traffic flow, like the flow of water,has several parameters associated with it. The traffic stream parameters provide informationregarding the nature of traffic flow, which helps the analyst in detecting any variation in flowcharacteristicis. Understanding traffic behavior requires a thorough knowledge of traffic streamparameters and their mutual relationships. In this chapter the basic concepts of traffic flow ispresented.

8.2 Traffic stream parameters

The traffic stream includes a combination of driver and vehicle behavior. The driver or humanbehavior being non-uniform, traffic stream is also non-uniform in nature. It is influenced notonly by the individual characteristics of both vehicle and human but also by the way a groupof such units interacts with each other. Thus a flow of traffic through a street of definedcharacteristics will vary both by location and time corresponding to the changes in the humanbehavior.

The traffic engineer, but for the purpose of planning and design, assumes that these changesare within certain ranges which can be predicted. For example, if the maximum permissiblespeed of a highway is 60 kmph, the whole traffic stream can be assumed to move on an averagespeed of 40 kmph rather than 100 or 20 kmph.

Thus the traffic stream itself is having some parameters on which the characteristics canbe predicted. The parameters can be mainly classified as : measurements of quantity, whichincludes density and flow of traffic and measurements of quality which includes speed. Thetraffic stream parameters can be macroscopic which characterizes the traffic as a whole ormicroscopic which studies the behavior of individual vehicle in the stream with respect to eachother.

As far as the macroscopic characteristics are concerned, they can be grouped as measurementof quantity or quality as described above, i.e. flow, density, and speed. While the microscopiccharacteristics include the measures of separation, i.e. the headway or separation betweenvehicles which can be either time or space headway. The fundamental stream characteristics



are speed, flow, and density and are discussed below.

8.3 Speed

Speed is considered as a quality measurement of travel as the drivers and passengers will beconcerned more about the speed of the journey than the design aspects of the traffic. It isdefined as the rate of motion in distance per unit of time. Mathematically speed or velocity vis given by,

v =d

t(8.1)

where, v is the speed of the vehicle in m/s, d is distance traveled in m in time t seconds. Speedof different vehicles will vary with respect to time and space. To represent these variation,several types of speed can be defined. Important among them are spot speed, running speed,journey speed, time mean speed and space mean speed. These are discussed below.

8.3.1 Spot Speed

Spot speed is the instantaneous speed of a vehicle at a specified location. Spot speed can beused to design the geometry of road like horizontal and vertical curves, super elevation etc.Location and size of signs, design of signals, safe speed, and speed zone determination, requirethe spot speed data. Accident analysis, road maintenance, and congestion are the modern fieldsof traffic engineer, which uses spot speed data as the basic input. Spot speed can be measuredusing an enoscope, pressure contact tubes or direct timing procedure or radar speedometer orby time-lapse photographic methods. It can be determined by speeds extracted from videoimages by recording the distance traveling by all vehicles between a particular pair of frames.

8.3.2 Running speed

Running speed is the average speed maintained over a particular course while the vehicle ismoving and is found by dividing the length of the course by the time duration the vehicle wasin motion. i.e. this speed doesn’t consider the time during which the vehicle is brought to astop, or has to wait till it has a clear road ahead. The running speed will always be more thanor equal to the journey speed, as delays are not considered in calculating the running speed

8.3.3 Journey speed

Journey speed is the effective speed of the vehicle on a journey between two points and is thedistance between the two points divided by the total time taken for the vehicle to complete thejourney including any stopped time. If the journey speed is less than running speed, it indicatesthat the journey follows a stop-go condition with enforced acceleration and deceleration. Thespot speed here may vary from zero to some maximum in excess of the running speed. Auniformity between journey and running speeds denotes comfortable travel conditions.



8.3.4 Time mean speed and space mean speed

Time mean speed is defined as the average speed of all the vehicles passing a point on a highwayover some specified time period. Space mean speed is defined as the average speed of all thevehicles occupying a given section of a highway over some specified time period. Both meanspeeds will always be different from each other except in the unlikely event that all vehiclesare traveling at the same speed. Time mean speed is a point measurement while space meanspeed is a measure relating to length of highway or lane, i.e. the mean speed of vehicles overa period of time at a point in space is time mean speed and the mean speed over a space at agiven instant is the space mean speed.

8.4 Flow

There are practically two ways of counting the number of vehicles on a road. One is flow orvolume, which is defined as the number of vehicles that pass a point on a highway or a givenlane or direction of a highway during a specific time interval. The measurement is carried outby counting the number of vehicles, nt, passing a particular point in one lane in a defined periodt. Then the flow q expressed in vehicles/hour is given by

q =nt

t(8.2)

Flow is expressed in planning and design field taking a day as the measurement of time.

8.4.1 Variations of Volume

The variation of volume with time, i.e. month to month, day to day, hour to hour and within ahour is also as important as volume calculation. Volume variations can also be observed fromseason to season. Volume will be above average in a pleasant motoring month of summer, butwill be more pronounced in rural than in urban area. But this is the most consistent of all thevariations and affects the traffic stream characteristics the least.

Weekdays, Saturdays and Sundays will also face difference in pattern. But comparing daywith day, patterns for routes of a similar nature often show a marked similarity, which is usefulin enabling predictions to be made.

The most significant variation is from hour to hour. The peak hour observed during morn-ings and evenings of weekdays, which is usually 8 to 10 per cent of total daily flow or 2 to 3times the average hourly volume. These trips are mainly the work trips, which are relativelystable with time and more or less constant from day to day.

8.4.2 Types of volume measurements

Since there is considerable variation in the volume of traffic, several types of measurements ofvolume are commonly adopted which will average these variations into a single volume countto be used in many design purposes.



1. Average Annual Daily Traffic(AADT) : The average 24-hour traffic volume at agiven location over a full 365-day year, i.e. the total number of vehicles passing the sitein a year divided by 365.

2. Average Annual Weekday Traffic(AAWT) : The average 24-hour traffic volumeoccurring on weekdays over a full year. It is computed by dividing the total weekdaytraffic volume for the year by 260.

3. Average Daily Traffic(ADT) : An average 24-hour traffic volume at a given locationfor some period of time less than a year. It may be measured for six months, a season, amonth, a week, or as little as two days. An ADT is a valid number only for the periodover which it was measured.

4. Average Weekday Traffic(AWT) : An average 24-hour traffic volume occurring onweekdays for some period of time less than one year, such as for a month or a season.

The relationship between AAWT and AWT is analogous to that between AADT and ADT.Volume in general is measured using different ways like manual counting, detector/sensor count-ing, moving-car observer method, etc. Mainly the volume study establishes the importance ofa particular route with respect to the other routes, the distribution of traffic on road, and thefluctuations in flow. All which eventually determines the design of a highway and the relatedfacilities. Thus, volume is treated as the most important of all the parameters of traffic stream.

8.5 Density

Density is defined as the number of vehicles occupying a given length of highway or lane andis generally expressed as vehicles per km. One can photograph a length of road x, count thenumber of vehicles, nx, in one lane of the road at that point of time and derive the density kas,

k =nx

x(8.3)

This is illustrated in figure 8:1. From the figure, the density is the number of vehicles betweenthe point A and B divided by the distance between A and B. Density is also equally importantas flow but from a different angle as it is the measure most directly related to traffic demand.Again it measures the proximity of vehicles in the stream which in turn affects the freedom tomaneuver and comfortable driving.

8.6 Derived characteristics

From the fundamental traffic flow characteristics like flow, density, and speed, a few otherparameters of traffic flow can be derived. Significant among them are the time headway,distance headway and travel time. They are discussed one by one below.



B A

Figure 8:1: Illustration of density

8.6.1 Time headway

The microscopic character related to volume is the time headway or simply headway. Timeheadway is defined as the time difference between any two successive vehicles when they crossa given point. Practically, it involves the measurement of time between the passage of one rearbumper and the next past a given point. If all headways h in time period, t, over which flowhas been measured are added then,

nt∑

1

hi = t (8.4)

But the flow is defined as the number of vehicles nt measured in time interval t, that is,

q =nt

t=

nt∑nt

1 hi=

1

hav(8.5)

where, hav is the average headway. Thus average headway is the inverse of flow. Time headwayis often referred to as simply the headway.

8.6.2 Distance headway

Another related parameter is the distance headway. It is defined as the distance betweencorresponding points of two successive vehicles at any given time. It involves the measurementfrom a photograph, the distance from rear bumper of lead vehicle to rear bumper of followingvehicle at a point of time. If all th space headways in distance x over which the density hasbeen measured are added,

nx∑

1

si = x (8.6)

But the density (k) is the number of vehicles nx at a distance of x, that is

k =nx

x=

nx∑nx

1 si=

1

sav(8.7)



(a) (b)

(c)

time

dist

ance

dist

ance

time

dist

ance

time

Figure 8:2: Time space diagram for a single vehicle

Where, sav is average distance headway. The average distance headway is the inverse of densityand is sometimes called as spacing.

8.6.3 Travel time

Travel time is defined as the time taken to complete a journey. As the speed increases, traveltime required to reach the destination also decreases and viceversa. Thus travel time is inverselyproportional to the speed. However, in practice, the speed of a vehicle fluctuates over time andthe travel time represents an average measure.

8.7 Time-space diagram

Time space diagram is a convenient tool in understanding the movement of vehicles. It showsthe trajectory of vehicles in the form of a two dimensional plot. Time space diagram can beplotted for a single vehicle as well as multiple vehicles. They are discussed below.

8.7.1 Single vehicle

Taking one vehicle at a time, analysis can be carried out on the position of the vehicle withrespect to time. This analysis will generate a graph which gives the relation of its position ona road stretch relative to time. This plot thus will be between distance x and time t and xwill be a functions the position of the vehicle for every t along the road stretch. This graphicalrepresentation of x(t) in a (t, x) plane is a curve which is called as a trajectory. The trajectoryprovide an intuitive, clear, and complete summary of vehicular motion in one dimension.

In figure 8:2(a), the the distance x goes on increasing with respect to the origin as timeprogresses. The vehicle is moving at a smooth condition along the road way. In figure 8:2(b),



the vehicle at first moves with a smooth pace after reaching a position reverses its direction ofmovement. In figure 8:2(c), the vehicle in between becomes stationary and maintains the sameposition.

From the figure, steeply increasing section of x(t) denote a rapidly advancing vehicle andhorizontal portions of x(t) denote a stopped vehicle while shallow sections show a slow-movingvehicle. A straight line denotes constant speed motion and curving sections denote acceleratedmotion; and if the curve is concave downwards it denotes acceleration. But a curve which isconvex upwards denotes deceleration.

8.7.2 Miultiple Vehicles

Time-space diagram can also be used to determine the fundamental parameters of traffic flowlike speed, density and volume. It can also be used to find the derived characteristics like spaceheadway and time headway. Figure 8:3 shows the time-space diagram for a set of vehiclestraveling at constant speed. Density, by definition is the number of vehicles per unit length.From the figure, an observer looking into the stream can count 4 vehicles passing the stretchof road between x1 and x2 at time t. Hence, the density is given as

k =4 vehicles

x2 − x1

(8.8)

We can also find volume from this time-space diagram. As per the definition, volume is thenumber of vehicles counted for a particular interval of time. From the figure 8:3 we can seethat 6 vehicles are present between the time t1 and t2. Therefore, the volume q is given as

q =3 vehicles

t2 − t1(8.9)

Again the averages taken at a specific location (i.e., time ranging over an interval) are calledtime means and those taken at an instant over a space interval are termed as space means.

Another related definition which can be given based on the time-space diagram is the head-way. Space headway is defined as the distance between corresponding points of two successivevehicles at any given time. Thus, the vertical gap between any two consecutive lines representsspace headway. The reciprocal of density otherwise gives the space headway between vehiclesat that time.

Similarly, time headway is defined as the time difference between any two successive vehicleswhen they cross a given point. Thus, the horizontal gap between the vehicles represented by thelines gives the time headway. The reciprocal of flow gives the average time headway betweenvehicles at that point.

8.8 Summary

Speed, flow and density are the basic parameters of traffic flow. Different measures of speedare used in traffic flow analysis like spot speed, time mean speed, space mean speed etc. Time-



tTime

(h)headway

x1

x2

dist

ance

t1 t2

(s)spacing

Figure 8:3: Time space diagram for many vehicles

space diagram also can be used for determining these parameters. Speed and flow of the trafficstream can be computed using moving observer method.

8.9 Problems

1. The instantaneous speed of a vehicle at a specified location is called

(a) Spot speed

(b) Journey speed

(c) Running speed

(d) Time mean speed

2. Which of the following is not a derived characteristic?

(a) Time headway

(b) Distance headway

(c) Travel time

(d) Density

8.10 Solutions

1. The instantaneous speed of a vehicle at a specified location is called

(a) Spot speed√

(b) Journey speed

(c) Running speed


CE415 Transportation Engineering II 8. Fundamental relations of traffic flow

(d) Time mean speed

2. Which of the following is not a derived characteristic?

(a) Time headway

(b) Distance headway

(c) Travel time

(d) Density√



Chapter 9

Fundamental relations of traffic flow

9.1 Overview

Speed is one of the basic parameters of traffic flow and time mean speed and space meanspeed are the two representations of speed. Time mean speed and space mean speed and therelationship between them will be discussed in detail in this chapter. The relationship betweenthe fundamental parameters of traffic flow will also be derived. In addition, this relationshipcan be represented in graphical form resulting in the fundamental diagrams of traffic flow.

9.2 Time mean speed (vt)

As noted earlier, time mean speed is the average of all vehicles passing a point over a durationof time. It is the simple average of spot speed. Time mean speed vt is given by,

vt =1

n

n∑

i=1

vi, (9.1)

where v is the spot speed of ith vehicle, and n is the number of observations. In many speedstudies, speeds are represented in the form of frequency table. Then the time mean speed isgiven by,

vt =

∑ni=1 qivi∑n

i=1 qi

, (9.2)

where qi is the number of vehicles having speed vi, and n is the number of such speed categories.

9.3 Space mean speed (vs)

The space mean speed also averages the spot speed, but spatial weightage is given instead oftemporal. This is derived as below. Consider unit length of a road, and let vi is the spot speedof ith vehicle. Let ti is the time the vehicle takes to complete unit distance and is given by 1

vi.

If there are n such vehicles, then the average travel time ts is given by,

ts =Σtin

=1

nΣ

1

vi(9.3)



No. speed range average speed (vi) volume of flow (qi) viqiqi

vi

1 2-5 3.5 1 3.5 2.292 6-9 7.5 4 30.0 0.543 10-13 11.5 0 0 04 14-17 15.5 7 108.5 0.45

total 12 142 3.28

If tav is the average travel time, then average speed vs = 1ts

. Therefore from the above equation,

vs =n

∑ni=1

1vi

(9.4)

This is simply the harmonic mean of the spot speed. If the spot speeds are expressed as afrequency table, then,

vs =

∑ni=1 qi

∑ni=1

qi

vi

(9.5)

where qi vehicle will have vi speed and ni is the number of such observations.

Example 1

If the spot speeds are 50, 40, 60,54 and 45, then find the time mean speed and space meanspeed.

Solution Time mean speed vt is the average of spot speed. Therefore, vt = Σvi

n= 50+40+60+54+45

5=

2495

= 49.8 Space mean speed is the harmonic mean of spot speed.Therefore, vs = nΣ 1

vi

=

5150

+ 140

+ 160

+ 154

+ 145

= 50.12

= 48.82

Example 2

The results of a speed study is given in the form of a frequency distribution table. Find thetime mean speed and space mean speed.

speed range frequency2-5 16-9 4

10-13 014-17 7

Solution The time mean speed and space mean speed can be found out from the frequencytable given below. First, the average speed is computed, which is the mean of the speed range.For example, for the first speed range, average speed, vi = 2+5

2= 3.5 seconds. The volume of

flow qi for that speed range is same as the frequency. The terms vi.qi and qi

viare also tabulated,



10 m/s 10 m/s 10 m/s 10 m/s 10 m/s

20 m/s20 m/s20 m/s

100 100

5050 5050

hs = 50/20 = 5sec ns = 60/5 = 12 ks = 1000/50 = 20

hf = 100/20 = 5sec nf = 60/5 = 12 kf = 1000/100 = 10

Figure 9:1: Illustration of relation between time mean speed and space mean speed

and their summations in the last row. Time mean speed can be computed as, vt = Σqivi

Σqi=

14212

= 11.83 Similarly, space mean speed can be computed as, vs = Σqi

Σqivi

= 123.28

= 3.65

9.4 Illustration of mean speeds

Inorder to understand the concept of time mean speed and space mean speed, following illus-tration will help. Let there be a road stretch having two sets of vehicle as in figure 9:1. Thefirst vehicle is traveling at 10m/s with 50 m spacing, and the second set at 20m/s with 100 mspacing. Therefore, the headway of the slow vehicle hs will be 50 m divided by 10 m/s whichis 5 sec. Therefore, the number of slow moving vehicles observed at A in one hour ns willbe 60/5 = 12 vehicles. The density K is the number of vehicles in 1 km, and is the inverseof spacing. Therefore, Ks = 1000/50 = 20 vehicles/km. Therefore, by definition, time meanspeed vt is given by vt = 12×10+12×20

24= 15 m/s. Similarly, by definition, space mean speed is

the mean of vehicle speeds over time. Therefore, vs = 20×10+10×2030

= 13.3 m/s This is same asthe harmonic mean of spot speeds obtained at location A; ie vs = 24

12× 110

+12× 120

= 13.3 m/s. It

may be noted that since harmonic mean is always lower than the arithmetic mean, and also asobserved , space mean speed is always lower than the time mean speed. In other words, spacemean speed weights slower vehicles more heavily as they occupy the road stretch for longerduration of time. For this reason, in many fundamental traffic equations, space mean speed ispreferred over time mean speed.

9.5 Relation between time mean speed and space mean

speed

The relation between time mean speed and space mean speed can be derived as below. Considera stream of vehicles with a set of substream flow q1,q2, . . . qi, . . . qn having speed v1,v2, . . . vi,. . . vn. The fundamental relation between flow(q), density(k) and mean speed vs is,

q = k × vs (9.6)



Therefore for any substream qi, the following relationship will be valid.

qi = ki × vi (9.7)

The summation of all substream flows will give the total flow q.

Σqi = q (9.8)

Similarly the summation of all substream density will give the total density k.

Σki = k (9.9)

Let fi denote the proportion of substream density ki to the total density k,

fi =ki

k(9.10)

Space mean speed averages the speed over space. Therefore, if ki vehicles has vi speed, thenspace mean speed is given by,

vs =Σkivi

k(9.11)

Time mean speed averages the speed over time.Therefore,

vt =Σqivi

q(9.12)

Substituting in 9.7 vt can be written as,

vt =Σkivi

2

q(9.13)

Rewriting the above equation and substituting 9.11, and then substituting 11.7, we get,

vt = kΣki

kv2

i

=kΣfivi

2

q

=Σfivi

2

vs

By adding and subtracting vs and doing algebraic manipulations, vt can be written as,

vt =Σfi(vs + (vi − vs))

2

vs(9.14)



8

Bv km

67 5 4 3 2 1

A

Figure 9:2: Illustration of relation between fundamental parameters of traffic flow

=Σfi(vs)

2 + (vi − vs)2 + 2.vs.(vi − vs)

vs

(9.15)

=Σfivs

2

vs+

Σfi(vi − vs)2

vs+

2.vs.Σfi(vi − vs)

vs(9.16)

The third term of the equation will be zero because Σfi(vi − vs) will be zero, since vs is themean speed of vi. The numerator of the second term gives the standard deviation of vi. Σfi

by definition is 1.Therefore,

vt = vsΣfi +σ2

vs+ 0 (9.17)

vt = vs +σ2

vs(9.18)

Hence, time mean speed is space mean speed plus standard deviation of the spot speed dividedby the space mean speed. Time mean speed will be always greater than space mean speed sincestandard deviation cannot be negative. If all the speed of the vehicles are the same, then spotspeed, time mean speed and space mean speed will also be same.

9.6 Fundamental relations of traffic flow

The relationship between the fundamental variables of traffic flow, namely speed, volume, anddensity is called the fundamental relations of traffic flow. This can be derived by a simpleconcept. Let there be a road with length v km, and assume all the vehicles are moving with vkm/hr.(Fig 9:2). Let the number of vehicles counted by an observer at A for one hour be n1.By definition, the number of vehicles counted in one hour is flow(q). Therefore,

n1 = q (9.19)

Similarly, by definition, density is the number of vehicles in unit distance. Therefore numberof vehicles n2 in a road stretch of distance v1 will be density × distance.Therefore,

n2 = k × v (9.20)

Since all the vehicles have speed v, the number of vehicles counted in 1 hour and the number



flow

(q)

C

B

A

q

O

density (k)

ED

qmax

k0 k1 kmax k2 kjam

Figure 9:3: Flow density curve

of vehicles in the stretch of distance v will also be same.(ie n1 = n2). Therefore,

q = k × v (9.21)

This is the fundamental equation of traffic flow. Please note that, v in the above equation refersto the space mean speed.

9.7 Fundamental diagrams of traffic flow

The relation between flow and density, density and speed, speed and flow, can be representedwith the help of some curves. They are referred to as the fundamental diagrams of traffic flow.They will be explained in detail one by one below.

9.7.1 Flow-density curve

The flow and density varies with time and location. The relation between the density and thecorresponding flow on a given stretch of road is referred to as one of the fundamental diagramof traffic flow. Some characteristics of an ideal flow-density relationship is listed below:

1. When the density is zero, flow will also be zero,since there is no vehicles on the road.

2. When the number of vehicles gradually increases the density as well as flow increases.

3. When more and more vehicles are added, it reaches a situation where vehicles can’t move.This is referred to as the jam density or the maximum density. At jam density, flow willbe zero because the vehicles are not moving.

4. There will be some density between zero density and jam density, when the flow is maxi-mum. The relationship is normally represented by a parabolic curve as shown in figure 9:3



spee

d u

density (k)k0 kjam

uf

Figure 9:4: Speed-density diagram

The point O refers to the case with zero density and zero flow. The point B refers to themaximum flow and the corresponding density is kmax. The point C refers to the maximumdensity kjam and the corresponding flow is zero. OA is the tangent drawn to the parabola at O,and the slope of the line OA gives the mean free flow speed, ie the speed with which a vehiclecan travel when there is no flow. It can also be noted that points D and E correspond to sameflow but has two different densities. Further, the slope of the line OD gives the mean speed atdensity k1 and slope of the line OE will give mean speed at density k2. Clearly the speed atdensity k1 will be higher since there are less number of vehicles on the road.

9.7.2 Speed-density diagram

Similar to the flow-density relationship, speed will be maximum, referred to as the free flowspeed, and when the density is maximum, the speed will be zero. The most simple assumptionis that this variation of speed with density is linear as shown by the solid line in figure 9:4.Corresponding to the zero density, vehicles will be flowing with their desire speed, or free flowspeed. When the density is jam density, the speed of the vehicles becomes zero. It is alsopossible to have non-linear relationships as shown by the dotted lines. These will be discussedlater.

9.7.3 Speed flow relation

The relationship between the speed and flow can be postulated as follows. The flow is zeroeither because there is no vehicles or there are too many vehicles so that they cannot move.At maximum flow, the speed will be in between zero and free flow speed. This relationship isshown in figure 9:5. The maximum flow qmax occurs at speed u. It is possible to have twodifferent speeds for a given flow.



flow q

q

usp

eed

u

Qmaxu0

u1

u2

uf

Figure 9:5: Speed-flow diagram

spee

d u

flow q

flow

q

density k

spee

d u

density k qmax

Figure 9:6: Fundamental diagram of traffic flow

9.7.4 Combined diagrams

The diagrams shown in the relationship between speed-flow, speed-density, and flow-densityare called the fundamental diagrams of traffic flow. These are as shown in figure 9:6

9.8 Summary

Time mean speed and space mean speed are two important measures of speed. It is possible tohave a relation between them and was derived in this chapter. Also, time mean speed will bealways greater than or equal to space mean speed. The fundamental diagrams of traffic floware vital tools which enables analysis of fundamental relationships. There are three diagrams -speed-density, speed-flow and flow-density. They can be together combined in a single diagramas discussed in the last section of the chapter.


CE415 Transportation Engineering II 9. Traffic stream models

9.9 Problems

1. Space mean speed is

(a) the harmonic mean of spot speeds

(b) the sum of spot speeds

(c) the arithmetic mean of spot speeds

(d) the sum of journey speeds

2. Which among the following is the fundamental equation of traffic flow?

(a) q = kv

(b) q = k × v

(c) v = q × k

(d) q = k2 × v

9.10 Solutions

1. Space mean speed is

(a) the harmonic mean of spot speeds√

(b) the sum of spot speeds

(c) the arithmetic mean of spot speeds

(d) the sum of journey speeds

2. Which among the following is the fundamental equation of traffic flow?

(a) q = kv

(b) q = k × v√

(c) v = q × k

(d) q = k2 × v



Chapter 10

Traffic stream models

10.1 Overview

To figure out the exact relationship between the traffic parameters, a great deal of researchhas been done over the past several decades. The results of these researches yielded manymathematical models. Some important models among them will be discussed in this chapter.

10.2 Greenshield’s macroscopic stream model

Macroscopic stream models represent how the behaviour of one parameter of traffic flow changeswith respect to another. Most important among them is the relation between speed and density.The first and most simple relation between them is proposed by Greenshield. Greenshieldassumed a linear speed-density relationship as illustrated in figure 10:1 to derive the model.The equation for this relationship is shown below.

v = vf −[

vf

kj

]

.k (10.1)

where v is the mean speed at density k, vf is the free speed and kj is the jam density. Thisequation ( 10.1) is often referred to as the Greenshields’ model. It indicates that when density

density (k)

spee

d u

kjam

uf

k0

Figure 10:1: Relation between speed and density



spee

d, u u

flow, qq qmax

uf

u0

Figure 10:2: Relation between speed and flow

flow

(q)

C

B

A

q

O

density (k)

ED

qmax

k0 k1 kmax k2 kjam

Figure 10:3: Relation between flow and density 1

becomes zero, speed approaches free flow speed (ie. v → vf when k → 0). Once the relationbetween speed and flow is established, the relation with flow can be derived. This relationbetween flow and density is parabolic in shape and is shown in figure 10:3. Also, we know that

q = k.v (10.2)

Now substituting equation 10.1 in equation 10.2, we get

q = vf .k −[

vf

kj

]

k2 (10.3)

Similarly we can find the relation between speed and flow. For this, put k = qv

in equation 10.1and solving, we get

q = kj .v −[

kj

vf

]

v2 (10.4)



This relationship is again parabolic and is shown in figure 10:2. Once the relationship betweenthe fundamental variables of traffic flow is established, the boundary conditions can be derived.The boundary conditions that are of interest are jam density, freeflow speed, and maximumflow. To find density at maximum flow, differentiate equation 10.3 with respect to k and equateit to zero. ie.,

dq

dk= 0

vf − vf

kj

.2k = 0

k =kj

2

Denoting the density corresponding to maximum flow as k0,

k0 =kj

2(10.5)

Therefore, density corresponding to maximum flow is half the jam density Once we get k0, wecan derive for maximum flow, qmax. Substituting equation 10.5 in equation 10.3

qmax = vf .kj

2− vf

kj.

[

kj

2

]2

= vf .kj

2− vf .

kj

4

=vf .kj

4

Thus the maximum flow is one fourth the product of free flow and jam density. Finally to getthe speed at maximum flow, v0, substitute equation 10.5 in equation 10.1 and solving we get,

v0 = vf −vf

kj.kj

2

v0 =vf

2(10.6)

Therefore, speed at maximum flow is half of the free speed.

10.3 Calibration of Greenshield’s model

Inorder to use this model for any traffic stream, one should get the boundary values, especiallyfree flow speed (vf ) and jam density (kj). This has to be obtained by field survey and this iscalled calibration process. Although it is difficult to determine exact free flow speed and jamdensity directly from the field, approximate values can be obtained from a number of speed and



density observations and then fitting a linear equation between them. Let the linear equationbe y = a + bx such that y is density k and x denotes the speed v. Using linear regressionmethod, coefficients a and b can be solved as,

b =n∑n

i=1 xiyi −∑n

i=1 xi.∑n

i=1 yi

n.∑n

i=1 xi2 − (

∑ni=1 xi)2

(10.7)

a = y − bx (10.8)

Alternate method of solving for b is,

b =

∑ni=1(xi − x)(yi − y)∑n

i=1 (xi − x)2 (10.9)

where xi and yi are the samples, n is the number of samples, and x and y are the mean of xi

and yi respectively.

Problem

For the following data on speed and density, determine the parameters of the Greenshields’model. Also find the maximum flow and density corresponding to a speed of 30 km/hr.

k v171 5129 1520 4070 25

Solution Denoting y = v and x = k, solve for a and b using equation 10.8 and equation 10.9.The solution is tabulated as shown below.

x(k) y(v) (xi − x) (yi − y) (xi − x)(yi − y) (xi − x2)171 5 73.5 -16.3 -1198.1 5402.3129 15 31.5 -6.3 -198.5 992.320 40 -77.5 18.7 -1449.3 6006.370 25 -27.5 3.7 -101.8 756.3390 85 -2947.7 13157.2

x = Σxn

= 3904

= 97.5, y = Σyn

= 854

= 21.3. From equation 10.9, b = −2947.713157.2

= -0.2 a = y − bx= 21.3 + 0.2×97.5 = 40.8 So the linear regression equation will be,

v = 40.8 − 0.2k (10.10)

Here vf = 40.8 andvf

kj= 0.2 This implies, kj = 40.8

0.2= 204 veh/km The basic parameters of

Greenshield’s model are free flow speed and jam density and they are obtained as 40.8 kmphand 204 veh/km respectively. To find maximum flow, use equation 10.6, i.e., qmax = 40.8×204

4=



density, k

spee

d, v

Figure 10:4: Greenberg’s logarithmic model

2080.8 veh/hr Density corresponding to the speed 30 km/hr can be found out by substitutingv = 30 in equation 10.10. i.e, 30 = 40.8 - 0.2 × k Therefore, k = 40.8−30

0.2= 54 veh/km

10.4 Other macroscopic stream models

In Greenshield’s model, linear relationship between speed and density was assumed. But infield we can hardly find such a relationship between speed and density. Therefore, the validityof Greenshields’ model was questioned and many other models came up. Prominent amongthem are Greenberg’s logarithmic model, Underwood’s exponential model, Pipe’s generalizedmodel, and multiregime models. These are briefly discussed below.

10.4.1 Greenberg’s logarithmic model

Greenberg assumed a logarithmic relation between speed and density. He proposed,

v = v0 lnkj

k(10.11)

This model has gained very good popularity because this model can be derived analytically.(This derivation is beyond the scope of this notes). However, main drawbacks of this model isthat as density tends to zero, speed tends to infinity. This shows the inability of the model topredict the speeds at lower densities.

10.4.2 Underwood’s exponential model

Trying to overcome the limitation of Greenberg’s model, Underwood put forward an exponentialmodel as shown below.

v = vf .e−kk0 (10.12)



spee

d, v

Density, k

Figure 10:5: Underwood’s exponential model

where vfThe model can be graphically expressed as in figure 10:5. is the free flow speedm andko is the optimum density, i.e. the densty corresponding to the maximum flow. In this model,speed becomes zero only when density reaches infinity which is the drawback of this model.Hence this cannot be used for predicting speeds at high densities.

10.4.3 Pipes’ generalized model

Further developments were made with the introduction of a new parameter (n) to provide for amore generalised modelling approach. Pipes proposed a model shown by the following equation.

v = vf [1 −(

k

kj

)n

] (10.13)

When n is set to one, Pipe’s model resembles Greenshields’ model. Thus by varying the valuesof n, a family of models can be developed.

10.4.4 Multiregime models

All the above models are based on the assumption that the same speed-density relation isvalid for the entire range of densities seen in traffic streams. Therefore, these models arecalled single-regime models. However, human behaviour will be different at different densities.This is corraborated with field observations which shows different relations at different rangeof densities. Therefore, the speed-density relation will also be different in different zones ofdensities. Based on this concept, many models were proposed generally called multi-regimemodels. The most simple one is called a two-regime model, where separate equations are usedto represent the speed-density relation at congested and uncongested traffic.



qA, vA, kA qB, vB, kB

Figure 10:6: Shock wave: Stream characteristics

density

A

B

flow

kjkBkA

qB

vA

vBqA

Figure 10:7: Shock wave: Flow-density curve

10.5 Shock waves

The flow of traffic along a stream can be considered similar to a fluid flow. Consider a stream oftraffic flowing with steady state conditions, i.e., all the vehicles in the stream are moving witha constant speed, density and flow. Let this be denoted as state A (refer figure 10:6. Suddenlydue to some obstructions in the stream (like an accident or traffic block) the steady statecharacteristics changes and they acquire another state of flow, say state B. The speed, densityand flow of state A is denoted as vA, kA, and qA, and state B as vB, kB, and qB respectively.The flow-density curve is shown in figure 10:7. The speed of the vehicles at state A is givenby the line joining the origin and point A in the graph. The time-space diagram of the trafficstream is also plotted in figure 10:8. All the lines are having the same slope which implies thatthey are moving with constant speed. The sudden change in the characteristics of the streamleads to the formation of a shock wave. There will be a cascading effect of the vehicles in theupstream direction. Thus shock wave is basically the movement of the point that demarcatesthe two stream conditions. This is clearly marked in the figure 10:7. Thus the shock wavesproduced at state B are propagated in the backward direction. The speed of the vehicles atstate B is the line joining the origin and point B of the flow-density curve. Slope of the line ABgives the speed of the shock wave (refer figure 10:7). If speed of the shock-wave is representedasωAB, then

ωAB =qA − qB

kA − kB

(10.14)

The above result can be analytically solved by equating the expressions for the number vehiclesleaving the upstream and joining the downstream of the shock wave boundary (this assumption



time

dist

ance A

B

Figure 10:8: Shock wave : time-distance diagram

is true since the vehicles cannot be created or destroyed. Let NA be the number of vehiclesleaving the section A. Then, NA = qB t. The relative speed of these vehicles with respect tothe shock wave will be vA − ωAB. Hence,

NA = kA (vA − ωAB) t (10.15)

Similarly, the vehicles entering the state B is given as

NB = kA (vB − ωAB) t (10.16)

Equating equations 10.15 and 10.16, and solving for ωAB as follows will yeild to:

NA = NB

kA (vA − ωAB) t = kB (vB − ωAB) t

kA vA t − kA ωAB t = kB vB t − kBωAB t

kAωAB t − kBωAB t = kA vA − kB vB

ωAB (kA − kB) = qA − qB

This will yeild the following expression for the shock-wave speed.

ωAB =qA − qB

kA − kB(10.17)

In this case, the shock wave move against the direction of traffic and is therefore called abackward moving shock wave. There are other possibilities of shockwaves such as forwardmoving shockwaves and stationary shockwaves. The forward moving shockwaves are formedwhen a stream with higher density and higher flow meets a stream with relatively lesser densityand flow. For example, when the width of the road increases suddenly, there are chances fora forward moving shockwave. Stationary shockwaves will occur when two streams having thesame flow value but different densities meet.



10.6 Macroscopic flow models

If one looks into traffic flow from a very long distance, the flow of fairly heavy traffic appearslike a stream of a fluid. Therefore, a macroscopic theory of traffic can be developed with thehelp of hydrodynamic theory of fluids by considering traffic as an effectively one-dimensionalcompressible fluid. The behaviour of individual vehicle is ignored and one is concerned onlywith the behaviour of sizable aggregate of vehicles. The earliest traffic flow models began bywriting the balance equation to address vehicle number conservation on a road. Infact, alltraffic flow models and theories must satisfy the law of conservation of the number of vehicleson the road. Assuming that the vehicles are flowing from left to right, the continuity equationcan be written as

∂k(x, t)

∂t+

∂q(x, t)

∂x= 0 (10.18)

where x denotes the spatial coordinate in the direction of traffic flow, t is the time, k is thedensity and q denotes the flow. However, one cannot get two unknowns, namely k(x, t) byand q(x, t) by solving one equation. One possible solution is to write two equations from tworegimes of the flow, say before and after a bottleneck. In this system the flow rate before andafter will be same, or

k1v1 = k2v2 (10.19)

From this the shockwave velocity can be derived as

v(to)p =q2 − q1

k2 − k1(10.20)

This is normally referred to as Stock’s shockwave formula. An alternate possibility whichLighthill and Whitham adopted in their landmark study is to assume that the flow rate q isdetermined primarily by the local density k, so that flow q can be treated as a function of onlydensity k. Therefore the number of unknown variables will be reduced to one. Essentially thisassumption states that k(x,t) and q (x,t) are not independent of each other. Therefore thecontinuity equation takes the form

∂k(x, t)

∂t+

∂q(k(x, t))

∂x= 0 (10.21)

However, the functional relationship between flow q and density k cannot be calculated fromfluid-dynamical theory. This has to be either taken as a phenomenological relation derived fromthe empirical observation or from microscopic theories. Therefore, the flow rate q is a functionof the vehicular density k; q = q(k). Thus, the balance equation takes the form

∂k(x, t)

∂t+

∂q(k(x, t))

∂x= 0 (10.22)

Now there is only one independent variable in the balance equation, the vehicle density k. Ifinitial and boundary conditions are known, this can be solved. Solution to LWR models are



kinematic waves moving with velocitydq(k)

dk(10.23)

This velocity vk is positive when the flow rate increases with density, and it is negative whenthe flow rate decreases with density. In some cases, this function may shift from one regime tothe other, and then a shock is said to be formed. This shockwave propagate at the velocity

vs =q(k2) − q(k1)

k2 − k1(10.24)

where q(k2) and q(k1) are the flow rates corresponding to the upstream density k2 and down-stream density k1 of the shockwave. Unlike Stock’s shockwave formula there is only one variablehere.

10.6.1 Extented Topics

1. Multi regine model (formulation of both two and three regime models)

2. Catastrophe Theory

3. Three dimentional models and plots

10.7 Summary

Traffic stream models attempt to establish a better relationship between the traffic parameters.These models were based on many assumptions, for instance, Greenshield’s model assumed alinear speed-density relationship. Other models were also discussed in this chapter. The modelsare used for explaining several phenomena in connection with traffic flow like shock wave.

10.8 Problems

1. Stationary shockwaves will occur

(a) when two streams having the same flow value but different densities meet.

(b) when two streams having the different flow value but same densities meet.

(c) when two streams having the same flow value and densities meet.

(d) when two streams with different speeds meet.

2. Linear relationship between speed and density was assumed in

(a) Greenberg’s model

(b) Greenshield’s model


CE415 Transportation Engineering II 10. Traffic data collection

(c) Pipe’s generalized model

(d) Underwood’s model

10.8.1 Solutions

1. Stationary shockwaves will occur when two streams having the same flow value but dif-ferent densities meet.

2. Linear relationship between speed and density was assumed in Greenshield’s model



Chapter 11

Traffic data collection

11.1 Overview

Unlike many other disciplines of the engineering, the situations that are interesting to a trafficengineer cannot be reproduced in a laboratory. Even if road and vehicles could be set up in largelaboratories, it is impossible to simulate the behavior of drivers in the laboratory. Therefore,traffic stream characteristics need to be collected only from the field. There are several methodsof data collection depending on the need of the study and some important ones are describedin this chapter.

11.2 Data requirements

The most important traffic characteristics to be collected from the field includes sped, traveltime, flow and density. Some cases, spacing and headway are directly measured. In addition,the occupancy, ie percentage of time a point on the road is occupied by vehicles is also ofinterest. The measurement procedures can be classified based on the geographical extent ofthe survey into five categories: (a) measurement at point on the road, (b) measurement over ashort section of the road (less than 500 metres) (c) measurement over a length of the road (morethan about 500 metres) (d) wide area samples obtained from number of locations, and (e) theuse of an observer moving in the traffic stream. In each category, numerous data collection arethere. However, important and basic methods will be discussed.

11.2.1 Measurements at a point

The most important point measurement is the vehicle volume count. Data can be collectedmanually or automatically. In manual method, the observer will stand at the point of interestand count the vehicles with the help of hand tallies. Normally, data will be collected for shortinterval of 5 minutes or 15 minutes etc. and for each types of vehicles like cars, two wheelers,three wheelers, LCV, HCV, multi axle trucks, nonmotorised traffic like bullock cart, hand cartetc. From the flow data, flow and headway can be derived.

Modern methods include the use of inductive loop detector, video camera, and many othertechnologies. These methods helps to collect accurate information for long duration. In videocameras, data is collected from the field and is then analyzed in the lab for obtaining results.



x

Base length

observer enoscope

Figure 11:1: Illustration of measurement over short section using enoscope

Radars and microwave detectors are used to obtain the speed of a vehicle at a point. Since nolength is involved, density cannot be obtained by measuring at a point.

11.2.2 Measurements over short section

The main objective of this study is to find the spot speed of vehicles. Manual methods includethe use of enoscope. In this method a base length of about 30-90 metres is marked on theroad. Enoscope is placed at one end and observer will stand at the other end. He could see thevehicle passing the farther end through enoscope and starts the stop watch. Then he stops thestop watch when the vehicle passes in front of him. The working of the enoscope is shown infigure 11:1.

An alternative method is to use pressure contact tube which gives a pressure signal whenvehicle moves at either end. Another most widely used method is inductive loop detector whichworks on the principle of magnetic inductance. Road will be cut and a small magnetic loop isplaced. When the metallic content in the vehicle passes over it, a signal will be generated andthe count of the vehicle can be found automatically. The advantage of this detector is that thecounts can be obtained throughout the life time of the road. However, chances of errors arepossible because noise signals may be generated due to heavy vehicle passing adjacent lanes.When dual loops are used and if the spacing between them is known then speed also can becalculated in addition to the vehicle cost.

11.2.3 Measurements over long section

This is normally used to obtain variations in speed over a stretch of road. Usually the stretchwill be having a length more than 500 metres. We can also get density. Most traditionalmethod uses aerial photography. From a single frame, density can be measured, but not speedor volumes. In time lapse photography, several frames are available. If several frames areobtained over short time intervals, speeds can be measured from the distance covered betweenthe two frames and time interval between them.



l

Figure 11:2: Illustration of moving observer method

11.2.4 Moving observer method for stream measurement

Determination of any of the two parameters of the traffic flow will provide the third one bythe equation q = u.k. Moving observer method is the most commonly used method to getthe relationship between the fundamental stream characteristics. In this method, the observermoves in the traffic stream unlike all other previous methods.

Consider a stream of vehicles moving in the north bound direction. Two different cases ofmotion can be considered. The first case considers the traffic stream to be moving and theobserver to be stationary. If no is the number of vehicles overtaking the observer during aperiod, t, then flow q is n0

t, or

n0 = q × t (11.1)

The second case assumes that the stream is stationary and the observer moves with speed vo.If np is the number of vehicles overtaken by observer over a length l, then by definition, densityk is np

l, or

np = k × l (11.2)

ornp = k.vo.t (11.3)

where v0 is the speed of the observer and t is the time taken for the observer to cover the roadstretch. Now consider the case when the observer is moving within the stream. In that casemo vehicles will overtake the observer and mp vehicles will be overtaken by the observer in the



test vehicle. Let the difference m is given by m0 - mp, then from equation 11.1 and equation11.3,

m = m0 − mp = q t − k vo t (11.4)

This equation is the basic equation of moving observer method, which relates q, k to the countsm, t and vo that can be obtained from the test. However, we have two unknowns, q and k, butonly one equation. For generating another equation, the test vehicle is run twice once with thetraffic stream and another one against traffic stream, i.e.

mw = q tw − k vw tw (11.5)

= q tw − k l

ma = q ta + k va ta (11.6)

= q ta + k l

where, a, w denotes against and with traffic flow. It may be noted that the sign of equation 11.6is negative, because test vehicle moving in the opposite direction can be considered as a casewhen the test vehicle is moving in the stream with negative velocity. Further, in this case, allthe vehicles will be overtaking, since it is moving with negative speed. In other words, when thetest vehicle moves in the opposite direction, the observer simply counts the number of vehiclesin the opposite direction. Adding equation 11.5 and 11.6, we will get the first parameter ofthe stream, namely the flow(q) as:

q =mw + ma

tw + ta(11.7)

Now calculating space mean speed from equation 11.5,

mw

tw= q − kvw

= q − q

vvw

= q − q

v

[

l

tw

]

= q

(

1 − l

v× 1

tw

)

= q(

1 − tavg

tw

)

If vs is the mean stream speed, then average travel time is given by tavg = lvs

. Therefore,

mw

q= tw(1 − tavg

tw) = tw − tavg

tavg = tw − mw

q=

l

v,



Rewriting the above equation, we get the second parameter of the traffic flow, namely the meanspeed vs and can be written as,

vs =l

tw − mw

q

(11.8)

Thus two parameters of the stream can be determined. Knowing the two parameters the thirdparameter of traffic flow density (k) can be found out as

k =q

vs

(11.9)

For increase accuracy and reliability, the test is performed a number of times and the averageresults are to be taken.

Example 1

The length of a road stretch used for conducting the moving observer test is 0.5 km and the speedwith which the test vehicle moved is 20 km/hr. Given that the number of vehicles encounteredin the stream while the test vehicle was moving against the traffic stream is 107, number ofvehicles that had overtaken the test vehicle is 10, and the number of vehicles overtaken by thetest vehicle is 74, find the flow, density and average speed of the stream.

Solution Time taken by the test vehicle to reach the other end of the stream while it ismoving along with the traffic is tw = 0.5

20= 0.025 hr Time taken by the observer to reach

the other end of the stream while it is moving against the traffic is ta = tw = 0.025 hr Flowis given by equation, q = 107+(10−74)

0.025+0.025= 860 veh/hr Stream speed vs can be found out from

equationvs = 0.50.025− 10.74

860

= 5 km/hr Density can be found out from equation as k = 8605

=

172veh/km

Example 2

The data from four moving observer test methods are shown in the table. Column 1 givesthe sample number, column 2 gives the number of vehicles moving against the stream, column3 gives the number of vehicles that had overtaken the test vehicle, and last column gives thenumber of vehicles overtaken by the test vehicle. Find the three fundamental stream parametersfor each set of data. Also plot the fundamental diagrams of traffic flow.

Sample no. 1 2 31 107 10 742 113 25 413 30 15 54 79 18 9

Solution From the calculated values of flow, density and speed, the three fundamental dia-grams can be plotted as shown in figure 11:3.



Sample no. ma mo mp m(mo − mp) ta tw q = ma+mw

ta+twu = l

tw−maq

k = qv

1 107 10 74 -64 0.025 0.025 860 5.03 1712 113 25 41 -16 0.025 0.025 1940 15.04 1293 30 15 5 10 0.025 0.025 800 40 204 79 18 9 9 0.025 0.025 1760 25.14 70

density k

flow

q

20 70 129171

800

1760

1940

flow q

spee

d u

860

40

density k5.03

15.0425.14

spee

d u

Figure 11:3: Fundamental diagrams of traffic flow



11.3 Summary

Traffic engineering studies differ from other studies in the fact that they require extensive datafrom the field which cannot be exactly created in any laboratory. Speed data are collectedfrom measurements at a point or over a short section or over an area. Traffic flow data arecollected at a point. Moving observer method is one in which both speed and traffic flow dataare obtained by a single experiment.

11.4 Problems

1. In the moving observer experiment, if the density is k, speed of the observer is vo , lengthof the test stretch is l, t is the time taken by the observer to cover the road stretch, thenumber of vehicles overtaken by the observer np is given by,

(a) np = k.t

(b) np = k.l

(c) np = kvo.t

(d) np = k.vo.t

2. If the length of the road stretch taken for conducting moving observer experiment is 0.4km, time taken by the observer to move with the traffic is 5 seconds, number of vehiclesmoving with the test vehicle in the same direction is 10, flow is 10 veh/sec, find the meanspeed.

(a) 50 m/s

(b) 100 m/s

(c) 150 m/s

(d) 200 m/s

11.5 Solutions

1. In the moving observer experiment, if the density is k, speed of the observer is vo , lengthof the test stretch is l, t is the time taken by the observer to cover the road stretch, thenumber of vehicles overtaken by the observer np is given by,

(a) np = k.t

(b) np = k.l

(c) np = kvo.t

(d) np = k.vo.t√


CE415 Transportation Engineering II 11. Microscopic traffic flow modeling

2. If the length of the road stretch taken for conducting moving observer experiment is 0.4km, time taken by the observer to move with the traffic is 5 seconds, number of vehiclesmoving with the test vehicle in the same direction is 10, flow is 10 veh/sec, find the meanspeed.

(a) 50 m/s

(b) 100 m/s√

(c) 150 m/s

(d) 200 m/s

Solution: Given that l=0.4 km, tw=5seconds, mw=10,q=10 veh/sec, substituting inequation,vs = l

tw−mwq

=100m/s.



Chapter 12

Microscopic traffic flow modeling

12.1 Overview

Macroscopic modeling looks at traffic flow from a global perspective, whereas microscopic mod-eling, as the term suggests, gives attention to the details of traffic flow and the interactionstaking place within it. This chapter gives an overview of microscopic approach to modelingtraffic and then elaborates on the various concepts associated with it.

A microscopic model of traffic flow attempts to analyze the flow of traffic by modelingdriver-driver and driver-road interactions within a traffic stream which respectively analyzesthe interaction between a driver and another driver on road and of a single driver on thedifferent features of a road. Many studies and researches were carried out on driver’s behaviorin different situations like a case when he meets a static obstacle or when he meets a dynamicobstacle. Several studies are made on modeling driver behavior in another following car andsuch studies are often referred to as car following theories of vehicular traffic.

12.2 Notation

Longitudinal spacing of vehicles are of particular importance from the points of view of safety,capacity and level of service. The longitudinal space occupied by a vehicle depend on thephysical dimensions of the vehicles as well as the gaps between vehicles. For measuring thislongitudinal space, two microscopic measures are used- distance headway and distance gap.Distance headway is defined as the distance from a selected point (usually front bumper) onthe lead vehicle to the corresponding point on the following vehicles. Hence, it includes thelength of the lead vehicle and the gap length between the lead and the following vehicles. Beforegoing in to the details, various notations used in car-following models are discussed here withthe help of figure 12:1. The leader vehicle is denoted as n and the following vehicle as (n + 1).Two characteristics at an instant t are of importance; location and speed. Location and speedof the lead vehicle at time instant t are represented by xt

n and vtn respectively. Similarly, the

location and speed of the follower are denoted by xtn+1 and vt

n+1 respectively. The followingvehicle is assumed to accelerate at time t + ∆T and not at t, where ∆T is the interval of timerequired for a driver to react to a changing situation. The gap between the leader and thefollower vehicle is therefore xt

n − xtn+1.



Follower

Direction of traffic

Leader

nvn+1

n + 1

vn

xn − xn+1

xn

xn+1

Figure 12:1: Notation for car following model

12.3 Car following models

Car following theories describe how one vehicle follows another vehicle in an uninterrupted flow.Various models were formulated to represent how a driver reacts to the changes in the relativepositions of the vehicle ahead. Models like Pipes, Forbes, General Motors and Optimal velocitymodel are worth discussing.

12.3.1 Pipe’s model

The basic assumption of this model is “A good rule for following another vehicle at a safedistance is to allow yourself at least the length of a car between your vehicle and the vehicleahead for every ten miles per hour of speed at which you are traveling” According to Pipe’scar-following model, the minimum safe distance headway increases linearly with speed. Adisadvantage of this model is that at low speeds, the minimum headways proposed by thetheory are considerably less than the corresponding field measurements.

12.3.2 Forbes’ model

In this model, the reaction time needed for the following vehicle to perceive the need to deceler-ate and apply the brakes is considered. That is, the time gap between the rear of the leader andthe front of the follower should always be equal to or greater than the reaction time. Therefore,the minimum time headway is equal to the reaction time (minimum time gap) and the timerequired for the lead vehicle to traverse a distance equivalent to its length. A disadvantage ofthis model is that, similar to Pipe’s model, there is a wide difference in the minimum distanceheadway at low and high speeds.

12.3.3 General Motors’ model

The General Motors’ model is the most popular of the car-following theories because of thefollowing reasons:

1. Agreement with field data; the simulation models developed based on General motors’car following models shows good correlation to the field data.



2. Mathematical relation to macroscopic model; Greenberg’s logarithmic model for speed-density relationship can be derived from General motors car following model.

In car following models, the motion of individual vehicle is governed by an equation, whichis analogous to the Newton’s Laws of motion. In Newtonian mechanics, acceleration can beregarded as the response of the particle to stimulus it receives in the form of force which includesboth the external force as well as those arising from the interaction with all other particles inthe system. This model is the widely used and will be discussed in detail later.

12.3.4 Optimal velocity model

The concept of this model is that each driver tries to achieve an optimal velocity based on thedistance to the preceding vehicle and the speed difference between the vehicles. This was analternative possibility explored recently in car-following models. The formulation is based onthe assumption that the desired speed vndesired

depends on the distance headway of the nthvehicle. i.e.vt

ndesired= vopt(∆xt

n) where vopt is the optimal velocity function which is a functionof the instantaneous distance headway ∆xt

n. Therefore atn is given by

atn = [1/τ ][V opt(∆xt

n) − vtn] (12.1)

where 1τ

is called as sensitivity coefficient. In short, the driving strategy of nth vehicle is that,it tries to maintain a safe speed which inturn depends on the relative position, rather thanrelative speed.

12.4 General motor’s car following model

12.4.1 Basic Philosophy

The basic philosophy of car following model is from Newtonian mechanics, where the acceler-ation may be regarded as the response of a matter to the stimulus it receives in the form ofthe force it receives from the interaction with other particles in the system. Hence, the basicphilosophy of car-following theories can be summarized by the following equation

[Response]n α [Stimulus]n (12.2)

for the nth vehicle (n=1, 2, ...). Each driver can respond to the surrounding traffic conditionsonly by accelerating or decelerating the vehicle. As mentioned earlier, different theories on car-following have arisen because of the difference in views regarding the nature of the stimulus.The stimulus may be composed of the speed of the vehicle, relative speeds, distance headwayetc, and hence, it is not a single variable, but a function and can be represented as,

atn = fsti(vn, ∆xn, ∆vn) (12.3)



where fsti is the stimulus function that depends on the speed of the current vehicle, relativeposition and speed with the front vehicle.

12.4.2 Follow-the-leader model

The car following model proposed by General motors is based on follow-the leader concept. Thisis based on two assumptions; (a) higher the speed of the vehicle, higher will be the spacingbetween the vehicles and (b) to avoid collision, driver must maintain a safe distance with thevehicle ahead.

Let ∆xtn+1 is the gap available for (n+1)th vehicle, and let ∆xsafe is the safe distance, vt

n+1

and vtn are the velocities, the gap required is given by,

∆xtn+1 = ∆xsafe + τvt

n+1 (12.4)

where τ is a sensitivity coefficient. The above equation can be written as

xn − xtn+1 = ∆xsafe + τvt

n+1 (12.5)

Differentiating the above equation with respect to time, we get

vtn − vt

n+1 = τ.atn+1

atn+1 =

1

τ[vt

n − vtn+1]

General Motors has proposed various forms of sensitivity coefficient term resulting in five gen-erations of models. The most general model has the form,

atn+1 =

[

αl,m(vtn+1)

m

(xtn − xt

n+1)l

]

[

vtn − vt

n+1

]

(12.6)

where l is a distance headway exponent and can take values from +4 to -1, m is a speed exponentand can take values from -2 to +2, and α is a sensitivity coefficient. These parameters are tobe calibrated using field data. This equation is the core of traffic simulation models.

In computer, implementation of the simulation models, three things need to be remembered:

1. A driver will react to the change in speed of the front vehicle after a time gap called thereaction time during which the follower perceives the change in speed and react to it.

2. The vehicle position, speed and acceleration will be updated at certain time intervalsdepending on the accuracy required. Lower the time interval, higher the accuracy.

3. Vehicle position and speed is governed by Newton’s laws of motion, and the accelerationis governed by the car following model.

Therefore, the governing equations of a traffic flow can be developed as below. Let ∆T is



the reaction time, and ∆t is the updation time, the governing equations can be written as,

vtn = vt−∆t

n + at−∆tn × ∆t (12.7)

xtn = xt−∆t

n + vt−∆tn × ∆t +

1

2at−∆t

n ∆t2 (12.8)

atn+1 =

[

αl,m(vtn+1)

m

(xt−∆Tn − xt−∆T

n+1 )l

]

(vt−∆Tn − vt−∆T

n+1 ) (12.9)

The equation 12.7 is a simulation version of the Newton’s simple law of motion v = u + at andequation 12.8 is the simulation version of the Newton’s another equation s = ut + 1

2at2. The

acceleration of the follower vehicle depends upon the relative velocity of the leader and thefollower vehicle, sensitivity coefficient and the gap between the vehicles.

Problem

Let a leader vehicle is moving with zero acceleration for two seconds from time zero. Then heaccelerates by 1 m/s2 for 2 seconds, then decelerates by 1m/s2for 2 seconds. The initial speedis 16 m/s and initial location is 28 m from datum. A vehicle is following this vehicle with initialspeed 16 m/s, and position zero. Simulate the behavior of the following vehicle using GeneralMotors’ Car following model (acceleration, speed and position) for 7.5 seconds. Assume theparameters l=1, m=0 , sensitivity coefficient (αl,m) = 13, reaction time as 1 second and scaninterval as 0.5 seconds.

Solution The first column shows the time in seconds. Column 2, 3, and 4 shows the accel-eration, velocity and distance of the leader vehicle. Column 5,6, and 7 shows the acceleration,velocity and distance of the follower vehicle. Column 8 gives the difference in velocities betweenthe leader and follower vehicle denoted as dv. Column 9 gives the difference in displacementbetween the leader and follower vehicle denoted as dx. Note that the values are assumed to bethe state at the beginning of that time interval. At time t=0, leader vehicle has a velocity of16 m/s and located at a distance of 28 m from a datum. The follower vehicle is also having thesame velocity of 16 m/s and located at the datum. Since the velocity is same for both, dv =0. At time t = 0, the leader vehicle is having acceleration zero, and hence has the same speed.The location of the leader vehicle can be found out from equation as, x = 28+16×0.5 = 36m. Similarly, the follower vehicle is not accelerating and is maintaining the same speed. Thelocation of the follower vehicle is, x = 0+16×0.5 = 8 m. Therefore, dx = 36-8 =28m. Thesesteps are repeated till t = 1.5 seconds. At time t = 2 seconds, leader vehicle accelerates at therate of 1 m/s2 and continues to accelerate for 2 seconds. After that it decelerates for a periodof two seconds. At t= 2.5 seconds, velocity of leader vehicle changes to 16.5 m/s. Thus dvbecomes 0.5 m/s at 2.5 seconds. dx also changes since the position of leader changes. Since thereaction time is 1 second, the follower will react to the leader’s change in acceleration at 2.0seconds only after 3 seconds. Therefore, at t=3.5 seconds, the follower responds to the leaderschange in acceleration given by equation i.e., a = 13×0.5

28.23= 0.23 m/s2. That is the current ac-

celeration of the follower vehicle depends on dv and reaction time ∆ of 1 second. The follower



Follower

Leader

Time(seconds)

Vel

ocity

15 20 25 10 5 0

17

18

19

20

16

15

30

Figure 12:2: Velocity vz Time

FollowerLeader

25 20 15 10 5 0

−0.5

−1

0

0.5

1

1.5

−1.5 30

Time(seconds)

Acc

eler

atio

n

Figure 12:3: Acceleration vz Time

will change the speed at the next time interval. i.e., at time t = 4 seconds. The speed of thefollower vehicle at t = 4 seconds is given by equation as v= 16+0.231×0.5 = 16.12 The locationof the follower vehicle at t = 4 seconds is given by equation as x = 56+16×0.5+1

2×0.231×0.52

= 64.03 These steps are followed for all the cells of the table.The earliest car-following models considered the difference in speeds between the leader and

the follower as the stimulus. It was assumed that every driver tends to move with the samespeed as that of the corresponding leading vehicle so that

atn =

1

τ(vt+1

n − vtn+1) (12.10)

where τ is a parameter that sets the time scale of the model and 1τ

can be considered as ameasure of the sensitivity of the driver. According to such models, the driving strategy is tofollow the leader and, therefore, such car-following models are collectively referred to as thefollow the leader model. Efforts to develop this stimulus function led to five generations of



Table 12:1: Car-following examplet a(t) v(t) x(t) a(t) v(t) x(t) dv dx

(1) (2) (3) (4) (5) (6) (7) (8) (9)t a(t) v(t) x(t) a(t) v(t) x(t) dv dx

0.00 0.00 16.00 28.00 0.00 16.00 0.00 0.00 28.000.50 0.00 16.00 36.00 0.00 16.00 8.00 0.00 28.001.00 0.00 16.00 44.00 0.00 16.00 16.00 0.00 28.001.50 0.00 16.00 52.00 0.00 16.00 24.00 0.00 28.002.00 1.00 16.00 60.00 0.00 16.00 32.00 0.00 28.002.50 1.00 16.50 68.13 0.00 16.00 40.00 0.50 28.133.00 1.00 17.00 76.50 0.00 16.00 48.00 1.00 28.503.50 1.00 17.50 85.13 0.23 16.00 56.00 1.50 29.134.00 -1.00 18.00 94.00 0.46 16.12 64.03 1.88 29.974.50 -1.00 17.50 102.88 0.67 16.34 72.14 1.16 30.735.00 -1.00 17.00 111.50 0.82 16.68 80.40 0.32 31.105.50 -1.00 16.50 119.88 0.49 17.09 88.84 -0.59 31.036.00 0.00 16.00 128.00 0.13 17.33 97.45 -1.33 30.556.50 0.00 16.00 136.00 -0.25 17.40 106.13 -1.40 29.877.00 0.00 16.00 144.00 -0.57 17.28 114.80 -1.28 29.207.50 0.00 16.00 152.00 -0.61 16.99 123.36 -0.99 28.648.00 0.00 16.00 160.00 -0.57 16.69 131.78 -0.69 28.228.50 0.00 16.00 168.00 -0.45 16.40 140.06 -0.40 27.949.00 0.00 16.00 176.00 -0.32 16.18 148.20 -0.18 27.809.50 0.00 16.00 184.00 -0.19 16.02 156.25 -0.02 27.7510.00 0.00 16.00 192.00 -0.08 15.93 164.24 0.07 27.7610.50 0.00 16.00 200.00 -0.01 15.88 172.19 0.12 27.8111.00 0.00 16.00 208.00 0.03 15.88 180.13 0.12 27.8711.50 0.00 16.00 216.00 0.05 15.90 188.08 0.10 27.9212.00 0.00 16.00 224.00 0.06 15.92 196.03 0.08 27.9712.50 0.00 16.00 232.00 0.05 15.95 204.00 0.05 28.0013.00 0.00 16.00 240.00 0.04 15.98 211.98 0.02 28.0213.50 0.00 16.00 248.00 0.02 15.99 219.98 0.01 28.0214.00 0.00 16.00 256.00 0.01 16.00 227.98 0.00 28.0214.50 0.00 16.00 264.00 0.00 16.01 235.98 -0.01 28.0215.00 0.00 16.00 272.00 0.00 16.01 243.98 -0.01 28.0215.50 0.00 16.00 280.00 0.00 16.01 251.99 -0.01 28.0116.00 0.00 16.00 288.00 -0.01 16.01 260.00 -0.01 28.0016.50 0.00 16.00 296.00 0.00 16.01 268.00 -0.01 28.0017.00 0.00 16.00 304.00 0.00 16.00 276.00 0.00 28.0017.50 0.00 16.00 312.00 0.00 16.00 284.00 0.00 28.0018.00 0.00 16.00 320.00 0.00 16.00 292.00 0.00 28.0018.50 0.00 16.00 328.00 0.00 16.00 300.00 0.00 28.0019.00 0.00 16.00 336.00 0.00 16.00 308.00 0.00 28.0019.50 0.00 16.00 344.00 0.00 16.00 316.00 0.00 28.0020.00 0.00 16.00 352.00 0.00 16.00 324.00 0.00 28.0020.50 0.00 16.00 360.00 0.00 16.00 332.00 0.00 28.00



car-following models, and the most general model is expressed mathematically as follows.

at+∆Tn+1 =

αl,m [vt−∆Tn+1 ]m

[xt−∆Tn − xt−∆T

n+1 ]l(vt−∆T

n − vt−∆Tn+1 ) (12.11)

where l is a distance headway exponent and can take values from +4 to -1, m is a speed exponentand can take values from -2 to +2, and α is a sensitivity coefficient. These parameters are tobe calibrated using field data.

12.5 Simulation Models

Simulation modeling is an increasingly popular and effective tool for analyzing a wide variety ofdynamical problems which are difficult to be studied by other means. Usually, these processesare characterized by the interaction of many system components or entities.

12.5.1 Applications of simulation

Traffic simulations models can meet a wide range of requirements:

1. Evaluation of alternative treatments

2. Testing new designs

3. As an element of the design process

4. Embed in other tools

5. Training personnel

6. Safety Analysis

12.5.2 Need for simulation models

Simulation models are required in the following conditions

1. Mathematical treatment of a problem is infeasible or inadequate due to its temporal orspatial scale

2. The accuracy or applicability of the results of a mathematical formulation is doubtful,because of the assumptions underlying (e.g., a linear program) or an heuristic procedure(e.g., those in the Highway Capacity Manual)

3. The mathematical formulation represents the dynamic traffic/control environment as asimpler quasi steady state system.

4. There is a need to view vehicle animation displays to gain an understanding of how thesystem is behaving



5. Training personnel

6. Congested conditions persist over a significant time.

12.5.3 Classification of Simulation Model

Simulation models are classified based on many factors like

1. Continuity

(a) Continuous model

(b) Discrete model

2. Level of detail

(a) Macroscopic models

(b) Mesoscopic models

(c) Microscopic models

3. Based on Processes

(a) Deterministic

(b) Stochastic

12.6 Summary

Microscopic traffic flow modeling focuses on the minute aspects of traffic stream like vehicle tovehicle interaction and individual vehicle behavior. They help to analyze very small changesin the traffic stream over time and space. Car following model is one such model where in thestimulus-response concept is employed. Optimal models and simulation models were brieflydiscussed.

12.7 Problems

1. The minimum safe distance headway increases linearly with speed. Which model followsthis assumption?

(a) Forbe’s model

(b) Pipe’s model

(c) General motor’s model

(d) Optimal velocity model

2. The most popular of the car following models is


CE415 Transportation Engineering II 12. Modeling Traffic Characteristics

(a) Forbe’s model

(b) Pipe’s model



12.8 Solutions

1. The minimum safe distance headway increases linearly with speed. Which model followsthis assumption?

(a) Forbe’s model

(b) Pipe’s model√



2. The most popular of the car following models is

(a) Forbe’s model

(b) Pipe’s model

(c) General motor’s model√




Chapter 13

Modeling Traffic Characteristics

13.1 Modeling time-headway’s

Modeling inter arrival time or time headway or simply headway is he time interval between thesuccessive arrival of two vehicles at a given point. This is a continuous variable and can betreated as a random variable.

13.1.1 Classification of headway distribution

One can observe three types of flow in the field:

1. Low volume flow

(a) Headway follow a random process as there is no interaction between the arrival oftwo vehicles.

(b) The arrival of one vehicle is independent of the arrival of other vehicle.

(c) The minimum headway is governed by the safety criteria.

(d) A negative exponential distribution can be used to model such flow

2. High volume flow

(a) This is characterized by ’near’ constant headway

(b) The flow is very high and is near to the capacity

(c) The mean is very low and so is the variance

(d) A normal distribution can used to model such flow

3. Intermediate flow

(a) Some vehicle travel independently and some vehicle has interaction

(b) More difficult to analyzed and has more application in the field.

(c) Pearson Type III Distribution can be used which is a very general case of negativeexponential distribution and normal distribution.



13.1.2 Negative exponential distribution

• PDF of negative exponential function

f(t) =1

βe

−tβ (13.1)

= e−t if β = 1

• The pb of h/w greater than of equal to 0

p(h ≥ 0) =∫ ∞

0e−t dt = −e−t

∣

∣

∣

∞

0= 1 (13.2)

• The pb of h/w greater than of equal to h

p(h ≥ t) = 1 − p(h < t)

= 1 − p

[

∫ h

0e−t dt

]

= 1 −[

−e−t∣

∣

∣

h

0

]

= e−h

• From the PDF of possson distribution we have the probability of x vehicle arriving intime interval t as p(x) and is given by

p(x) =mx e−m

x!(13.3)

where m is the mean arrival rate in the time interval t. The mean arrival rate is given asm = V

3600t where V is the hourly flow rate.

• If the probability of no vehicle in the interval t is given as p(0), then this probability issame as the probability that the headway greater than or equal to t. Therefore,

p(x = 0) = e−m (13.4)

= p(h ≥ t) (13.5)

= e−tµ (13.6)

where µ = 1V

3600

whcih is same as the mean headway.

• p[t ≤ h ≤ t + δt] = p[h ≥ t] − p[h ≥ t + δt]

13.1.3 Normal distribution

• for high volume traffic



• probability density function for normal distribution is

f(t) =1√2πσ

e−(t−µ)2

2σ2 (13.7)

• The probability for headway less than a given time

p(h ≤ t) =∫ t

−∞

1√2πσ

e−(t−µ)2

2σ2 (13.8)

p(h ≤ t + δt) =∫ t+δt

−∞

1√2πσ

e−(t+δt−µ)2

2σ2

p(t ≤ h ≤ t + δt) = p(h ≤ t + δt) − p(h ≤ t)

• The normal distribution has two parameters; µ and σ

• The µ is simple mean headway or inverse of average flow rate given by

µ =1

(3600V

)(13.9)

• Since headway’s cannot be negative, σ has to be calculated, not from observation, butempirically. First specify the theoretical minimum headway possible, say α and set it asmean minus twice standard deviation.

α = µ − 2 × σ (13.10)

σ =µ − α

2(13.11)

For example, if the hourly flow rate is 1500 vehicles and the minimum expected headwaysis 0.5 second, then,

σ =36001500

− 0.5

2= 0.95 (13.12)

• The integration of the pdf of normal distribution is not available in a closed form solution.Therefore, one has to numerically integrate or normalize the given distribution havingmean µ and sd σ to a standard normal distribution (µ = 0 and sd σ = 1) whose valuesare available as standard tables. The transformation is give as:

p[t ≤ h ≤ (t + δt)] = p

[

t − µ

σ≤ h − µ

σ≤ (t + δt) − µ

σ

]

(13.13)

= p

[

h ≤ (t + δt) − µ

σ

]

− p[

h ≤ t − µ

σ

]

(13.14)



• For example if q=1600 veh/hr, then µ = 36001600

= 2.25. Therefore σ = 2.25−0.52

= 0.875(assuming α = 0.5 sec).

p[1.5 ≤ h ≤ 2.0] = p[h ≤ 2.0] − p[h ≤ 1.5] (13.15)

= p[

h ≤ 2.0 − 2.25

0.875

]

− p[

h ≤ 1.5 − 2.25

0.875

]

(13.16)

= p [h ≤ −0.29] − p [h ≤ −0.86] (13.17)

= 0.3859 − 0.1949 (from tables) (13.18)

= 0.191. (13.19)

13.1.4 Pearson type III distribution

• The pdf is given by

f(t) = λΓ(K)

[λ(t − α)]K−1 e−λ(t−α) K, α ∈ R Pearson

= λΓ(K)

[λt]K−1 e−λt if α = 0 Gamma

= λ(K−1)!

[λt]K−1 e−λt if K ∈ I Erlang

= λe−λt if K = 1 Neg. Exp.

where λ is parameter that is function of µ, K, and α; K is a user specified parameterbetween 0 and ∞; α is a user selected parameter greater than 0 and called as the shiftpatmeter; and Γ() is the gamma function (Γ(K) = (K − 1)!).

• Then

p(h ≥ t) =∫ ∞

tf(t)dt (13.20)

• Prob of h/w lying b/w t and t + δt is

p(t ≤ h < (t + δt)) =∫ ∞

tf(t)dt −

∫ ∞

(t+δt)f(t)dt (13.21)

• Approximating

p(t ≤ h < (t + δt)) =

[

f(t) + f(t + δt)

2

]

δt (13.22)

• The solution steps

1. Input µ, σ

2. α = 0.5 ensuring h/w is greater than 0.5

3. K = µ−ασ

4. λ = Kµ−α


CE415 Transportation Engineering II 13. Macroscopic traffic flow modeling

5. Determination of Γ(K)

Γ(K) =

{

∫∞0 e−xxK−1dx, if 1 ≤ K ≤ 2

(K − 1)Γ(K − 1) if K > 2(13.23)

6. Solve for f(t) using Equation 13.20

7. Solve for p(t ≤ h < (t + δt)) using Equation 13.22

Example

An obseravtion from 3424 samples is given table below. Mean headway observed was 3.5seconds and the standard deviation 2.6 seconds. Fit a (i) negatie expoetial distribution, (ii)normal distrbution and (iii) Person Type III Distribution.

Table 13:1: Obsered headway distributiont t+dt p (obs)

0.0 0.5 0.0120.5 1.0 0.0641.0 1.5 0.1141.5 2.0 0.1592.0 2.5 0.1572.5 3.0 0.1303.0 3.5 0.0883.5 4.0 0.0654.0 4.5 0.0434.5 5.0 0.0335.0 5.5 0.0225.5 6.0 0.0196.0 6.5 0.0146.5 7.0 0.0107.0 7.5 0.0127.5 8.0 0.0088.0 8.5 0.0058.5 9.0 0.0079.0 9.5 0.0059.5 > 0.033

Total 1.00

Solutions



Table 13:2: Solution using negative exponential distributiont t+dt p (obs) Pobs=p*N p(h >= t) N-exp (p) P=p*N

0.0 0.5 0.012 41.1 1.000 0.133 455.80.5 1.0 0.064 219.1 0.867 0.115 395.11.0 1.5 0.114 390.3 0.751 0.100 342.51.5 2.0 0.159 544.4 0.651 0.087 296.92.0 2.5 0.157 537.6 0.565 0.075 257.42.5 3.0 0.130 445.1 0.490 0.065 223.13.0 3.5 0.088 301.3 0.424 0.056 193.43.5 4.0 0.065 222.6 0.368 0.049 167.74.0 4.5 0.043 147.2 0.319 0.042 145.44.5 5.0 0.033 113.0 0.276 0.037 126.05.0 5.5 0.022 75.3 0.240 0.032 109.25.5 6.0 0.019 65.1 0.208 0.028 94.76.0 6.5 0.014 47.9 0.180 0.024 82.16.5 7.0 0.010 34.2 0.156 0.021 71.27.0 7.5 0.012 41.1 0.135 0.018 61.77.5 8.0 0.008 27.4 0.117 0.016 53.58.0 8.5 0.005 17.1 0.102 0.014 46.48.5 9.0 0.007 24.0 0.088 0.012 40.29.0 9.5 0.005 17.1 0.076 0.010 34.89.5 > 0.033 113.0 0.066 0.066 226.8

Total 1.000 3424 1.000 3424

13.1.5 Comparison of distributions

13.1.6 Evaluating the selected distribution

13.2 Modeling vehicle arrival

13.3 Modeling speed

13.4 Estimation of population and sample means



Table 13:3: Solution using normal distributionh t+dt p (obs) P=p*N p(h <= t) p(t < h < t + 0.5) P=p*N

0.0 0.5 0.012 41.09 0.010 0.013 44.2890.5 1.0 0.064 219.14 0.023 0.025 85.7381.0 1.5 0.114 390.34 0.048 0.043 148.6731.5 2.0 0.159 544.42 0.091 0.067 230.9282.0 2.5 0.157 537.57 0.159 0.094 321.2992.5 3.0 0.13 445.12 0.252 0.117 400.4333.0 3.5 0.088 301.31 0.369 0.131 447.0333.5 4.0 0.065 222.56 0.500 0.131 447.0334.0 4.5 0.043 147.23 0.631 0.117 400.4334.5 5.0 0.033 112.99 0.748 0.094 321.2995.0 5.5 0.022 75.33 0.841 0.067 230.9285.5 6.0 0.019 65.06 0.909 0.043 148.6736.0 6.5 0.014 47.94 0.952 0.025 85.7386.5 7.0 0.01 34.24 0.977 0.013 44.2897.0 7.5 0.012 41.09 0.990 0.006 20.4927.5 8.0 0.008 27.39 0.996 0.002 8.4938.0 8.5 0.005 17.12 0.999 0.001 3.1538.5 9.0 0.007 23.97 1.000 0.000 1.0489.0 9.5 0.005 17.12 1.000 0.000 0.3129.5 > 0.033 112.99 1.000 0.010 33.716

3424



Table 13:4: Solution using Pearson type III distributionh t+dt p (obs) P=p*N f(t) p(t < h < t + 0.5) P=p*N

0.0 0.5 0.012 41.09 0.000 0.000.5 1.0 0.064 219.14 0.000 0.066 225.911.0 1.5 0.114 390.34 0.264 0.127 433.261.5 2.0 0.159 544.42 0.242 0.114 389.452.0 2.5 0.157 537.57 0.213 0.099 339.132.5 3.0 0.13 445.12 0.183 0.085 291.123.0 3.5 0.088 301.31 0.157 0.072 247.863.5 4.0 0.065 222.56 0.133 0.061 209.904.0 4.5 0.043 147.23 0.112 0.052 177.084.5 5.0 0.033 112.99 0.095 0.044 148.975.0 5.5 0.022 75.33 0.079 0.037 125.045.5 6.0 0.019 65.06 0.067 0.031 104.786.0 6.5 0.014 47.94 0.056 0.026 87.676.5 7.0 0.01 34.24 0.047 0.021 73.287.0 7.5 0.012 41.09 0.039 0.018 61.187.5 8.0 0.008 27.39 0.033 0.015 51.048.0 8.5 0.005 17.12 0.027 0.012 42.548.5 9.0 0.007 23.97 0.023 0.010 35.449.0 9.5 0.005 17.12 0.019 0.009 29.519.5 > 0.033 112.99 0.016 0.102 350.83

x 3424



Chapter 14

Macroscopic traffic flow modeling

14.1 Introdcution

If one looks into traffic flow from a very long distance, the flow of fairly heavy traffic appearslike a stream of a fluid. Therefore, a macroscopic theory of traffic can be developed with thehelp of hydrodynamic theory of fluids by considering traffic as an effectively one-dimensionalcompressible fluid. The behaviour of individual vehicle is ignored and one is concerned onlywith the behaviour of sizable aggregate of vehicles. The earliest traffic flow models began bywriting the balance equation to address vehicle number conservation on a road. Infact, alltraffic flow models and theories must satisfy the law of conservation of the number of vehicleson the road. Assuming that the vehicles are flowing from left to right, the continuity equationcan be written as

∂k(x, t)

∂t+

∂q(x, t)

∂x= 0 (14.1)

where x denotes the spatial coordinate in the direction of traffic flow, t is the time, k is thedensity and q denotes the flow. However, one cannot get two unknowns, namely k(x, t) byand q(x, t) by solving one equation. One possible solution is to write two equations from tworegimes of the flow, say before and after a bottleneck. In this system the flow rate before andafter will be same, or

k1v1 = k2v2 (14.2)

From this the shockwave velocity can be derived as

v(to)p =q2 − q1

k2 − k1(14.3)

This is normally referred to as Stock’s shockwave formula. An alternate possibility whichLighthill and Whitham adopted in their landmark study is to assume that the flow rate q isdetermined primarily by the local density k, so that flow q can be treated as a function of onlydensity k. Therefore the number of unknown variables will be reduced to one. Essentially thisassumption states that k(x,t) and q (x,t) are not independent of each other. Therefore thecontinuity equation takes the form

∂k(x, t)

∂t+

∂q(k(x, t))

∂x= 0 (14.4)


CE415 Transportation Engineering II 14. Cell transmission models

However, the functional relationship between flow q and density k cannot be calculated fromfluid-dynamical theory. This has to be either taken as a phenomenological relation derived fromthe empirical observation or from microscopic theories. Therefore, the flow rate q is a functionof the vehicular density k; q = q(k). Thus, the balance equation takes the form

∂k(x, t)

∂t+

∂q(k(x, t))

∂x= 0 (14.5)

Now there is only one independent variable in the balance equation, the vehicle density k.If initial and boundary conditions are known, this can be solved. Solution to LWR models arekinematic waves moving with velocity

dq(k)

dk(14.6)

This velocity vk is positive when the flow rate increases with density, and it is negative whenthe flow rate decreases with density. In some cases, this function may shift from one regime tothe other, and then a shock is said to be formed. This shockwave propagate at the velocity

vs =q(k2) − q(k1)

k2 − k1

(14.7)

where q(k2) and q(k1) are the flow rates corresponding to the upstream density k2 and down-stream density k1 of the shockwave. Unlike Stock’s shockwave formula there is only one variablehere.



Chapter 15

Cell transmission models

15.1 Introduction

In the classical methods to explain macroscopic behaviour of traffic, like hydrodynamic theory,differential equations need to be solved to predict traffic evolution. However in situations ofsudden high density variations, like bottlenecking, the hydrodynamic model calls for a shockwave (an ad-hoc). Hence these equations are essentially piecewise continuous which are difficultto solve. Cell transmission models are developed as a discrete analogue of these differentialequations in the form of difference equations which are easy to solve and also take care of highdensity changes.In this lecture note the hydrodynamic model and cell transmission model and their equivalenceis discussed. The cell transmission model is explained in two parts, first with only a sourceand a sink, and then it is extended to a network. In the first part, the concepts of basic flowadvancement equations of CTM and a generalized form of CTM are presented. In addition,the phenomenon of instability is also discussed. In the second part, the network representationand topologies are established, after which the model is discussed in terms of a linear programformulation for merging and diverging.

15.2 CTM: Single source and a sink

The cell transmission model simulates traffic conditions by proposing to simulate the systemwith a time-scan strategy where current conditions are updated with every tick of a clock. Theroad section under consideration is divided into homogeneous sections called cells, numberedfrom i = 1 to I. The lengths of the sections are set equal to the distances travelled in lighttraffic by a typical vehicle in one clock tick. Under light traffic condition, all the vehicles in acell can be assumed to advance to the next with each clock tick. i.e,

ni+1(t + 1) = ni(t) (15.1)

where, ni(t) is the number of vehicles in cell i at time t. However equation 15.1 is not reasonablewhen flow exceeds the capacity. Hence a more robust set of flow advancement equations arepresented next.



ni(t)t ni+1(t)

Qi−1(t), Ni−1(t) Qi(t), Ni(t) Qi+1(t), Ni+1(t)

t + 1 ni(t + 1)

ni−1(t)

Figure 15:1: Flow advancement

15.2.1 Flow advancement equations

First, two constants associated with each cell are defined, namely,Ni(t) and Qi(t)

1. Ni(t) is the maximum number of vehicles that can be present in cell i at time t, it is theproduct of the cell’s length and its jam density.

2. Qi(t) is the maximum number of vehicles that can flow into cell i when the clock advancesfrom t to t+1 (time interval t), it is the minimum of the capacity of cells from i−1 and i.It is called the capacity of cell i. It represents the maximum flow that can be transferredfrom i − 1 to i.

Now the flow advancement equation can be written as:

ni(t + 1) = ni(t) + yi(t) − yi+1(t) (15.2)

where, ni(t + 1) is the cell occupancy at time t + 1, ni(t) the cell occupance at time t, yi(t)is the inflow at time t, yi+1(t) is the outflow at time t. The flows are related to the currentconditions at time t as indicated below:

yi(t) = min [ni−1(t), Qi(t), Ni(t) − ni(t)] (15.3)

where, ni−1(t) is the number of vehicles in cell i − 1 at time t, Qi(t) is the capacity flow into ifor time interval t, and Ni(t) - ni(t) is the amount of empty space in cell i at time t.

15.2.2 Boundary conditions

Boundary conditions are specified by means of input and output cells. The output cell, a sink forall exiting traffic, should have infinite size (NI+1 = ∞) and a suitable, possibly time-varying,capacity. Input cells are a cell pair. A source cell numbered 00 with an infinite number ofvehicles (n00(O) = ∞) that discharges into an empty gate cell 00 of infinite size, N0(t) = ∞.The inflow capacity Q0(t) of the gate cell is set equal to the desired link input flow for timeinterval t + 1.



-v

kbka

vkj/2

qmax

Figure 15:2: Flow-density relationship for the basic cell-transmission model

15.2.3 Equivalence with Hydrodynamic theory

Consider equation 15.2 and equation 15.3, they are discrete approximations to the hydrody-namic model with a density- flow (k-q) relationship in the shape of an isoscaled trapezoid, asin Fig.15:2. This relationship can be expressed as:

q = min [vk, qmax, v(kj − k)], for 0 ≤ k ≤ kj, (15.4)

Flow conservation is given by,∂q(x, t)

∂x=

∂k(x, t)

∂t(15.5)

To demonstrate the equivalence of the discrete and continuous approaches, the clock tick setto be equal to ∂t and choose the unit of distance such that v∂t = 1. Then the cell length is 1,v is also 1, and the following equivalences hold:

x ≡ i, kj ≡ N , qmax ≡ Q, and k(x, t) ≡ ni(t) with these conventions, it can be easily seen thatthe equation 15.4 & equation 15.3 are equivalent. Equation 15.6 can be equivalently writtenas:

yi(t) − yi+1(t) = −ni(t) + ni+1(t + 1) (15.6)

This represents change in flow over space equal to change in occupancy over time. Rearrangingterms of this equation we can arrive at equation 15.3, which is the same as the basic flowadvancement equation of the cell transmission model. .

15.2.4 Generalized CTM

Generalized CTM is an extension of the cell transmission model that would approximate thehydrodynamic model for an equation of state that allows backward waves with speed w ≤ v(see Fig. 15:3 ). This is a realistic model, since on many occassions speed of backward wavewill not be same as the free flow speed.



kj1v+ 1

w

qmax

−wF

low

ka kb Density

Figure 15:3: Flow-density relationship for the generalized CTM

yi(t) = min [ni−1(t), Qi(t), w/v[Ni(t) − ni(t)]] (15.7)

A small modification is made in the above equation to avoid the error caused due to numericalspreading. Equation 15.7 is rewritten as

yi(t) = min [ni−1(t), Qi(t), α[Ni(t) − ni(t)]] (15.8)

where,

15.2.5 Numerical example 1

Consider a 1.25 km homogeneous road with v = 50 kmph, kj = 180 vpkm and qmax = 3000vph. Initially traffic is flowing undisturbed at 80% of capacity: q = 2400 VPH. Then, a partiallane blockage lasting 2 min occurs one third of the distance from the end of the road. Theblockage effectively restricts flow to 20% of the maximum. Clearly, a queue is going to buildand dissipate behind the restriction. Predict the evolution of the traffic. Take one clock tick as30 seconds.

Solution :

Initialization of the table:Choosing clock tick as 30seconds.(i.e1/120th of an hr.)Cell length=50/120=5/12th of a km.Number of cells=1.25*12/5=3.Cell constants:N= 180*5/12=75Q=3000/120=25.2min=120s=4 clock ticks.Initial occupancy=2400/120=20.



t=1

t=2

2020 20

35

Q=25,N=75 Q=25,N=75 Q=5,N=75

520

t=1

t=2

2020 20

Q=25,N=75

20 20

Q=20,N=75 Q=25,N=75

20

Table at the start of the simulation. (see Fig. ??)Note: Simulation need not be started in any specific order, it can be started from any cell inthe row corresponding to the current clock tick. For illustration, consider cell 2 at time 2 inthe final table (see Fig.??), its entry depends on the cells marked with rectangles.

By flow conservation law:

Occupancy = Storage + Inflow-Outflow

Here,Storage = 20.For inflow use equation 15.3Inflow= min [20,min(25,25),(75-20)]= 20Outflow= min [20,min(25,5),(75-20)]= 5Occupancy= 20+20-5=35.For cell 1 at time 2,

Inflow= min [20,min(25,25),(75-20)]= 20Outflow= min [20,min(25,25),(75-20)]= 20Occupancy= 20+20-20=20.For cell 3 at time 2,

Inflow= min [20, min (25,5),(75-20)]= 5Outflow= 20 (:.sink cell takes all the vehicles in previous cell)Occupancy= 20+5-20=5.



|Λ−1(j)| = 0 |Λ(j)| = 1

Figure 15:4: Source Cell

|Λ(j)| = 0|Λ−1(j)|=1

Figure 15:5: Sink Cell

15.2.6 Numerical example 2

Consider a 1.25 km homogeneous road with v = 50 kmph, kj = 180 vpkm and qmax = 3000vph. Initially traffic is flowing undisturbed at 80% of capacity: q = 2400 VPH. Then, a partiallane blockage lasting 2 min occurs l/3 of the distance from the end of the road. The blockageeffectively restricts flow to 20% of the maximum. Clearly, a queue is going to build and dissi-pate behind the restriction. Predict the evolution of the traffic. Take one clock tick as 6 seconds.

Solution:

This problem is same as the earlier problem, only change being the clock tick. The simu-lation is done for this smaller clock tick; the results are shown in Fig. ?? One can clearlyobserve the pattern in which the cells are getting updated.

After the decrease in capacity on last one-third segment queuing is slowly building up andthe backward wave can be appreciated through the first arrow. The second arrow shows thedissipation of queue and one can see that queue builds up at a faster than it dissipates. Thissimple illustration shows how CTM mimics the traffic conditions.

15.3 CTM: Network Traffic

15.3.1 General

As sequel to his first paper on CTM, Daganzo (1995) published first paper on CTM applied tonetwork traffic. In this section application of CTM to network traffic considering merging anddiverging is discussed. Some basic notations: (The notations used from here on, are adoptedfrom Ziliaskopoulos (2000)) Γ−1 = Set of predecessor cells. Γ = Set of successor cells.

15.3.2 Network topologies

The notations introduced in previous section are applied to different types of cells, as shown inFig. 15:4 Some valid and invalid representations in a network are shown in Fig 6.



Cell j

|Λ−1(j)| = 1 |Λ(j)| = 1

Figure 15:6: Ordinary Cell

Cell j

|Λ−1(j)| > 1 |Λ(j)| = 1

Figure 15:7: Merging Cell

Cell j

|Λ−1(j)| = 1 |Λ(j)| > 1

Figure 15:8: Diverging Cell

k

k

Figure 15:9: Invalid representations

Figure 15:10: Valid representations



E_{k}

B_{k}k

Figure 15:11: Ordinary Link

15.3.3 Ordinary link

Consider an ordinary link with a beginning cell and ending cell, which gives the flow betweentwo cells is simplified as explained below.

yk(t) = min(nBk(t), min[QBk(t), QEk(t)], δEk[NEk(t) − nEk(t)]) (15.9)

SI(t) = min(QI, nI) (15.10)

RI(t) = min(QI , δI , [NI − nI ]) (15.11)

yk(t) = min(SBk, REk) (15.12)

From equations one can see that a simplification is done by splitting yk(t) in to SBk andREk terms. ’S’ represents sending capacity and ’R’ represents receiving capacity. Duringtime periods when SBk < REk the flow on link k is dictated by upstream traffic conditions-as would be predicted from the forward moving characteristics of the Hydrodynamic model.Conversely, when SBk > REk, flow is dictated by downstream conditions and backward movingcharacteristics.

15.3.4 Merging and diverging

Consider two cells merging, here we have a beginning cell and its complimentary merging intoending cell, the constraints on the flow that can be sent and received are given by equation 15.15and equation 15.15.

yk(t) ≤ SBk (15.13)

yck(t) ≤ SCk (15.14)

yk(t) + yck(t) ≤ REk (15.15)

where, SI(t) is the min (QI , nI), and RI(t) is the min (QI , δI, [NI −nI ]). A number of combina-tions of yk(t) + yck(t) are possible satisfying the above said constraints. Similarly for diverginga number of possible outflows to different links is possible satisfying corresponding constraints,hence this calls for an optimization problem. Ziliaskopoulos (2000), has given this LP formu-lations for both merging and diverging, this has been discussed later



Ek

Bk

Ck

k

ck

15.4 Conclusion

15.4.1 Summary

• CTM is a discrete approximation of hydrodynamic model. System evolution is based ondifference equations.

• Unlike hydrodynamic model, it explains the phenomenon of Instability.

• Lesser the time per clock tick lesser are the size of cells and more accurate results wouldbe obtained. But a compromise is needed between accuracy and computational effort.Largest possible cell size which would sufficiently give the details needed must be chosen.

• CTM has many applications in DTA, NDP, traffic operations, emergency evacuations etc.

• There is a vast scope for improvement and applications of this model.

15.4.2 Advantages and applications

• CTM is consistent with hydrodynamic theory, which is a widely used model for studyingmacroscopic behavior of the traffic.

• It is simple and sufficiently accurate for planning purposes.

• Here two efficient modeling approaches are used, one is Linear Programming, whosecharacteristics well known and the other is Parallel computing.These are important fromthe perspective of computational speed and efficiency.

• CTM can be used to provide ”real time” information to the drivers.

• CTM has been used in developing a system optimal signal optimization formulation.

• CTM based Dynamic Traffic Assignment have shown good results.

• CTM has its application in Network Design Problems.(NDP)

• Dynamic network models based on CTM can also be applied to evaluate the performanceof emergency evacuation plans.


CE415 Transportation Engineering II 15. Traffic intersections

15.4.3 Limitations

• CTM is for a ”typical vehicle” in network traffic, work is needed for the multi-modalrepresentation of traffic.

• Cell length cannot be varied. Change in the model has to be brought without loss ofgenerality, to relax the constraint on arbitrary selection of cell length, which will facilitatethe modelers to select variable cell lengths that are best aligned with the geometry.

• CTM is largely deterministic, stochastic variables are needed to be introduced to representthe random human behavior.

• Mesoscopic features can be introduced so that concepts like lane changing behavior etc.can be studied.



Chapter 16

Traffic intersections

16.1 Overview

Intersection is an area shared by two or more roads. This area is designated for the vehiclesto turn to different directions to reach their desired destinations. Its main function is toguide vehicles to their respective directions. Traffic intersections are complex locations on anyhighway. This is because vehicles moving in different direction wan to occupy same space at thesame time. In addition, the pedestrians also seek same space for crossing. Drivers have to makesplit second decision at an intersection by considering his route, intersection geometry, speedand direction of other vehicles etc. A small error in judgment can cause severe accidents. It alsocauses delay and it depends on type, geometry, and type of control. Overall traffic flow dependson the performance of the intersections. It also affects the capacity of the road. Therefore,both from the accident perspective and the capacity perspective, the study of intersections veryimportant for the traffic engineers especially in the case of urban scenario.

16.2 Conflicts at an intersection

Conflicts at an intersection are different for different types of intersection. Consider a typicalfour-legged intersection as shown in figure. The number of conflicts for competing throughmovements are 4, while competing right turn and through movements are 8. The conflictsbetween right turn traffics are 4, and between left turn and merging traffic is 4. The conflictscreated by pedestrians will be 8 taking into account all the four approaches. Diverging trafficalso produces about 4 conflicts. Therefore, a typical four legged intersection has about 32different types of conflicts. This is shown in figure 16:1.

The essence of the intersection control is to resolve these conflicts at the intersection forthe safe and efficient movement of both vehicular traffic and pedestrians. Two methods ofintersection controls are there: time sharing and space sharing. The type of intersection controlthat has to be adopted depends on the traffic volume, road geometry, cost involved, importanceof the road etc.



P

P

P P

P

P

PP

P 8 Pedestrian

Conflicts in a traffic signal

8 Right turn−Through

Total = 32 Conflicts

4 Merging

4 Right turn

4 Through traffic

4 Diverging

Figure 16:1: Conflicts at an intersection

16.3 Levels of intersection control

The control of an intersection can be exercised at different levels. They can be either passivecontrol, semi control, or active control. In passive control, there is no explicit control on thedriver . In semi control, some amount of control on the driver is there from the traffic agency.Active control means the movement of the traffic is fully controlled by the traffic agency andthe drivers cannot simply maneuver the intersection according to his choice.

16.3.1 Passive control

When the volume of traffic is less, no explicit control is required. Here the road users arerequired to obey the basic rules of the road. Passive control like traffic signs, road markingsetc. are used to complement the intersection control. Some of the intersection control that areclassified under passive control are as follows:

1. No control If the traffic coming to an intersection is low, then by applying the basicrules of the road like driver on the left side of the road must yield and that throughmovements will have priority than turning movements. The driver is expected to obeythese basic rules of the road.

2. Traffic signs: With the help of warning signs, guide signs etc. it is able to providesome level of control at an intersection. Give way control, two-way stop control, andall-way stop control are some examples. The GIVE WAY control requires the driver inthe minor road to slow down to a minimum speed and allow the vehicle on the majorroad to proceed. Two way stop control requires the vehicle drivers on the minor streetsshould see that the conflicts are avoided. Finally an all-way stop control is usually usedwhen it is difficult to differentiate between the major and minor roads in an intersection.In such a case, STOP sign is placed on all the approaches to the intersection and thedriver on all the approaches are required to stop the vehicle. The vehicle at the rightside will get priority over the left approach. The traffic control at ’at-grade’ intersection



may be uncontrolled in cases of low traffic. Here the road users are required to obey thebasic rules of the road. Passive control like traffic signs, road markings etc. are used tocomplement the intersection control.

3. Traffic signs plus marking: In addition to the traffic signs, road markings also comple-ment the traffic control at intersections. Some of the examples include stop line marking,yield lines, arrow marking etc.

16.3.2 Semi control

In semi control or partial control, the drivers are gently guided to avoid conflicts. Channelizationand traffic rotaries are two examples of this.

1. Channelization: The traffic is separated to flow through definite paths by raising aportion of the road in the middle usually called as islands distinguished by road markings.The conflicts in traffic movements are reduced to a great extent in such a case. Inchannelized intersections, as the name suggests, the traffic is directed to flow throughdifferent channels and this physical separation is made possible with the help of somebarriers in the road like traffic islands, road markings etc.

2. Traffic rotaries: It is a form of intersection control in which the traffic is made to flowalong one direction around a traffic island. The essential principle of this control is toconvert all the severe conflicts like through and right turn conflicts into milder conflictslike merging, weaving and diverging. It is a form of ‘at-grade’ intersection laid out for themovement of traffic such that no through conflicts are there. Free-left turn is permittedwhere as through traffic and right-turn traffic is forced to move around the central islandin a clock-wise direction in an orderly manner. Merging, weaving and diverging operationsreduces the conflicting movements at the rotary.

16.3.3 Active control

Active control implies that the road user will be forced to follow the path suggested by thetraffic control agencies. He cannot maneuver according to his wish. Traffic signals and gradeseparated intersections come under this classification.

1. Traffic signals: Control using traffic signal is based on time sharing approach. At agiven time, with the help of appropriate signals, certain traffic movements are restrictedwhere as certain other movements are permitted to pass through the intersection. Two ormore phases may be provided depending upon the traffic conditions of the intersection.When the vehicles traversing the intersection is very large, then the control is done withthe help of signals. The phases provided for the signal may be two or more. If more thantwo phases are provided, then it is called multiphase signal.

The signals can operate in several modes. Most common are fixed time signals and vehicleactuated signals. In fixed time signals, the cycle time, phases and interval of each signal



is fixed. Each cycle of the signal will be exactly like another. But they cannot caterto the needs of the fluctuating traffic. On the other hand, vehicle actuated signals canrespond to dynamic traffic situations. Vehicle detectors will be placed on the streetsapproaching the intersection and the detector will sense the presence of the vehicle andpass the information to a controller. The controller then sets the cycle time and adjuststhe phase lengths according to the prevailing traffic conditions.

2. Grade separated intersections: The intersections are of two types. They are at-gradeintersections and grade-separated intersections. In at-grade intersections, all roadwaysjoin or cross at the same vertical level. Grade separated intersections allows the traffic tocross at different vertical levels. Sometimes the topography itself may be helpful in con-structing such intersections. Otherwise, the initial construction cost required will be veryhigh. Therefore, they are usually constructed on high speed facilities like expressways,freeways etc. These type of intersection increases the road capacity because vehicles canflow with high speed and accident potential is also reduced due to vertical separation oftraffic.

16.4 Grade separated intersections

As we discussed earlier, grade-separated intersections are provided to separate the traffic inthe vertical grade. But the traffic need not be those pertaining to road only. When a railwayline crosses a road, then also grade separators are used. Different types of grade-separators areflyovers and interchange. Flyovers itself are subdivided into overpass and underpass. Whentwo roads cross at a point, if the road having major traffic is elevated to a higher grade forfurther movement of traffic, then such structures are called overpass. Otherwise, if the majorroad is depressed to a lower level to cross another by means of an under bridge or tunnel, it iscalled under-pass.

Interchange is a system where traffic between two or more roadways flows at different levelsin the grade separated junctions. Common types of interchange include trumpet interchange,diamond interchange , and cloverleaf interchange.

1. Trumpet interchange: Trumpet interchange is a popular form of three leg interchange.If one of the legs of the interchange meets a highway at some angle but does not crossit, then the interchange is called trumpet interchange. A typical layout of trumpet inter-change is shown in figure 16:2.

2. Diamond interchange: Diamond interchange is a popular form of four-leg interchangefound in the urban locations where major and minor roads crosses. The important featureof this interchange is that it can be designed even if the major road is relatively narrow.A typical layout of diamond interchange is shown in figure 16:3.

3. Clover leaf interchange: It is also a four leg interchange and is used when two highwaysof high volume and speed intersect each other with considerable turning movements. Themain advantage of cloverleaf intersection is that it provides complete separation of traffic.



Figure 16:2: Trumpet interchange

Figure 16:3: Diamond interchange



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Figure 16:4: Cloverleaf interchange

In addition, high speed at intersections can be achieved. However, the disadvantage isthat large area of land is required. Therefore, cloverleaf interchanges are provided mainlyin rural areas. A typical layout of this type of interchange is shown in figure 16:4.

16.5 Channelized intersection

Vehicles approaching an intersection are directed to definite paths by islands, marking etc. andthis method of control is called channelization. Channelized intersection provides more safetyand efficiency. It reduces the number of possible conflicts by reducing the area of conflictsavailable in the carriageway. If no channelizing is provided the driver will have less tendency toreduce the speed while entering the intersection from the carriageway. The presence of trafficislands, markings etc. forces the driver to reduce the speed and becomes more cautious whilemaneuvering the intersection. A channelizing island also serves as a refuge for pedestrians andmakes pedestrian crossing safer. Channelization of traffic through a three-legged intersection(refer figure 16:5) and a four-legged intersection (refer figure 16:6) is shown in the figure.

16.6 Summary

Traffic intersections are problem spots on any highway, which contribute to a large share ofaccidents. For safe operation, these locations should be kept under some level of control de-pending upon the traffic quantity and behavior. Based on this, intersections and interchangesare constructed, the different types of which were discussed in the chapter.



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Figure 16:5: Channelization of traffic through a three-legged intersection

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Figure 16:6: Channelization of traffic through a four-legged intersection


CE415 Transportation Engineering II 16. Traffic signs

16.7 Problems

1. The GIVE WAY control

(a) requires the driver in the minor road to slow down to a minimum speed and allowthe vehicle on the major road to proceed.

(b) requires the driver in the major road to slow down to a minimum speed and allowthe vehicle on the minor road to proceed.

(c) requires the drivers on both minor and major roads to stop.

(d) is similar to one way control.

2. Traffic signal is an example of

(a) Passive control

(b) No control

(c) Active control

(d) none of these

16.8 Solutions

1. The GIVE WAY control

(a) requires the driver in the minor road to slow down to a minimum speed and allowthe vehicle on the major road to proceed.

√

(b) requires the driver in the major road to slow down to a minimum speed and allowthe vehicle on the minor road to proceed.

(c) requires the drivers on both minor and major roads to stop.

(d) is similar to one way control.

2. Traffic signal is an example of

(a) Passive control

(b) No control

(c) Active control√

(d) none of these



Chapter 17

Traffic signs

17.1 Overview

Traffic control device is the medium used for communicating between traffic engineer and roadusers. Unlike other modes of transportation, there is no control on the drivers using the road.Here traffic control devices comes to the help of the traffic engineer. The major types oftraffic control devices used are- traffic signs, road markings , traffic signals and parking control.This chapter discusses traffic control signs. Different types of traffic signs are regulatory signs,warning signs and informatory signs.

17.2 Requirements of traffic control devices

1. The control device should fulfill a need : Each device must have a specific purposefor the safe and efficient operation of traffic flow. The superfluous devices should not beused.

2. It should command attention from the road users: This affects the design of signs.For commanding attention, proper visibility should be there. Also the sign should bedistinctive and clear. The sign should be placed in such a way that the driver requires noextra effort to see the sign.

3. It should convey a clear, simple meaning: Clarity and simplicity of message isessential for the driver to properly understand the meaning in short time. The use ofcolor, shape and legend as codes becomes important in this regard. The legend should bekept short and simple so that even a less educated driver could understand the messagein less time.

4. Road users must respect the signs: Respect is commanded only when the drivers areconditioned to expect that all devices carry meaningful and important messages. Overuse,misuse and confusing messages of devices tends the drivers to ignore them.

5. The control device should provide adequate time for proper response from theroad users: This is again related to the design aspect of traffic control devices. The signboards should be placed at a distance such that the driver could see it and gets sufficient



time to respond to the situation. For example, the STOP sign which is always placedat the stop line of the intersection should be visible for atleast one safe stopping sightdistance away from the stop line.

17.3 Communication tools

A number of mechanisms are used by the traffic engineer to communicate with the road user.These mechanisms recognize certain human limitations, particularly eyesight. Messages areconveyed through the following elements.

1. Color: It is the first and most easily noticed characteristics of a device. Usage of differentcolors for different signs are important. The most commonly used colors are red, green,yellow, black, blue, and brown . These are used to code certain devices and to reinforcespecific messages. Consistent use of colors helps the drivers to identify the presence ofsign board ahead.

2. Shape : It is the second element discerned by the driver next to the color of the device.The categories of shapes normally used are circular, triangular, rectangular, and diamondshape. Two exceptional shapes used in traffic signs are octagonal shape for STOP signand use of inverted triangle for GIVE WAY (YIELD) sign. Diamond shape signs are notgenerally used in India.

3. Legend : This is the last element of a device that the drive comprehends. This is animportant aspect in the case of traffic signs. For the easy understanding by the driver,the legend should be short, simple and specific so that it does not divert the attention ofthe driver. Symbols are normally used as legends so that even a person unable to readthe language will be able to understand that. There is no need of it in the case of trafficsignals and road markings.

4. Pattern: It is normally used in the application of road markings, complementing trafficsigns. Generally solid, double solid and dotted lines are used. Each pattern conveys dif-ferent type of meaning. The frequent and consistent use of pattern to convey informationis recommended so that the drivers get accustomed to the different types of markings andcan instantly recognize them.

17.4 Types of traffic signs

There are several hundreds of traffic signs available covering wide variety of traffic situations.They can be classified into three main categories.

1. Regulatory signs: These signs require the driver to obey the signs for the safety ofother road users.



2. Warning signs:These signs are for the safety of oneself who is driving and advice thedrivers to obey these signs.

3. Informative signs: These signs provide information to the driver about the facilitiesavailable ahead, and the route and distance to reach the specific destinations

In addition special type of traffic sign namely work zone signs are also available. These typeof signs are used to give warning to the road users when some construction work is going onthe road. They are placed only for short duration and will be removed soon after the work isover and when the road is brought back to its normal condition. The first three signs will bediscussed in detail below.

17.4.1 Regulatory signs

These signs are also called mandatory signs because it is mandatory that the drivers must obeythese signs. If the driver fails to obey them, the control agency has the right to take legal actionagainst the driver. These signs are primarily meant for the safety of other road users. Thesesigns have generally black legend on a white background. They are circular in shape with redborders. The regulatory signs can be further classified into :

1. Right of way series: These include two unique signs that assign the right of way tothe selected approaches of an intersection. They are the STOP sign and GIVE WAY signFor example, when one minor road and major road meets at an intersection, preferenceshould be given to the vehicles passing through the major road. Hence the give way signboard will be placed on the minor road to inform the driver on the minor road that heshould give way for the vehicles on the major road. In case two major roads are meeting,then the traffic engineer decides based on the traffic on which approach the sign boardhas to be placed. Stop sign is another example of regulatory signs that comes in right ofway series which requires the driver to stop the vehicle at the stop line.

2. Speed series: Number of speed signs may be used to limit the speed of the vehicle onthe road. They include typical speed limit signs, truck speed, minimum speed signs etc.Speed limit signs are placed to limit the speed of the vehicle to a particular speed formany reasons. Separate truck speed limits are applied on high speed roadways whereheavy commercial vehicles must be limited to slower speeds than passenger cars for safetyreasons. Minimum speed limits are applied on high speed roads like expressways, freewaysetc. where safety is again a predominant reason. Very slow vehicles may present hazardto themselves and other vehicles also.

3. Movement series: They contain a number of signs that affect specific vehicle maneuvers.These include turn signs, alignment signs, exclusion signs, one way signs etc. Turn signsinclude turn prohibitions and lane use control signs. Lane use signs make use of arrowsto specify the movements which all vehicles in the lane must take. Turn signs are used tosafely accommodate turns in unsignalized intersections.



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STOPGIVEWAY

Figure 17:1: Examples of regulatory signs ( stop sign, give way sign, signs for no entry, signindicating prohibition for right turn, vehicle width limit sign, speed limit sign)

4. Parking series: They include parking signs which indicate not only parking prohibitionsor restrictions, but also indicate places where parking is permitted, the type of vehicle tobe parked, duration for parking etc.

5. Pedestrian series: They include both legend and symbol signs. These signs are meantfor the safety of pedestrians and include signs indicating pedestrian only roads, pedestriancrossing sites etc.

6. Miscellaneous: Wide variety of signs that are included in this category are: a ”KEEPOF MEDIAN” sign, signs indicating road closures, signs restricting vehicles carryinghazardous cargo or substances, signs indicating vehicle weight limitations etc.

Some examples of the regulatory signs are shown in figure 17:1. They include a stop sign, giveway sign, signs for no entry, sign indicating prohibition for right turn, vehicle width limit sign,speed limit sign etc.

17.4.2 Warning signs

Warning signs or cautionary signs give information to the driver about the impending roadcondition. They advice the driver to obey the rules. These signs are meant for the own safetyof drivers. They call for extra vigilance from the part of drivers. The color convention used forthis type of signs is that the legend will be black in color with a white background. The shapeused is upward triangular or diamond shape with red borders. Some of the examples for thistype of signs are given in fig 17:2 and includes right hand curve sign board, signs for narrowroad, sign indicating railway track ahead etc.

17.4.3 Informative signs

Informative signs also called guide signs, are provided to assist the drivers to reach their desireddestinations. These are predominantly meant for the drivers who are unfamiliar to the place.The guide signs are redundant for the users who are accustomed to the location.



��

��

Figure 17:2: Examples of cautionary signs ( right hand curve sign board, signs for narrow road,sign indicating railway track ahead)

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

TOLL BOOTH

AHEAD

��

��

NH

��

��

8

Figure 17:3: Examples of informative signs (route markers, destination signs, mile posts, servicecentre information etc)

Some of the examples for these type of signs are route markers, destination signs, mile posts,service information, recreational and cultural interest area signing etc. Route markers are usedto identify numbered highways. They have designs that are distinctive and unique. They arewritten black letters on yellow background. Destination signs are used to indicate the directionto the critical destination points, and to mark important intersections. Distance in kilometersare sometimes marked to the right side of the destination. They are, in general, rectangularwith the long dimension in the horizontal direction. They are color coded as white letters withgreen background.

Mile posts are provided to inform the driver about the progress along a route to reach hisdestination. Service guide signs give information to the driver regarding various services suchas food, fuel, medical assistance etc. They are written with white letters on blue background.Information on historic, recreational and other cultural area is given on white letters with brownbackground. In the figure 17:3 we can see some examples for informative signs which includeroute markers, destination signs, mile posts, service centre information etc..

17.5 Summary

Traffic signs are means for exercising control on or passing information to the road users. Theymay be regulatory, warning, or informative. Among the design aspects of the signs, the size,shape, color and location matters. Some of the signs along with examples were discussed inthis chapter. A few web sites discussing on traffic signs are giben below:

www.aptransport.org/html/signs.htm, www.indiacar.com/infobank/Traffic-signs.htm.


CE415 Transportation Engineering II 17. Road markings

17.6 Problems

1. Regulatory signs are also called

(a) Mandatory signs

(b) Cautionary signs

(c) Informative signs

(d) Warning signs

2. Stop sign comes under

(a) Regulatory signs



(d) none of these

17.7 Solutions

1. Regulatory signs are also called

(a) Mandatory signs√



(d) Warning signs

2. Stop sign comes under

(a) Regulatory signs√



(d) none of these



Chapter 18

Road markings

18.1 Overview

The essential purpose of road markings is to guide and control traffic on a highway. Theysupplement the function of traffic signs. The markings serve as a psychological barrier andsignify the delineation of traffic path and its lateral clearance from traffic hazards for the safemovement of traffic. Hence they are very important to ensure the safe, smooth and harmoniousflow of traffic. Various types of road markings like longitudinal markings, transverse markings,object markings and special markings to warn the driver about the hazardous locations in theroad etc. will be discussed in detail in this chapter.

18.2 Classification of road markings

The road markings are defined as lines, patterns, words or other devices, except signs, setinto applied or attached to the carriageway or kerbs or to objects within or adjacent to thecarriageway, for controlling, warning, guiding and informing the users. The road markingsare classified as longitudinal markings, transverse markings, object markings, word messages,marking for parkings, marking at hazardous locations etc.

18.3 Longitudinal markings

Longitudinal markings are placed along the direction of traffic on the roadway surface, for thepurpose of indicating to the driver, his proper position on the roadway. Some of the guidingprinciples in longitudinal markings are also discussed below.

Longitudinal markings are provided for separating traffic flow in the same direction and thepredominant color used is white. Yellow color is used to separate the traffic flow in oppositedirection and also to separate the pavement edges. The lines can be either broken, solid ordouble solid. Broken lines are permissive in character and allows crossing with discretion, iftraffic situation permits. Solid lines are restrictive in character and does not allow crossingexcept for entry or exit from a side road or premises or to avoid a stationary obstruction.Double solid lines indicate severity in restrictions and should not be crossed except in caseof emergency. There can also be a combination of solid and broken lines. In such a case, a



150

4.5 m3 m

Figure 18:1: Centre line marking for a two lane road

1.5m 3m

3m 4.5 m

Figure 18:2: Centre line and lane marking for a four lane road

solid line may be crossed with discretion, if the broken line of the combination is nearer to thedirection of travel. Vehicles from the opposite directions are not permitted to cross the line.Different types of longitudinal markings are centre line, traffic lanes, no passing zone, warninglines, border or edge lines, bus lane markings, cycle lane markings.

18.3.1 Centre line

Centre line separates the opposing streams of traffic and facilitates their movements. Usuallyno centre line is provided for roads having width less than 5 m and for roads having morethan four lanes. The centre line may be marked with either single broken line, single solid line,double broken line, or double solid line depending upon the road and traffic requirements. Onurban roads with less than four lanes, the centre line may be single broken line segments of 3 mlong and 150 mm wide. The broken lines are placed with 4.5 m gaps (figure 18:1). On curvesand near intersections, gap shall be reduced to 3 metres. On undivided urban roads with atleast two traffic lanes in each direction, the centre line marking may be a single solid line of150 mm wide as in figure 18:2, or double solid line of 100 mm wide separated by a space of100 mm as shown in figure 18:3. The centre barrier line marking for four lane road is shownin figure 18:4.

18.3.2 Traffic lane lines

The subdivision of wide carriageways into separate lanes on either side of the carriage way helpsthe driver to go straight and also curbs the meandering tendency of the driver. At intersections,these traffic lane lines will eliminate confusion and facilitates turning movements. Thus traffic



100

100

1.5m 3m

Figure 18:3: Double solid line for a two lane road

150 mm

100 mm

3m1.5m

Figure 18:4: Centre barrier line marking for four lane road

lane markings help in increasing the capacity of the road in addition ensuring more safety. Thetraffic lane lines are normally single broken lines of 100 mm width. Some examples are shownin figure 18:5 and figure 18:6.

18.3.3 No passing zones

No passing zones are established on summit curves, horizontal curves, and on two lane andthree lane highways where overtaking maneuvers are prohibited because of low sight distance.It may be marked by a solid yellow line along the centre or a double yellow line. In the case ofa double yellow line, the left hand element may be a solid barrier line, the right hand may be aeither a broken line or a solid line . These solid lines are also called barrier lines. When a solid

3.0 m

100

150

1.5m

Figure 18:5: Lane marking for a four lane road with solid barrier line



100

150

1.5 m 3.0 m

3.0 m 4.5 m

Figure 18:6: Traffic lane marking for a four lane road with broken centre line

yellow single/double line

Figure 18:7: Barrier line marking for a four lane road

line is to the right of the broken line, the passing restriction shall apply only to the opposingtraffic. Some typical examples are shown in figure 18:7 and figure 18:8. In the latter case, theno passing zone is staggered for each direction.

18.3.4 Warning lines

Warning lines warn the drivers about the obstruction approaches. They are marked on hori-zontal and vertical curves where the visibility is greater than prohibitory criteria specified forno overtaking zones. They are broken lines with 6 m length and 3 m gap. A minimum of sevenline segments should be provided. A typical example is shown in figure 18:9

18.3.5 Edge lines

Edge lines indicate edges of rural roads which have no kerbs to delineate the limits upto whichthe driver can safely venture. They should be at least 150 mm from the actual edge of thepavement. They are painted in yellow or white.

All the lines should be preferably light reflective, so that they will be visible during nightalso. Improved night visibility may also be obtained by the use of minute glass beads embeddedin the pavement marking materials to produce a retroreflective surface.



Barrier li

ne

Figure 18:8: No passing zone marking at horizontal curves

3m6m

Figure 18:9: Warning line marking for a two lane road



300200

STOP

150

Figure 18:10: Stop line marking near an intersection

18.4 Transverse markings

Transverse markings are marked across the direction of traffic. They are marked at intersectionsetc. The site conditions play a very important role. The type of road marking for a particularintersection depends on several variables such as speed characteristics of traffic, availability ofspace etc. Stop line markings, markings for pedestrian crossing, direction arrows, etc. are someof the markings on approaches to intersections.

18.4.1 Stop line

Stop line indicates the position beyond which the vehicles should not proceed when required tostop by control devices like signals or by traffic police. They should be placed either parallel tothe intersecting roadway or at right angles to the direction of approaching vehicles. An examplefor a stop line marking is shown in figure 18:10.

18.4.2 Pedestrian crossings

Pedestrian crossings are provided at places where the conflict between vehicular and pedestriantraffic is severe. The site should be selected that there is less inconvenience to the pedestriansand also the vehicles are not interrupted too much. At intersections, the pedestrian crossingsshould be preceded by a stop line at a distance of 2 to 3m for unsignalized intersections and at adistance of one metre for signalized intersections. Most commonly used pattern for pedestriancrossing is Zebra crossing consisting of equally spaced white strips of 500 mm wide. A typicalexample of an intersection illustrating pedestrian crossings is shown in figure 18:11.

18.4.3 Directional arrows

In addition to the warning lines on approaching lanes, directional arrows should be used to guidethe drivers in advance over the correct lane to be taken while approaching busy intersections.Because of the low angle at which the markings are viewed by the drivers, the arrows should



Figure 18:11: Pedestrian marking near an intersection

0.3m0.3m

3.5m

0.2m

1.25m

0.4m0.55 m

0.4m

1.2

m

0.5m

3.5m

1.2

m

Figure 18:12: Directional arrow marking

be elongated in the direction of traffic for adequate visibility. The dimensions of these arrowsare also very important. A typical example of a directional arrow is shown in figure 18:12.

18.5 Object marking

Physical obstructions in a carriageway like traffic island or obstructions near carriageway likesignal posts, pier etc. cause serious hazard to the flow of traffic and should be adequatelymarked. They may be marked on the objects adjacent to the carriageway.



Figure 18:13: Marking for objects adjacent to the road way

18.5.1 Objects within the carriageway

The obstructions within the carriageway such as traffic islands, raised medians, etc. may bemarked by not less than five alternate black and yellow stripes. The stripes should slope forwardat an angle of 45◦ with respect to the direction of traffic. These stripes shall be uniform andshould not be less than 100 m wide so as to provide sufficient visibility.

18.5.2 Objects adjacent to carriageway

Sometimes objects adjacent to the carriageway may pose some obstructions to the flow of traffic.Objects such as subway piers and abutments, culvert head walls etc. are some examples forsuch obstructions. They should be marked with alternate black and white stripes at a forwardangle of 45◦ with respect to the direction of traffic. Poles close to the carriageway should bepainted in alternate black and white up to a height of 1.25 m above the road level. Otherobjects such as guard stones, drums, guard rails etc. where chances of vehicles hitting them areonly when vehicle runs off the carriageway should be painted in solid white. Kerbs of all islandslocated in the line of traffic flow shall be painted with either alternating black and white stripesof 500 mm wide or chequered black and white stripes of same width. The object marking forcentral pier and side walls of an underpass is illustrated in figure 18:13.

18.6 Word messages

Information to guide, regulate, or warn the road user may also be conveyed by inscriptionof word message on road surface. Characters for word messages are usually capital letters.The legends should be as brief as possible and shall not consist of more than three words for



313

7826

0

1250

Figure 18:14: Typical dimension of the character T used in road marking

any message. Word messages require more and important time to read and comprehend thanother road markings. Therefore, only few and important ones are usually adopted. Some ofthe examples of word messages are STOP, SLOW, SCHOOL, RIGHT TUN ONLY etc. Thecharacter of a road message is also elongated so that driver looking at the road surface at a lowangle can also read them easily. The dimensioning of a typical alphabet is shown in figure 18:14.

18.7 Parking

The marking of the parking space limits on urban roads promotes more efficient use of theparking spaces and tends to prevent encroachment on places like bus stops, fire hydrant zonesetc. where parking is undesirable. Such parking space limitations should be indicated withmarkings that are solid white lines 100 mm wide. Words TAXI, CARS, SCOOTERS etc. mayalso be written if the parking area is specific for any particular type of vehicle. To indicateparking restriction, kerb or carriage way marking of continuous yellow line 100 mm wide coveringthe top of kerb or carriageway close to it may be used.



LLL

Figure 18:15: Approach marking for obstructions on the road way

18.8 Hazardous location

Wherever there is a change in the width of the road, or any hazardous location in the road,the driver should be warned about this situation with the help of suitable road markings.Road markings showing the width transition in the carriageway should be of 100 mm width.Converging lines shall be 150 mm wide and shall have a taper length of not less than twentytimes the off-set distance. Typical carriageway markings showing transition from wider tonarrower sections and vice-versa is shown in figure 18:15. In the figure, the driver is warnedabout the position of the pier through proper road markings.

18.9 Summary

Road markings are aids to control traffic by exercising psychological control over the roadusers. They are made use of in delineating the carriage way as well as marking obstructions, toensure safe driving. They also assist safe pedestrian crossing. Longitudinal markings which areprovided along the length of the road and its various classifications were discussed. Transversemarkings are provided along the width of the road. Road markings also contain word messages,but since it is time consuming to understand compared to other markings there are only very fewof them. Markings are also used to warn the driver about the hazardous locations ahead. Thusroad markings ensure smooth flow of traffic providing safety also to the road users. The followingweb link give further insight in to the road markings: mutcd.fhwa.dot.gov/pdfs/200311/pdf-index.htm.

18.10 Problems

1. Broken lines

(a) allows crossing with discretion

(b) does not allow crossing except for entry or exit from a side road

(c) allows crossing only in case of extreme emergency

(d) are not at all used as road markings.


CE415 Transportation Engineering II 18. Traffic rotaries

2. Stop line comes under

(a) Longitudinal markings

(b) Object markings

(c) Transverse markings

(d) None of these

18.11 Solutions

1. Broken lines

(a) allows crossing with discretion√

(b) does not allow crossing except for entry or exit from a side road

(c) allows crossing only in case of extreme emergency

(d) are not at all used as road markings.

2. Stop line comes under

(a) Longitudinal markings

(b) Object markings

(c) Transverse markings√

(d) None of these



Chapter 19

Traffic rotaries

19.1 Overview

Rotary intersections or round abouts are special form of at-grade intersections laid out for themovement of traffic in one direction around a central traffic island. Essentially all the majorconflicts at an intersection namely the collision between through and right-turn movements areconverted into milder conflicts namely merging and diverging. The vehicles entering the rotaryare gently forced to move in a clockwise direction in orderly fashion. They then weave out ofthe rotary to the desired direction. The benefits, design principles, capacity of rotary etc. willbe discussed in this chapter.

19.2 Advantages and disadvantages of rotary

The key advantages of a rotary intersection are listed below:

1. Traffic flow is regulated to only one direction of movement, thus eliminating severe con-flicts between crossing movements.

2. All the vehicles entering the rotary are gently forced to reduce the speed and continue tomove at slower speed. Thus, none of the vehicles need to be stopped,unlike in a signalizedintersection.

3. Because of lower speed of negotiation and elimination of severe conflicts, accidents andtheir severity are much less in rotaries.

4. Rotaries are self governing and do not need practically any control by police or trafficsignals.

5. They are ideally suited for moderate traffic, especially with irregular geometry, or inter-sections with more than three or four approaches.

Although rotaries offer some distinct advantages, there are few specific limitations for rotarieswhich are listed below.

1. All the vehicles are forced to slow down and negotiate the intersection. Therefore, thecumulative delay will be much higher than channelized intersection.



2. Even when there is relatively low traffic, the vehicles are forced to reduce their speed.

3. Rotaries require large area of relatively flat land making them costly at urban areas.

4. The vehicles do not usually stop at a rotary. They accelerate and exit the rotary atrelatively high speed. Therefore, they are not suitable when there is high pedestrianmovements.

19.3 Guidelines for the selection of rotaries

Because of the above limitation, rotaries are not suitable for every location. There are fewguidelines that help in deciding the suitability of a rotary. They are listed below.

1. Rotaries are suitable when the traffic entering from all the four approaches are relativelyequal.

2. A total volume of about 3000 vehicles per hour can be considered as the upper limitingcase and a volume of 500 vehicles per hour is the lower limit.

3. A rotary is very beneficial when the proportion of the right-turn traffic is very high;typically if it is more than 30 percent.

4. Rotaries are suitable when there are more than four approaches or if there is no separatelanes available for right-turn traffic. Rotaries are ideally suited if the intersection geometryis complex.

19.4 Traffic operations in a rotary

As noted earlier, the traffic operations at a rotary are three; diverging, merging and weaving.All the other conflicts are converted into these three less severe conflicts.

1. Diverging: It is a traffic operation when the vehicles moving in one direction is separatedinto different streams according to their destinations.

2. Merging: Merging is the opposite of diverging. Merging is referred to as the process ofjoining the traffic coming from different approaches and going to a common destinationinto a single stream.

3. Weaving: Weaving is the combined movement of both merging and diverging movementsin the same direction.

These movements are shown in figure 19:1. It can be observed that movements from eachdirection split into three; left, straight, and right turn.



��

��

��

��

��

��

��

��

��

��

Figure 19:1: Traffic operations in a rotary

19.4.1 Design elements

The design elements include design speed, radius at entry, exit and the central island, weavinglength and width, entry and exit widths. In addition the capacity of the rotary can alsobe determined by using some empirical formula. A typical rotary and the important designelements are shown in figure 19:2

19.4.2 Design speed

All the vehicles are required to reduce their speed at a rotary. Therefore, the design speedof a rotary will be much lower than the roads leading to it. Although it is possible to designroundabout without much speed reduction, the geometry may lead to very large size incurringhuge cost of construction. The normal practice is to keep the design speed as 30 and 40 kmphfor urban and rural areas respectively.

19.4.3 Entry, exit and island radius

The radius at the entry depends on various factors like design speed, super-elevation, andcoefficient of friction. The entry to the rotary is not straight, but a small curvature is introduced.This will force the driver to reduce the speed. The entry radius of about 20 and 25 metres isideal for an urban and rural design respectively.

The exit radius should be higher than the entry radius and the radius of the rotary island sothat the vehicles will discharge from the rotary at a higher rate. A general practice is to keepthe exit radius as 1.5 to 2 times the entry radius. However, if pedestrian movement is higherat the exit approach, then the exit radius could be set as same as that of the entry radius.

The radius of the central island is governed by the design speed, and the radius of the entrycurve. The radius of the central island, in practice, is given a slightly higher radius so that the



splitter island

the inscribed circle

radius of

entry

exit radius

radius

entrywidth

circulation width

weavin

g len

gth

GIVE WAY line

approachwidth

widthexit

weaving width

radius of the centralisland

Rentry = 20

speed = 30 −

Rexit = Rentr

RCentralIsland

Figure 19:2: Design of a rotary



movement of the traffic already in the rotary will have priority. The radius of the central islandwhich is about 1.3 times that of the entry curve is adequate for all practical purposes.

19.4.4 Width of the rotary

The entry width and exit width of the rotary is governed by the traffic entering and leaving theintersection and the width of the approaching road. The width of the carriageway at entry andexit will be lower than the width of the carriageway at the approaches to enable reduction ofspeed. IRC suggests that a two lane road of 7 m width should be kept as 7 m for urban roadsand 6.5 m for rural roads. Further, a three lane road of 10.5 m is to be reduced to 7 m and7.5 m respectively for urban and rural roads.

The width of the weaving section should be higher than the width at entry and exit. Nor-mally this will be one lane more than the average entry and exit width. Thus weaving widthis given as,

wweaving =(

e1 + e2

2

)

+ 3.5m (19.1)

where e1 is the width of the carriageway at the entry and e2 is the carriageway width at exit.Weaving length determines how smoothly the traffic can merge and diverge. It is decided

based on many factors such as weaving width, proportion of weaving traffic to the non-weavingtraffic etc. This can be best achieved by making the ratio of weaving length to the weavingwidth very high. A ratio of 4 is the minimum value suggested by IRC. Very large weavinglength is also dangerous, as it may encourage over-speeding.

19.5 Capacity

The capacity of rotary is determined by the capacity of each weaving section. Transportationroad research lab (TRL) proposed the following empirical formula to find the capacity of theweaving section.

Qw =280w[1 + e

w][1 − p

3]

1 + wl

(19.2)

where e is the average entry and exit width, i.e, (e1+e2)2

, w is the weaving width, l is the lengthof weaving, and p is the proportion of weaving traffic to the non-weaving traffic. Figure 19:3shows four types of movements at a weaving section, a and d are the non-weaving traffic and band c are the weaving traffic. Therefore,

p =b + c

a + b + c + d(19.3)

This capacity formula is valid only if the following conditions are satisfied.

1. Weaving width at the rotary is in between 6 and 18 metres.

2. The ratio of average width of the carriage way at entry and exit to the weaving width isin the range of 0.4 to 1.



d

b

c

a

d

c

b

a

Figure 19:3: Weaving operation in a rotary

1405

400

505

510

375 408650

500

600

250

1260

1140

420350

370

1433

N

EW

S

Figure 19:4: Traffic approaching the rotary

3. The ratio of weaving width to weaving length of the roundabout is in between 0.12 and0.4.

4. The proportion of weaving traffic to non-weaving traffic in the rotary is in the range of0.4 and 1.

5. The weaving length available at the intersection is in between 18 and 90 m.

Example

The width of a carriage way approaching an intersection is given as 15 m. The entry and exitwidth at the rotary is 10 m. The traffic approaching the intersection from the four sides isshown in the figure 19:4 below. Find the capacity of the rotary using the given data.

Solution

• The traffic from the four approaches negotiating through the roundabout is illustrated infigure 19:5.



505

500

350 510650+

370+

600 350

+650+

370+

420

400

250

408

E

N

S

W

370 510

600 375 500

510505+

+375 +600

375

Figure 19:5: Traffic negotiating a rotary

• Weaving width is calculated as, w = [e1+e2

2] + 3.5 = 13.5 m

• Weaving length, l is calculated as = 4×w = 54 m

• The proportion of weaving traffic to the non-weaving traffic in all the four approaches isfound out first.

• It is clear from equation,that the highest proportion of weaving traffic to non-weavingtraffic will give the minimum capacity. Let the proportion of weaving traffic to the non-weaving traffic in West-North direction be denoted as pWN , in North-East direction aspNE, in the East-South direction as pES, and finally in the South-West direction as pSW .

• The weaving traffic movements in the East-South direction is shown in figure 19:6. Thenusing equation,pES = 510+650+500+600

510+650+500+600+250+375=2260

2885=0.783

pWN = 505+510+350+600505+510+350+600+400+370

=19652735

=0.718

pNE = 650+375+505+370650+375+505+370+510+408

=19002818

=0.674

pSW = 350+370+500+375350+370+500+375+420+600

=15952615

=0.6099

• Thus the proportion of weaving traffic to non-weaving traffic is highest in the East-Southdirection.

• Therefore, the capacity of the rotary will be capacity of this weaving section. Fromequation,

QES =280 × 13.5[1 + 10

13.5][1 − 0.783

3]

1 + 13.554

= 2161.164veh/hr. (19.4)



d

b

c

a

d

c

b

a

375

250

510+650

500+600

Figure 19:6: Traffic weaving in East-South direction

19.6 Summary

Traffic rotaries reduce the complexity of crossing traffic by forcing them into weaving operations.The shape and size of the rotary are determined by the traffic volume and share of turningmovements. Capacity assessment of a rotary is done by analyzing the section having the greatestproportion of weaving traffic. The analysis is done by using the formula given by TRL.

19.7 Problems

1. The width of approaches for a rotary intersection is 12 m. The entry and exit width atthe rotary is 10 m. Table below gives the traffic from the four approaches, traversing theintersection. Find the capacity of the rotary.

Approach Left turn Straight Right turnNorth 400 700 300South 350 370 420East 200 450 550West 350 500 520

Solution

• The traffic from the four approaches negotiating through the roundabout is illustrated infigure 19:7.

• Weaving width is calculated as, w = [e1+e2

2] + 3.5 = 13.5 m

• Weaving length can be calculated as, l = 4×w = 54 m

• The proportion of weaving traffic to the non-weaving traffic in all the four approaches isfound out first.

• It is clear from equation,that the highest proportion of weaving traffic to non-weavingtraffic will give the minimum capacity. Let the proportion of weaving traffic to the non-weaving traffic in West-North direction be denoted as pWN , in North-East direction as


CE415 Transportation Engineering II 19. Traffic signal design-I

420500

520 520

N

S

W E

450+550300450

370

420

550+300

520700+

420+

550+

700 300

+

+ 500+

350

350

200

400370

Figure 19:7: Traffic negotiating a rotary

pNE, in the East-South direction as pES, and finally in the South-West direction as pSW .Then using equation,pES = 450+550+700+520

200+450+550+700+520+300=2220

2720=0.816

pWN = 370+550+500+520350+370+550+500+520+420

=17402510

=0.69

pNE = 420+500+700+300520+400+420+500+700+300

=19202840

=0.676

pSW = 450+300+370+420550+450+400+370+420+350

=15402540

=0.630

• Thus the proportion of weaving traffic to non-weaving traffic is highest in the East-Southdirection.

• Therefore, the capacity of the rotary will be the capacity of this weaving section. From

equation,QES =280×13.5[1+ 10

13.5][1− 0.816

3]

1+ 13.554

= 380.56veh/hr.



Chapter 20

Traffic signal design-I

20.1 Overview

The conflicts arising from movements of traffic in different directions is solved by time sharingof the principle. The advantages of traffic signal includes an orderly movement of traffic, anincreased capacity of the intersection and requires only simple geometric design. However thedisadvantages of the signalized intersection are it affects larger stopped delays, and the designrequires complex considerations. Although the overall delay may be lesser than a rotary for ahigh volume, a user is more concerned about the stopped delay.

20.2 Definitions and notations

A number of definitions and notations need to be understood in signal design. They arediscussed below:

• Cycle: A signal cycle is one complete rotation through all of the indications provided.

• Cycle length: Cycle length is the time in seconds that it takes a signal to complete onefull cycle of indications. It indicates the time interval between the starting of of green forone approach till the next time the green starts. It is denoted by C.

• Interval: Thus it indicates the change from one stage to another. There are two types ofintervals - change interval and clearance interval. Change interval is also called the yellowtime indicates the interval between the green and red signal indications for an approach.Clearance interval is also called all red is included after each yellow interval indicating aperiod during which all signal faces show red and is used for clearing off the vehicles inthe intersection.

• Green interval: It is the green indication for a particular movement or set of movementsand is denoted by Gi. This is the actual duration the green light of a traffic signal is turnedon.

• Red interval: It is the red indication for a particular movement or set of movements andis denoted by Ri. This is the actual duration the red light of a traffic signal is turned on.



• Phase: A phase is the green interval plus the change and clearance intervals that followit. Thus, during green interval, non conflicting movements are assigned into each phase.It allows a set of movements to flow and safely halt the flow before the phase of anotherset of movements start.

• Lost time: It indicates the time during which the intersection is not effectively utilizedfor any movement. For example, when the signal for an approach turns from red to green,the driver of the vehicle which is in the front of the queue, will take some time to perceivethe signal (usually called as reaction time) and some time will be lost here before hemoves.

20.3 Phase design

The signal design procedure involves six major steps. They include the (1) phase design, (2)determination of amber time and clearance time, (3) determination of cycle length, (4)appor-tioning of green time, (5) pedestrian crossing requirements, and (6) the performance evaluationof the above design. The objective of phase design is to separate the conflicting movementsin an intersection into various phases, so that movements in a phase should have no conflicts.If all the movements are to be separated with no conflicts, then a large number of phases arerequired. In such a situation, the objective is to design phases with minimum conflicts or withless severe conflicts.

There is no precise methodology for the design of phases. This is often guided by the geom-etry of the intersection, flow pattern especially the turning movements, the relative magnitudesof flow. Therefore, a trial and error procedure is often adopted. However, phase design is veryimportant because it affects the further design steps. Further, it is easier to change the cycletime and green time when flow pattern changes, where as a drastic change in the flow patternmay cause considerable confusion to the drivers. To illustrate various phase plan options, con-sider a four legged intersection with through traffic and right turns. Left turn is ignored. Seefigure 20:1. The first issue is to decide how many phases are required. It is possible to havetwo, three, four or even more number of phases.

20.3.1 Two phase signals

Two phase system is usually adopted if through traffic is significant compared to the turningmovements. For example in figure 20:2, non-conflicting through traffic 3 and 4 are groupedin a single phase and non-conflicting through traffic 1 and 2 are grouped in the second phase.However, in the first phase flow 7 and 8 offer some conflicts and are called permitted right turns.Needless to say that such phasing is possible only if the turning movements are relatively low.On the other hand, if the turning movements are significant ,then a four phase system is usuallyadopted.



1

5

3

7

2

64

8

Figure 20:1: Four legged intersection

Phase 1 ( P1) Phase 1 ( P2)

2

5

6

1

4

8

3

7

Figure 20:2: Two phase signal



73

7

8

4

5

1

6

2

P1 P2

P3 P4

Figure 20:3: One way of providing four phase signals

20.3.2 Four phase signals

There are at least three possible phasing options. For example, figure 20:3 shows the most simpleand trivial phase plan. where, flow from each approach is put into a single phase avoiding allconflicts. This type of phase plan is ideally suited in urban areas where the turning movementsare comparable with through movements and when through traffic and turning traffic needto share same lane. This phase plan could be very inefficient when turning movements arerelatively low.

Figure 20:4 shows a second possible phase plan option where opposing through traffic areput into same phase. The non-conflicting right turn flows 7 and 8 are grouped into a thirdphase. Similarly flows 5 and 6 are grouped into fourth phase. This type of phasing is veryefficient when the intersection geometry permits to have at least one lane for each movement,and the through traffic volume is significantly high. Figure 20:5 shows yet another phase plan.However, this is rarely used in practice.

There are five phase signals, six phase signals etc. They are normally provided if theintersection control is adaptive, that is, the signal phases and timing adapt to the real timetraffic conditions.

20.4 Interval design

There are two intervals, namely the change interval and clearance interval, normally providedin a traffic signal. The change interval or yellow time is provided after green time for movement.The purpose is to warn a driver approaching the intersection during the end of a green timeabout the coming of a red signal. They normally have a value of 3 to 6 seconds.

The design consideration is that a driver approaching the intersection with design speedshould be able to stop at the stop line of the intersection before the start of red time. Institute oftransportation engineers (ITE) has recommended a methodology for computing the appropriate



7

6

43

1

5

7

8

2

P1 P2

P3 P4

Figure 20:4: Second possible way of providing a four phase signal

3

1

6 4

2 8

5

7

P1P2

P3 P4

Figure 20:5: Third possible way of providing a four-phase signal



1 32 N

Figure 20:6: Group of vehicles at a signalized intersection waiting for green signal

length of change interval which is as follows:

y = t +v85

2a + 19.6g(20.1)

where y is the length of yellow interval in seconds, t is the reaction time of the driver, v85 isthe 85th percentile speed of approaching vehicles in m/s, a is the deceleration rate of vehiclesin m/s2, g is the grade of approach expressed as a decimal. Change interval can also beapproximately computed as y = SSD

v, where SSD is the stopping sight distance and v is the

speed of the vehicle. The clearance interval is provided after yellow interval and as mentionedearlier, it is used to clear off the vehicles in the intersection. Clearance interval is optional in asignal design. It depends on the geometry of the intersection. If the intersection is small, thenthere is no need of clearance interval whereas for very large intersections, it may be provided.

20.5 Cycle time

Cycle time is the time taken by a signal to complete one full cycle of iterations. i.e. onecomplete rotation through all signal indications. It is denoted by C. The way in which thevehicles depart from an intersection when the green signal is initiated will be discussed now.Figure 20:6 illustrates a group of N vehicles at a signalized intersection, waiting for the greensignal. As the signal is initiated, the time interval between two vehicles, referred as headway,crossing the curb line is noted. The first headway is the time interval between the initiation ofthe green signal and the instant vehicle crossing the curb line. The second headway is the timeinterval between the first and the second vehicle crossing the curb line. Successive headwaysare then plotted as in figure 20:7. The first headway will be relatively longer since it includesthe reaction time of the driver and the time necessary to accelerate. The second headwaywill be comparatively lower because the second driver can overlap his/her reaction time withthat of the first driver’s. After few vehicles, the headway will become constant. This constantheadway which characterizes all headways beginning with the fourth or fifth vehicle, is definedas the saturation headway, and is denoted as h. This is the headway that can be achieved by astable moving platoon of vehicles passing through a green indication. If every vehicles requireh seconds of green time, and if the signal were always green, then s vehicles/per hour would



Vehicles in queue

h

Hea

dway

h

e1

e2 e3h1

Figure 20:7: Headways departing signal

pass the intersection. Therefore,

s =3600

h(20.2)

where s is the saturation flow rate in vehicles per hour of green time per lane, h is the saturationheadway in seconds. vehicles per hour of green time per lane. As noted earlier, the headwaywill be more than h particularly for the first few vehicles. The difference between the actualheadway and h for the ith vehicle and is denoted as ei shown in figure 20:7. These differencesfor the first few vehicles can be added to get start up lost time, l1 which is given by,

l1 =n∑

i=1

ei (20.3)

The green time required to clear N vehicles can be found out as,

T = l1 + h.N (20.4)

where T is the time required to clear N vehicles through signal, l1 is the start-up lost time, andh is the saturation headway in seconds.

20.5.1 Effective green time

Effective green time is the actual time available for the vehicles to cross the intersection. It isthe sum of actual green time (Gi) plus the yellow minus the applicable lost times. This losttime is the sum of start-up lost time (l1) and clearance lost time (l2) denoted as tL. Thuseffective green time can be written as,

gi = Gi + Yi − tL (20.5)



20.5.2 Lane capacity

The ratio of effective green time to the cycle length (gi

C)is defined as green ratio. We know

that saturation flow rate is the number of vehicles that can be moved in one lane in one hourassuming the signal to be green always. Then the capacity of a lane can be computed as,

ci = sigi

C(20.6)

where ci is the capacity of lane in vehicle per hour, si is the saturation flow rate in vehicle perhour per lane, C is the cycle time in seconds.

Problem

Let the cycle time of an intersection is 60 seconds, the green time for a phase is 27 seconds, andthe corresponding yellow time is 4 seconds. If the saturation headway is 2.4 seconds/vehicle,the start-up lost time is 2 seconds/phase, and the clearance lost time is 1 second/phase, findthe capacity of the movement per lane?

Solution Total lost time, tL = 2+1 = 3 seconds. From equation effective green time, gi =27+4-3 = 28 seconds. From equationsaturation flow rate, si = 3600

h= 3600

2.4= 1500 veh/hr.

Capacity of the given phase can be found out from equation,Ci = 1500× 2860

= 700 veh/hr/lane.

20.5.3 Critical lane

During any green signal phase, several lanes on one or more approaches are permitted to move.One of these will have the most intense traffic. Thus it requires more time than any other lanemoving at the same time. If sufficient time is allocated for this lane, then all other lanes willalso be well accommodated. There will be one and only one critical lane in each signal phase.The volume of this critical lane is called critical lane volume.

20.6 Determination of cycle length

The cycle length or cycle time is the time taken for complete indication of signals in a cycle.Fixing the cycle length is one of the crucial steps involved in signal design.

If tLi is the start-up lost time for a phase i, then the total start-up lost time per cycle,L =

∑Ni=1 tL, where N is the number of phases. If start-up lost time is same for all phases, then

the total start-up lost time is L = NtL. If C is the cycle length in seconds, then the numberof cycles per hour = 3600

CThe total lost time per hour is the number of cycles per hour times

the lost time per cycle and is = 3600C

.L Substituting as L = NtL, total lost time per hour can

be written as = 3600.N.tlC

The total effective green time Tg available for the movement in a hourwill be one hour minus the total lost time in an hour. Therefore,

Tg = 3600 − 3600.N.tLC

(20.7)



= 3600[

1 − N.tLC

]

(20.8)

(20.9)

Let the total number of critical lane volume that can be accommodated per hour is given by Vc,then Vc = Tg

hSubstituting for Tg, from equation 21.2 and si from the maximum sum of critical

lane volumes that can be accommodated within the hour is given by,

=Tg

h(20.10)

Vc =3600

h

[

1 − N.tLC

]

(20.11)

= si

[

1 − N.tLC

]

(20.12)

Therefore C =N.tL

1 − Vc

s

(20.13)

(20.14)

The expression for C can be obtained by rewriting the above equation. The above equationis based on the assumption that there will be uniform flow of traffic in an hour. To accountfor the variation of volume in an hour, a factor called peak hour factor, (PHF) which is theratio of hourly volume to the maximum flow rate, is introduced. Another ratio called v/c ratioindicating the quality of service is also included in the equation. Incorporating these two factorsin the equation for cycle length, the final expression will be,

C =N.tL

1 − Vc

si×PHF× vc

(20.15)

Highway capacity manual (HCM) has given an equation for determining the cycle length whichis a slight modification of the above equation. Accordingly, cycle time C is given by,

C =N.L.XC

XC − Σ(Vs)i

(20.16)

where N is the number of phases, L is the lost time per phase, (Vs)i is the ratio of volume to

saturation flow for phase i, XC is the quality factor called critical VC

ratio where V is the volumeand C is the capacity.

Problem

The traffic flow in an intersection is shown in the figure 20:8. Given start-up lost time is 3seconds, saturation head way is 2.3 seconds, compute the cycle length for that intersection.Assume a two-phase signal.



1150

900

1300

1800

Figure 20:8: Traffic flow in the intersection

1150

900

1800

1300

Figure 20:9: One way of providing phases

Solution

• If we assign two phases as shown below figure 20:9, then the critical volume for the firstphase which is the maximum of the flows in that phase = 1150 vph. Similarly criticalvolume for the second phase = 1800 vph. Therefore, total critical volume for the twosignal phases = 1150+1800 = 2950 vph.

• Saturation flow rate for the intersection can be found out from the equation as si = 36002.3

= 1565.2 vph. This means, that the intersection can handle only 1565.2 vph. However,the critical volume is 2950 vph . Hence the critical lane volume should be reduced andone simple option is to split the major traffic into two lanes. So the resulting phase planis as shown in figure ( 20:10).

• Here we are dividing the lanes in East-West direction into two, the critical volume in thefirst phase is 1150 vph and in the second phase it is 900 vph. The total critical volumefor the signal phases is 2050 vph which is again greater than the saturation flow rate andhence we have to again reduce the critical lane volumes.

• Assigning three lanes in East-West direction, as shown in figure 20:11, the critical volumein the first phase is 575 vph and that of the second phase is 600 vph, so that the totalcritical lane volume = 575+600 = 1175 vph which is lesser than 1565.2 vph.



1150

1300/2

1300/2

1800/2

1800/2

Figure 20:10: second way of providing phases

1800/31800/31800/3

1150/2 1150/2

Figure 20:11: Third way of providing phases

• Now the cycle time for the signal phases can be computed from equation,C = 2×31− 1175

1565.2

=

24 seconds.

20.7 Summary

Traffic signal is an aid to control traffic at intersections where other control measures fail.The signals operate by providing right of way to a certain set of movements in a cyclic order.Depending on the requirements they can be either fixed or vehicle actuated and two or multi-valued. The design procedure discussed in this chapter include interval design, determinationof cycle time, and computation of saturation flow making use of HCM guidelines.

20.8 Problems

1. Saturation flow rate can be computed as,

(a) 3600h

(b) h3600

(c) 3600×h

(d) none of these

2. Lane capacity is

(a) ci = si × gi

C


CE415 Transportation Engineering II 20. Traffic signal design-II

(b) ci = si × gi

(c) ci = si

C

(d) none of these

20.9 Solutions

1. Saturation flow rate can be computed as,

(a) 3600h

√

(b) h3600

(c) 3600×h

(d) none of these

2. Lane capacity is

(a) ci = si × gi

C

√

(b) ci = si × gi

(c) ci = si

C

(d) none of these



Chapter 21

Traffic signal design-II

21.1 Overview

In the previous chapter, a simple design of cycle time was discussed. Here we will discuss howthe cycle time is divided in a phase. The performance evaluation of a signal is also discussed.

21.2 Green splitting

Green splitting or apportioning of green time is the proportioning of effective green time ineach of the signal phase. The green splitting is given by,

gi =

[

Vci∑n

i=1 Vci

]

× tg (21.1)

where Vci is the critical lane volume and tg is the total effective green time available in a cycle.This will be cycle time minus the total lost time for all the phases. Therefore,

Tg = C − n.tL (21.2)

where C is the cycle time in seconds, n is the number of phases, and tL is the lost time perphase. If lost time is different for different phases, then cycle time can be computed as follows.

Tg = C −n∑

i=1

tLi (21.3)

where tLi is the lost time for phase i, n is the number of phases and C is the lost time inseconds. Actual greentime can be now found out as,

Gi = gi − yi + tLi (21.4)

where Gi is the actual green time, gi is the effective green time available, yi is the amber time,and Li is the lost time for phase i.



600

500

900

1000

Figure 21:1: Phase diagram for an intersection

Problem

The phase diagram with flow values of an intersection with two phases is shown in figure 21:1.The lost time and yellow time for the first phase is 2.5 and 3 seconds respectively. For thesecond phase the lost time and yellow time are 3.5 and 4 seconds respectively. If the cycle timeis 120 seconds, find the green time allocated for the two phases.

Solution

• Critical lane volume for the first phase, VC1 = 1000 vph.

• Critical lane volume for the second phase, VC2 = 600 vph.

• The sum of the critical lane volumes, VC = VC1 + VC2 = 1000+600 = 1600 vph.

• Effective green time can be found out from equationas Tg=120-(2.5-3.5)= 114 seconds.

• Green time for the first phase, g1 can be found out from equationas g1 = 10001600

× 114 =71.25 seconds.

• Green time for the second phase, g2 can be found out from equationas g2 = 6001600

× 114=42.75 seconds.

• Actual green time can be found out from equationThus actual green time for the firstphase, G1 = 71.25-3+2.5 = 70.75 seconds.

• Actual green time for the second phase, G2 = 42.75-3+2.5 = 42.25 seconds.

• The phase diagram is as shown in figure 21:2.

21.3 Pedestrian crossing requirements

Pedestrian crossing requirements can be taken care by two ways; by suitable phase designor by providing an exclusive pedestrian phase. It is possible in some cases to allocate timefor the pedestrians without providing an exclusive phase for them. For example, consider anintersection in which the traffic moves from north to south and also from east to west. If we



��

��

��

��

��

��

70.75 3 46.25

��

��

��

��

��

��

��

��

��

73.75 42.25 4

120

Figure 21:2: Timing diagram

are providing a phase which allows the traffic to flow only in north-south direction, then thepedestrians can cross in east-west direction and vice-versa. However in some cases, it maybe necessary to provide an exclusive pedestrian phase. In such cases, the procedure involvescomputation of time duration of allocation of pedestrian phase. Green time for pedestriancrossing Gp can be found out by,

Gp = ts +dx

uP(21.5)

where Gp is the minimum safe time required for the pedestrians to cross, often referred to asthe “pedestrian green time”, ts is the start-up lost time, dx is the crossing distance in metres,and up is the walking speed of pedestrians which is about 15th percentile speed. The start-uplost time ts can be assumed as 4.7 seconds and the walking speed can be assumed to be 1.2m/s.

21.4 Performance measures

Performance measures are parameters used to evaluate the effectiveness of the design. There aremany parameters involved to evaluate the effectiveness of the design and most common of theseinclude delay, queuing, and stops. Delay is a measure that most directly relates the driver’sexperience. It describes the amount of time that is consumed while traversing the intersection.The figure 21:3 shows a plot of distance versus time for the progress of one vehicle. The desiredpath of the vehicle as well as the actual progress of the vehicle is shown. There are three typesof delay as shown in the figure. They are stopped delay, approach delay and control delay.Stopped time delay includes only the time at which the vehicle is actually stopped waiting atthe red signal. It starts when the vehicle reaches a full stop, and ends when the vehicle begins toaccelerate. Approach delay includes the stopped time as well as the time lost due to accelerationand deceleration. It is measured as the time differential between the actual path of the vehicle,and path had there been green signal. Control delay is measured as the difference betweenthe time taken for crossing the intersection and time taken to traverse the same section, hadbeen no intersection. For a signalized intersection, it is measured at the stop-line as the vehicleenters the intersection. Among various types of delays, stopped delay is easy to derive andoften used as a performance indicator and will be discussed.

Vehicles are not uniformly coming to an intersection. i.e., they are not approaching the



Time

Desired path Actual path

Dis

tanc

e

D3

D2

D1

D3 = Travel time delayD2 = Approach delayD1 = Stopped time delay

Figure 21:3: Illustration of delay measures

intersection at constant time intervals. They come in a random manner. This makes themodeling of signalized intersection delay complex. Most simple of the delay models is Webster’sdelay model. It assumes that the vehicles are arriving at a uniform rate. Plotting a graph withtime along the x-axis and cumulative vehicles along the y-axis we get a graph as shown infigure 21:4. The delay per cycle is shown as the area of the hatched portion in the figure.Webster derived an expression for delay per cycle based on this, which is as follows.

di =C2[1 − gi

C]2

1 − Vi

S

(21.6)

where gi is the effective green time, C is the cycle length, Vi is the critical flow for that phase,and S is the saturation flow.

Delay is the most frequently used parameter of effectiveness for intersections. Other mea-sures like length of queue at any given time (QT ) and number of stops are also useful. Lengthof queue is used to determine when a given intersection will impede the discharge from anadjacent upstream intersection. The number of stops made is an important input parameter inair quality models.

Problem

The traffic flow for a four-legged intersection is as shown in figure 21:5. Given that the losttime per phase is 2.4 seconds, saturation headway is 2.2 seconds, amber time is 3 seconds perphase, find the cycle length, green time and performance measure(delay per cycle). Assumecritical v/c ratio as 0.9.

Solution

• The phase plan is as shown in figure 21:6. Sum of critical lane volumes is the sum ofmaximum lane volumes in each phase, ΣVCi = 433+417+233+215 = 1298 vph.



��

��

R

Ctime

cum

ulat

ive

num

ber

of v

ehic

les

gi

Vi S

Figure 21:4: Graph between time and cumulative number of vehicles at an intersection

��

��215400140

196367170

187

433

220120 417 233

Figure 21:5: Traffic flow for a typical four-legged intersection

P1

433

400

P2

417

367

P3

196

233

P4

215

187

Figure 21:6: Phase plan



Phase 1

Phase 2

Phase 4

Pedestrian phase

23 3 78.5

26

Phase 3

52

3

4

13

1268

3

23 3

17.583

21.5

36.5

52.5

104.5


• Saturation flow rate, Si from equation=36002.2

= 1637 vph. Vc

Si= 433

1637+ 417

1637+ 233

1637+ 1298

1637=

0.793.

• Cycle length can be found out from the equation as C=4×2.4×0.90.9− 1298

1637

= 80.68 seconds ≈ 80

seconds.

• The effective green time can be found out as Gi = VCi

VC× (C − L) = 80-(4×2.4)= 70.4

seconds, where L is the lost time for that phase = 4× 2.4.

• Green splitting for the phase 1 can be found out as g1 = 70.4 × [ 4831298

] = 22.88 seconds.

• Similarly green splitting for the phase 2,g2 = 70.4 × [ 4171298

] = 22.02 seconds.


] = 12.04 seconds.


] = 11.66 seconds.

• The actual green time for phase 1 from equationas G1= 22.88-3+2.4 ≈ 23 seconds.

• Similarly actual green time for phase 2, G2 = 22.02-3+2.4 ≈ 23 seconds.



• Pedestrian time can be found out from as Gp = 4 + 6×3.51.2

= 21.5 seconds. The phasediagram is shown in figure 21:7. The actual cycle time will be the sum of actual greentime plus amber time plus actual red time for any phase. Therefore, for phase 1, actualcycle time = 23+3+78.5 = 104.5 seconds.



• Delay at the intersection in the east-west direction can be found out from equationas

dEW =104.5

2[1 − 23−2.4+3

104.5]2

1 − 4331637

= 42.57sec/cycle. (21.7)

• Delay at the intersection in the west-east direction can be found out from equation,as

dWE =104.5

2[1 − 23−2.4+3

104.5]2

1 − 4001637

= 41.44sec/cycle. (21.8)

• Delay at the intersection in the north-south direction can be found out from equation,

dNS =104.5

2[1 − 23−2.4+3

104.5]2

1 − 3671637

= 40.36sec/cycle. (21.9)

• Delay at the intersection in the south-north direction can be found out from equation,

dSN =104.5

2[1 − 23−2.4+3

104.5]2

1 − 4171637

= 42.018sec/cycle. (21.10)

• Delay at the intersection in the south-east direction can be found out from equation,

dSE =104.5

2[1 − 13−2.4+3

104.5]2

1 − 2331637

= 46.096sec/cycle. (21.11)

• Delay at the intersection in the north-west direction can be found out from equation,

dNW =104.5

2[1 − 13−2.4+3

104.5]2

1 − 1961637

= 44.912sec/cycle. (21.12)

• Delay at the intersection in the west-south direction can be found out from equation,

dWS =104.5

2[1 − 12−2.4+3

104.5]2

1 − 2151637

= 46.52sec/cycle. (21.13)

• Delay at the intersection in the east-north direction can be found out from equation,

dEN =104.5

2[1 − 12−2.4+3

104.5]2

1 − 1871637

= 45.62sec/cycle. (21.14)



P1 P2

750

450

500

650

Figure 21:8: Phase diagram

21.5 Summary

Green splitting is done by proportioning the green time among various phases according to thecritical volume of the phase. Pedestrian phases are provided by considering the walking speedand start-up lost time. Like other facilities, signals are also assessed for performance, delaybeing th e important parameter used.

21.6 Problems

1. Table shows the traffic flow for a four-legged intersection. The lost time per phase is 2.4seconds, saturation headway is 2.2 seconds, amber time is 3 seconds per phase. Find thecycle length, green time and performance measure. Assume critical volume to capacityratio as 0.85. Draw the phasing and timing diagrams.

From To Flow(veh/hr)North South 750East West 650West East 500

Solution

• Given, saturation headway is 2.2 seconds, total lost time per phase (tL) is 2.4 seconds, sat-uration flow = 3600

2.2= 1636.36 veh/hr. Phasing diagram can be assumed as in figure 21:9.

• Cycle time C can be found from equation= 2×2.4×0.850.85− 750+650

1636.36

as negative.

• Hence the traffic flowing from north to south can be allowed to flow into two lanes.

• Now cycle time can be find out as 2×2.4×0.850.85− 450+650

1636.36

= 22.95 or 23 seconds.

• The effective green time , tg = C − (N × tL) = 23 − (2 × 2.4) = 18.2 seconds.



P1 P2

450

500

650

750/2750/2

Figure 21:9: Phase diagram

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

9.85 10.15 3

23

6.85 3 13.15

Phase 2

Phase 1


• This green time can be split into two phases as, For phase 1, g1 = 450×18.21100

= 7.45 seconds.For phase 2, g2 = 650×18.2

1100= 10.75 seconds. Now actual green time ,G1 = g1 minus amber

time plus lost time. Therefore, G1 = 7.45-3+2.4 = 6.85 seconds. G2 = 10.75-3+2.4 =10.15 seconds.

• Timing diagram is as shown in figure 21:10

• Delay at the intersection in the east-west direction can be found out from equationas

dEW =232[1 − 10.75−2.4+3

23]2

1 − 6501636.36

= 4.892sec/cycle. (21.15)

• Delay at the intersection in the west-east direction can be found out from equation,as

dWE =232[1 − 10.75−2.4+3

23]2

1 − 5001636.36

= 4.248sec/cycle. (21.16)

• Delay at the intersection in the north-south direction can be found out from equation,

dNS =232[1 − 7.45−2.4+3

23]2

1 − 7501636.36

= 8.97sec/cycle. (21.17)


CE415 Transportation Engineering II 21. Traffic signal design-III

• Delay at the intersection in the south-north direction can be found out from equation,

dSN =232[1 − 7.45−2.4+3

23]2

1 − 4501636.36

= 6.703sec/cycle. (21.18)



Chapter 22

Traffic signal design-III

22.1 Overview

Topic that will be covered in this chapter are:

1. Effect of right turning vehicles

2. Adjustments on saturatin flow

3. Clearence and change interval

4. Various delay models at signalized intersection

5. HCM procedure on signalized intersection capacity and level of service analysis

22.2 Effect of right-turning vehicles

1. A right-turnings vehicle will consume more effective green time traversing the intersectionthan a corresponding through vehicle.

2. Applicable especially at permitted right movements

3. right turn has great difficulty in manneouring and find a safe gap

4. right turn vehicle may block a through vehicle behind it

5. right turn vehciles may take 2, 4, or even 10 times the time to that of a through movement

6. The equivalency concept will answer how many through vehicles could pass the intersec-tion during the time utilized by a through movement.

7. If 3 through and 2 right turn movement takes place at some time duration in a given lane.Assume at the same time duration in another identical lane if 9 through vehicles moved,then vehicles, then

9 = 3 + 2 × eRT ,⇒ eRT =9 − 3

2= 3.0 (22.1)

Therefore, the right-turn adjustment factor under the current prevailing condition is 3.0.



8. This factor is normally applied in the saturation flow by adjusting its value.

hadj = hideal sec × (pRT × eRT + (1 − pRT ) × 1) (22.2)

For example, if there is 15 percent right-turn movement, eRT is 3, and saturation headwayis 2 sec, then the adjusted staturatin headway is computed as follows:

hadj = 2 sec × (0.15 × 3 + 0.85 × 1) = 2.6 sec/veh (22.3)

9. The saturation head way is increased thereby reducing the saturatin flow sadj = 3600hadj

=36002.6

= 1385 veh/hr.

10. The adjested saturation flow sadj can be written as

sadj = sideal × fRT (22.4)

11. From the Equation 22.2 and 22.4, following relation can be easily derived:

fRT =1

1 + pRT (eRT − 1)(22.5)

where fRT is the multiplicative right turn adjustment factor to the ideal stauration flow.

12. In the above example,

fRT =1

1 + 0.15 × (3 − 1)= 0.77 = 1386veh/hr. (22.6)

Therefore the adjusted saturatin flow is sadj = 1800 ∗ 0.77 veh/sec.

22.3 Change interval

Change interval or yellow or amber time is given after GREEN and before RED which allowsthe vehicles within a ’stopping sight distance’ from the stop line to leagally cross the intersectin.The amber time Y is calculated as

Y = t +v

2(gn + a))(22.7)

where t is the reaction time (about 1.0 sec), v is the velocity of the approaching vehicles, g isthe acceleration due to gravity (9.8 m/sec2), n is the grade of the approach in decimels and ais the deceleration of the vehicle (around 3 m/sec2).


CE415 Transportation Engineering II 22. HCM Method of Signal Design

22.4 Clearence interval

The clearence interval or all-red will facilitate a vehicle just crossed the stop line at the turnof red to clear the intersection with out being collided by a vehicle from the next phase. ITErecomends the following policy for the design of all read time, given as

RAR ==

w+Lv

if no pedestrians

max(

w+Lv

, Pv

)

if pedestrain corossingP+L

vif protected

(22.8)

where w is the width of the intersection from stop line to the farthest conflicting trafic, L isthe length of the vehicle (about 6 m), v is the speed of the vehicle, and P is the width of theintersection from STOP line to the farthest confliting pedestrain cross-walk.



Chapter 23

HCM Method of Signal Design

23.1 The HCM model

The HCM model for signalized intersection analysis is relatively straightforward. The modelbecomes complex when opposing right turns are invloved.

Input module:

The input module is simply a set of conditions that must be specified for analysis to proceed.It is the parametric description of the variables to be analyzed. Some of the variables involvedare discussed here.

Area Type: The location of the intersection must be classified as being in the central businessdistrict (CBD) or not. The calibration study conducted for the 1985 HCM [10] indicatedthat intersections in CBDs have saturation flow rates approximately 10% lower than similarintersections in other areas. If drivers are used to driving in a big city CBD, all locationsin satellite communities would be classified as “other”. In an isolated rural community, evena small business area would be classified as a CBD. The general theory is that the busierenvironment of the CBD causes drivers to be more cautious and less efficient than in otherareas.

Parking Conditions and Parking Activity: If a lane group has curb parking within 84mof the stop line, the existence of a parking lane is assumed. Any vehicle entering or leaving acurb parking space constitutes a “movement”. Where parking exists, the number of parkingmovements per hour occuring within 84m of the stop line is an important variable.

Conflicting Pedestrian Flow: Left-turning vehicles turn through the adjacent pedestriancrosswalk. The flow of pedestrians impedes left-turining vehicles and influences the saturationflow rate for the lane group in question. Pedestrian flows between 1700 ped/hr and 2100 ped/hrin a cross-walk have been shown to fully block left-turners during the green phase.

Local Bus Volume: In signalized intersection analysis, a “local bus” is one that stops topick up and/or discharge passengers within the intersection at either a near or a far side bus



stop. Stopped buses disrupt the flow of other vehicles and influence the saturation flow rate ofthe affected lane group. A bus that passes through the intersection without stopping to pickup or discharge passengers is considered to be a “heavy vehicle”.

Arrival type: The single most important factor influencing delay predictions is the quality ofprogression. The 1994 HCM model uses six “arrival types” to account for this impact. ArrivalType 1: Dense platoon, containing over 80% of the lane group volume, arriving at the start ofthe red phase. Represents very poor progression quality.Arrival Type 2: Moderately dense platoon arriving in the middle of the red phase or dispersedplatoon containing 40% to 80% of the lane group volume, arriving throughout the red phase.Represents unfavourable progression on two-way arterials.Arrival Type 3: Random arrivals in which the main platoon contains less than 40% of the lanegroup volume. Represents operations at isolated and non-interconnected signaliazed intersec-tions characterized by highly dispersed platoons.Arrival Type 4: Moderately dense platoon arriving at the middle of the green phase or dispersedplatoon, containing 40% to 80% of the lane group volume, arriving throughout the green phaseand represents favourable progression quality on a two-way arterial.Arrival Type 5: Dense to moderately dense platoon, containing over 80% of the lane group vol-ume, arriving at the start of the green phase. Represents higly favourable progression quality.Arrival Type 6: This arrival type is reserved for exceptional progression quality on routes withnear-ideal progression characteristics.

Volume adjustment module:

In the 1994 HCM module, all adjustments are applied to saturation flow rate, not to volumes.Several important determinations and calculations are done in this module.

Conversion of hourly volumes to peak rates of flow: The 1994 HCM model focusseson operational analysis of the peak 15-minute period within the hour of interest. Since demandvolumes are entered as full-hour volumes, each must be adjusted to reflect the peak 15-minuteinterval using a peak hour factor. This assumes that all the movements of the intersection ,peak during the same 15-minute period.

Establish lane group for analysis: Any set of lanes across which drivers may optimize theiroperation through unimpeded lane selection will operate in equlibrium conditions determinedby those drivers. Any such set of lanes is analyzed as a single cohesive lane group. An approachis considered to be a single lane group, except for the cases of exclusive left or right-turn lanes.Where an exclusive turning lane exists, it must be analyzed as a separate lane group for analysis.

Lane utilization adjustments: The lane adjustment made to volume is for unequal laneuse. Where lane groups have more than one lane, equilibrium may not imply equal use of lanes.The 1994 HCM allows for an optimal adjustment factor to account for this. The lane utilization



factor adjusts the total lane group flow rate such that when divided by the number of lanes inthe group, the result is the rate of flow expected is the most heavily-used lane. When a laneutilization adjustment is used, the resulting v/c ratios and delays reflect conditons in the mostheavily-used lane of the group. If the factor is not used, the resulting v/c ratios and delaysreflect average conditions over the lane group.

Worksheet: A worksheet is prepared for tabulating intersection movements,peak hour fac-tor,peak flow rates,lane groups for analysis,lane group flow rates, number of lanes,lane utiliza-tion factor and proportion of left- and right-turns in each lane.

Saturation flow rate module:

In this module, the prevailing total saturation flow rate for each lane group is estimated takinginto account eight adjustment factors. The adjustment factors each adjust the saturation flowrate to account for one prevailing condition that may differ from the defined ideal conditions.

Lane width adjustment factor: The ideal lane width is defined as 4m, and it is for thisvalue that the ideal saturation flow rate is defined. When narrower lanes exist, the increasedside-friction between adjacent vehicles causes drivers to be more cautious, and increases head-ways. If width is less than 4m, a negative adjustment occurs;if width is greater than 4m, apositive adjustment occurs and if the width is equal to 4m, the factor becomes 1.00.

Grade adjustment factor: The procedure involved assumes that the effect of grades is onthe operation of heavy vehicles only, and that it is the heavy vehicles that affect other vehiclesin the traffic stream. At signalized intersections, the grade adjustment deals with the impactof an approach grade on the saturation headway at which the vehicles cross the stop line.

Parking adjustment factor: The parking adjustment factor accounts for two deleteriouseffects on flow in a lane group containing a curb parking lane within 84 m of the stop line: (a)The existence of the parking lane creates additional side friction for vehicles in the adjacentlane, thereby affecting the saturation flow rate, and (b) Vehicles entering or leaving curb parkingspaces within 84m of the stop line will disrupt flow in the adjacent lane, which will further affectthe saturation flow. It is generally assumed that the primary effect of a parking lane is on flowin the immediately adjacent lane. If the number of lanes in the lane group is more than one, itis assumed that the adjustment factor for other lanes is 1.00.

Local bus blockage adjustment factor: A general adjustment factor is prescribed for themajority of “ordinary” bus stop situations. The model assumes that the only lane affected bylocal buses is the left most lane. For general cases, there is no differentiation between busesstopping in a travel lane and buses pulling into and out of a stop not in a travel lane. It isassumed that there is no effect on other lanes,i.e. the factor for other lanes is 1.00.



Area type adjustment factor: Data collected for preparation of 1985 HCM suggest thatsaturation flow rates in CBDs tended to be 10% less than similar intersections in other partsof the urban and suburban area. The data were, however, not statistically conclusive andthere is no algorithm for this adjustment as it depends only on the location of the signalizedintersection.

Left-turn adjustment factor: Left-turn vehicles, in general, conflict with pedestrians usingthe adjacent crosswalk. Left-turns may be handled under seven different scenarios.

1. Exclusive LT lane with protected LT phase (no pedestrians)

2. Exclusive LT lane with permitted LT phase

3. Exclusive LT lane with protected + permitted LT phase

4. Shared LT lane with protected LT phase

5. Shared LT lane with permitted LT phase

6. Shared LT lane with protected +permitted LT phase

7. Single lane approach

Right-turn adjustment factor: There are six ways in which right-turns may be handledat a signalized intersection:

1. Exclusive RT lane with protected RT phasing

2. Exclusive RT lane with permitted RT phasing

3. Exclusive RT lane with compound RT phasing

4. Shared RT lane with protected phasing

5. Shared RT lane with permitted phasing

6. Shared RT lane with compound phasing

Modelling permitted right turns:

In modelling the permitted right turns it is necessary to take into consideration,subdividing ofthe green phase,average time to arrival of first right turning vehicle in subject lane group,denotedby gf ,the average time for opposing standing queue clear the intersection from a multi lane ap-proach,denoted as gq, estimation of proportion of right turning vehicles in right lane, denoted byPL. These parameters are to be estimated for various combinations of multilane and single-lanesubject and opposing approaches.


CE415 Transportation Engineering II 23. Coordinated signal design

Modelling the right-turn adjustment factor for compound (protected/permitted)phasing:

The most complicated right-turn case to be modelled is the combination of protected andpermitted phasing. The factors that need to be considered are compound phasing in shared lanegroups, compound phasing in exclusive right-turn lane groups, the right-turn adjustment factorfor protected portion of compound right-turn phases,and the right-turn adjustment factor forthe permitted portion of a compound right-turn phase. A variety of base cases can be referredto when dealing with the analysis of protected + permitted or permitted + protected signalphasing. In applying these procedures, manual computation becomes extremely difficult andthe usage of software becomes the preferred way to implement these procedures.

Capacity analysis module:

Analysis of signalized intersection can be made through the capacity analysis module. Deter-mining the v/s ratios, determining critical lane groups and the sum of critical lane v/s ratios,determining lane group capacities and v/c ratios, modidfying signal timing based on v/s ratiosare outcomes of the procedure involved in the capacity analysis module.

Level of service module:

This involves the estimation of average individual stopped delays for each lane group. Thesevalues may be aggregated to find weighted average delays for each approach, and finally for theintersection as a whole. Once delays are determined, a level of service to each lane group canbe designated and the intersection as a whole. ?



Chapter 24

Coordinated signal design

24.1 Signal Coordination for Progressive and Congested

Conditions

For signals that are closely spaced, it is necessary to coordinate the green time so that vehiclesmay move efficiently through the set of signals. In some cases, two signals are so closely spacedthat they should be considered to be one signal. In other cases, the signals are so far apart thatthey may be considered independently. Vehicles released from a signal often maintain theirgrouping for well over 335m.

24.1.1 Factors affecting coordination

There are four major areas of consideration for signal coordination:

1. Benefits

2. Purpose of signal system

3. Factors lessening benefits

4. Exceptions to the coordinated scheme

The most complex signal plans require that all signals have the same cycle length. Fig. 24:1illustrates path (trajectory) that a vehicle takes as time passes. At t = t1, the first signal turns

secondsignal

Firstsignal

Signal offset

Time

t= $t_{2}$

signal green

yellow

red

$t_{1}$

Figure 24:1: Vehicle trajectory



green. After some lag, the vehicle starts and moves down the street. It reaches the secondsignal at some time t = t2. Depending on the indication of that signal, it either continues orstops. The difference between the two green initiation times is referred to as the signal offset, orsimply as the offset. In general, the offset is defined as the difference between green initiationtimes, measured in terms of the downstream green initiation relative to the upstream greeninitiation.

Benefits

It is common to consider the benefit of a coordination plan in terms of a “cost” or “penalty”function; a weighted combination of stops and delay, and other terms as given here:

cost = A × (total stops) + B × (total delay) + (other terms) (24.1)

The object is to make this disbenefit as small as possible. The weights A and B are coefficientsto be specified by the engineer or analyst. The values of A and B may be selected so as to reflectthe estimated economic cost of each stop and delay. The amounts by which various timing plansreduce the cost, can then be used in a cost-benefit analysis to evaluate alternative plans. Theconservation of energy and the preservation of the environment have grown in importance overthe years. Given that the vehicles must travel, fuel conservation and minimum air pollutionare achieved by keeping vehicles moving as smoothly as possible at efficient speeds. This canbe achieved by a good signal-coordination timing plan. Other benefits of signal coordinationinclude, maintenance of a preferred speed, possibility of sending vehicles through successiveintersections in moving platoons and avoiding stoppage of large number of vehicles.

Purpose of the signal system

The physical layout of the street system and the major traffic flows determine the purpose ofthe signal system. It is necessary to consider the type of system, whether one-way arterial,two-way arterial, one-way,two-way, or mixed network. the capacitites in both directions onsome streets, the movements to be progressed, determination of preferential paths

Factorslessening benefits

Among the factors limiting benefits of signal coordination are the following:

• inadequate roadway capacity

• existence of substantial side frictions, including parking, loading, double parking, andmultiple driveways

• wide variability in traffic speeds

• very short signal spacing

• heavy turn volumes, either into or out of the street



$N$ Distance (m) Northbound

vehicle

0 60 120

600

400

200

Time (sec)

Figure 24:2: Time space diagram

Exceptions of the coordinated scheme

All signals cannot be easily coordinated. When an intersection creating problems lies directly inthe way of the plan that has to be designed for signal coordination, then two separate systems,one on each side of this troublesome intersection, can be considered. A critical intersection isone that cannot handle the volumes delivered to it at any cycle length.

24.1.2 The time-space diagram and ideal offsets

The time-space diagram is simply the plot of signal indications as a function of time for two ormore signals. The diagram is scaled with respect to distance, so that one may easily plot vehicelpositions as a position of time. Fig. 24:2 is a time-space diagram for two intersections. Thestandard conventions are used in Fig. 24:2: a green signal indication is shown by a blank or thinline, amber by a shaded line and red by a solid line. For purpose of illustration of trajectoryin the time space diagram for intersections, a northbound vehicle going at a constnat speed of40fps is shown. The “ideal offset” is defined as the offset that will cause the specified objectiveto be best satisfied. For the objective of minimum delay, it is the offset that will cause minimumdelay. In Fig. 24:2, the ideal offset is 25 sec for that case and that objective. If it is assumedthat the platoon was moving as it went through the upstream intersection then the ideal offsetis given by

t(ideal) =L

S(24.2)

where: t(ideal) = ideal offset,sec, L = block length, m, S = vehicle speed, mps.

24.1.3 Signal progression on one-way streets

Determining ideal offsets

In Fig. 24:3 a one-way arterial is shown with the link lengths indicated. Assuming no vehiclesare queued at the signals, the ideal offsets can be determined if the platoon speed is known. Forthe purpose of illustration, a platoon speed of 60 fps is assumed. The offsets are determinedaccording to Eqn. 24.2. Next the time-space diagram is constructed according to the followingrules:



$N$ Distance (m) Northbound

vehicle

0 60 120

600

400

200

Time (sec)

Figure 24:3: Case study:progression on a one way street

60 120 180 24001

4

5

6

800

Time (sec)

400

1600

2000

Point 1

point 3

Point 2

Dis

tanc

e (m

)

2

3

1200

Figure 24:4: Time space diagram for case study

1. The vertical should be scaled so as to accomodate the dimensions of the arterial, and thehorizontal so as to accomodate atleast three to four cycle lengths.

2. The beginning intersection should be scaled first, usually with main street green initiationat t=0, followed by periods of green and red.

3. The main street green of the next downstream signal should be located next, relative tot=0 and at the proper distance fromt he first intersection. With this point located, theperiods of green, yellow and red for this signal are filled in.

4. This procedure is repeated for all other intersections working one at a time.

Fig. 24:4 shows the time-space diagram for the illustration mentioned previously. Fig. 24:5explores some features of the time-space diagram.

Effect of vehicles queued at signals

It sometimes happens that there are vehicles stored in block waiting for a green light. Thesemay be stragglers from the last platoon, vehicles that turned into the block, or vehicles thatcame out of parking lots or spots. The ideal offset must be adjusted to allow for these vehicles,so as to avoid unnecessary stops. The ideal offset can then be given as:

tideal =L

S− (Qh + Loss1) (24.3)



6

060 120 180

1600

2000

1200

800

400

2401

2

3

4

5

Dis

tanc

e (m

)

Time (sec)

Figure 24:5: Vehicle trajectory and green wave in a progressed movement

1

6

1200

800

400

60 120 180 240Time (sec)

2000

4

5

3

2

Dis

tanc

e (m

)

0

1600

Figure 24:6: Moving southbound

where, Q = number of vehicles queued per lane, veh, h= discharge headway of queued ve-hicle, sec/veh, and Loss1 = loss time associated with vehicles starting from rest at the firstdownstream signal.

A note on queue estimation

If it is known that there exists a queue and its size is known approximately, then the link offsetcan be set better than by pretending that no queue exists. There can be great cycle-to-cyclevariation in the actual queue size, although its average size may be estimated. Even then,queue estimation is a difficult and expensive task and should be viewed with caution.

24.1.4 Signal Progression - on two-way streets and in networks

Consider that the arterial shown in Fig. 24:3 is not a one-way but rather a two-way street.Fig. 24:6 shows the trajectory of a southbound vehicle on this arterial.

Offset determination on a two-way street

If any offset were changed in Fig. 24:6 to accomodate the southbound vehicle(s), then thenorthbound vehicle or platoon would suffer. The fact that offsets are interrelated presents oneof the most fundamental problems of signal optimization. The inspection of a typical cycle (asin Fig. 24:7) yields the conclusion that the offsets in two directions add to one cycle length.



$2C$

$t_{NB} + t_{SB} = C$

$C$

$L$

$t_{NB}$

$t_{SB}$

Figure 24:7: Offsets on 2 way arterials are not independent- One cycle length

$t_{NB}$

$C$

$L$

Distance $2C$

$t_{SB}$

$2C$

Figure 24:8: Offsets on 2 way arterials are not independent- Two cycle length

For longer lengths (as in Fig. 24:8) the offsets might add to two cycle lengths. When queueclearances are taken into account, the offsets might add to zero lengths. The general expressionfor the two offsets in a link on a two-way street can be written as

tNB,i + tSB,i = nC (24.4)

where the offsets are actual offsets, n is an integer and C is the cycle length. Any actual offsetcan be expressed as the desired “ideal” offset, plus an “error” or “discrepancy” term:

tactual(j,i) = tideal(j,i) + e(j,i) (24.5)

where j represents the direction and i represents the link.

Offset determination in a grid

A one-way street system has a number of advantages, not the least of which is traffic eliminationof left turns against opposing traffic. The total elimination of constraints imposed by the“closure” of loops within the network or grid is not possible. Fig. 24:9 highlights the factthat if the cycle length, splits, and three offsets are specified, the offset in the fourth link isdetermined and cannot be independently specified. Fig. 24:9 extends this to a grid of one-waystreets, in which all of the north-south streets are independently specified. The specificationof one east-west street then “locks in” all other east-west offsets. The key feature is that anopen tree of one-way links can be completely independently set, and that it is the closing or“closure” of the open tree which presents constraints on some links.



$C$

$D$

$B$

$A$

2

$N$

$N$

Streets with implied offsets

4

3

1

One way Progressions

Figure 24:9: Closure effect in grid

Distance

4

3

2

1

600

0

Northbound

Time (sec) 12060

200

400

(m) vehicle

Figure 24:10: Bandwidths on a time space diagram

24.1.5 The bandwidth concept and maximum bandwidth

The bandwidth concept is very popular in traffic engineering practice, because

1. the windows of green (through which platoons of vehicles can move) are easy visual imagesfor both working profeesionals and public presentations

2. good solutions can often be obtained manually, by trial and error.

Bandwidth and efficiency of a progression

The efficiency of a bandwidth (measured in seconds) is defined as the ratio of the bandwidthto the cycle length, expressed as a percentage:

efficiency =bandwidth

cycle length× 100% (24.6)

An efficiency of 40% to 50% is considered good. The bandwidth is limited by the minimumgreen in the direction of interest. Fig. 24:10 illustrates the bandwidths for one signal-timingplan. The northbound efficiency can be estimated as (17/60)100% = 28.4%. There is nobandwidth through the south-bound. The system is badly in need of retiming atleast on thebasis of the bandwidth objective. In terms of vehicles that can be put through this systemwithout stopping, note that the northbound bandwidth can carry 17/2.0 = 8.5 vehicles perlane per cycle in a nonstop path through the defined system. The northbound direction can



handle8.5veh

cycle× cycle

60sec× 3600sec

hr= 510vph per lane

very efficiently if they are organized into 8-vehicle platoons when they arrive at this system.If the per lane demand volume is less than 510vphpl and if the flows are so organized, thesystem will operate well in the northbound direction, even though better timing plans mightbe obtained. The computation can be formalized into an equation as follows:

nonstop volume =3600(BW )(L)

(h)(C)vph (24.7)

where: BW = measured or computed bandwidth, sec, L= number of through lanes in indicateddirection, h = headway in moving platoon, sec/veh,and C =cycle length.

Finding bandwidths: A trial-and-error approach and a case study

The engineer ususally wishes to design for maximum bandwidth in one direction, subject tosome relation between the bandwidths in the two directions. There are both trial-and-errorand somewhat elaborate manual techniques for establishing maximum bandwidths. Refer toFig. 24:11, which shows four signals and decent progressions in both the directions. For purposeof illustration, assume it is given that a signal with 50:50 split may be located midway betweenIntersections 2 and 3. The possible effect as it appears in Fig. 24:12 is that there is no way toinclude this signal without destroying one or the other through band, or cutting both in half.The offsets must be moved around until a more satisfactory timing plan develops. A change incycle length may even be required. The changes in offset may be explored by:

• copying the time-space diagram of Fig. 24:12

• cutting the copy horizontally into strips, one strip per intersection

• placing a guideline over the strips, so as to indicate the speed of the platoon(s) by theslope of the guideline

• sliding the strips relative to each other, until some improved offset pattern is identified

There is no need to produce new strips for each cycle length considered: all times can be maderelative to an arbitrary cycle length ‘C”. The only change necessary is to change the slope(s) ofthe guidelines representing the vehicle speeds. The northbound vehicle takes 3600/60 = 60secto travel from intersection 4 to intersection 2. If the cycle length C = 120sec, the vehicle wouldhave arrived at intersection 2 at C/2, or one half of the cycle length. To obtain a good solutionthrough trial-and-error attempt, the following should be kept in consideration:

• If the green initiation at Intersection 1 comes earlier, the southbound platoon is releasedsooner and gets stopped or disrupted at intersection 2.

• Likewise, intersection 2 cannot be northbound without harming the southbound.

• Nor can intersection 3 help the southbound without harming the northbound.



1

2

3

4

N2 lanes/directions

V = 20m/s

1500 vpl

600m

600m

600m

1500 vph

60 120 180

Distance (m)

Time (sec)

Figure 24:11: Case study:Four intersections with good progressions

18012060Time (sec)

Distance (m)

4

3

New

2

1Intersection

Figure 24:12: Effect of inserting a new signal into system

A historical perspective on the use of bandwidth

An elegant mathematical formulation requiring two hours of computation on a supercomputeris some-what irrelevant in most engineering offices. The determination of good progressions onan arterial must be viewed in this context:only 25 years ago, hand held calculators did not exist;20 years ago, calculators had only the most basic functions. 15 years ago, personal computerswere at best a new concept. Previously, engineers used slide rules. Optimization of progressionscould not depend on mathematical formulations simply because even one set of computationscould take days witht he tools available. Accordingly,graphical methods were developed. Thefirst optimization programs that took queues and other detaisl into account began to appear,leading to later developments that produced the signal-optimization programs in common usein late 1980s. As computers became more accessible and less expensive, the move to computersolutions accelerated in the 1970s. New work on the maximum-bandwidth solution followedwith greater computational power encouraging the new formulations.

24.1.6 Forward and reverse progressions

Simple progression is the name given to the progression in which all the signals are set so thata vehicle released from the first intersection will arrive at the downstream intersections just asthe signals at those intersections initiate green. As the simple progression results in a greenwave that advances with the vehicles, it is often called a forward progression. It may happenthat the simple progression is revised two or more times in a day, so as to conform to the



direction of the major flow, or to the flow level. In this case, the scheme may be referred to asa flexible progression. Under certain circumstances, the internal queues are sufficiently largethat the ideal offset is negative. The downstream signal must turn green before the upstreamsignal, to allow sufficient time for the queue to start moving before the arrival of the platoon.The visual image of such a pattern is of the green marching upstream, toward the drivers inthe platoon. This is referred to as reverse progression.

24.1.7 Effective progression on two-way streets

In certain geometries it is possible to obtain very effective progressions in both directions ontwo-way streets. The existence of these patterns presents the facts that:

• The system cycle length should be specified based primarily on the geometry and platoonspeed whenever possible, to enhance progressions.

• The task of good progression in both directions becomes easy if an appropriate combina-tion of cycle length, block length and platoon speed exist.

• Whenever possible the value of these appropriate combinations should be consideredexplicitly for they can greatly determine the qualityof flow for decades.

• In considering the installations of new signals on existing arterials, the same care shouldbe taken to preserve the appropriate combinations and/or to introduce them.

24.1.8 Insights from the importance of signal phasing and cyclelength

The traffic engineer may well be faced with a situation that looks intimidating, but for whichthe community seek to have smooth flow of traffic along an arterial or in a system. The orderlyapproach begins with first, appreciating the magnitude of the problem. The splits, offsets, andcycle length might be totally out of date for the existing traffic demand. Even if the plan isnot out of date, the settings in the field might be totally out of date, the settings in the fieldmight be totally different than those originally intended and/or set. Thus, a logical first stepis simply to ride the system and inspect it. Second, it would be very useful to sketch out howmuch of the system can be thought of as an “open tree” of one way links. A distinction shouldbe made among

• streets that are one way

• streets that can be treated as one-way, due to the actual or desired flow patterns

• streets that must be treated as two-ways

• larger grids in which streets interact because they form unavoidable “closed trees” andare each important in that they cannot be ignored for the sake of establishing a “mastergrid” which is an open tree


CE415 Transportation Engineering II 24. Area Traffic Control

• smaller grids in which the issue is not coordination but local land access and circulation

Downtown grids might well fall into the last category, at least in some cases. Third, attentionshould focus on the combination of cycle length, block length and platoon speed and theirinteraction. Fourth, if the geometry is not suitable, one can adapt and “fix up” the situationto a certain extent. Another issue to address, ofcourse, is whether the objective of progressedmovement of traffic should be maintained.

24.1.9 Oversaturated traffic

The problem of oversaturation is not just one of degree but of kind - extreme congestion ismarked by a new phenomenon: intersection blockage. The overall approach can be stated in alogical set of steps:

• Address the root causes of congestion

• Update the signalization, for poor signalization is frequently the cause of what looks likean incurable problem

• If the problem persists, use novel signalization to minimize the impact and spatial extentof the extreme congestion.

• Provide more space by use of turn bays and parking congestions.

• Develop site specific evaluations where there are conflicting goals.

24.1.10 Signal remedies

Signalization can be improved through measures like, reasonably short cycle lengths, properoffsets and proper splits. Sometimes when there is too much traffic then options such as equityoffsets(to aid cross flows) and different splits may be called upon. A metering plan involvingthe three types - internal, external and release - may be applied. Internal metering refers tothe use of control strategies within a congested network so as to influence the distribution ofvehicles arriving at or departing froma critical location. External metering refers to the controlof the major access points to the defined system, so that inflow rates into the system are limitedif the system if the system is already too congested. Release metering refers to the cases inwhich vehicles are stored in such locations as parking garages and lots, from which their releasecan be in principle controlled.


CE415 Transportation Engineering II 25. Area Traffic Control

Chapter 25

Area Traffic Control

25.0.1 Introduction

Digital computers are used to control traffic signals along arterials and in networks in manycities throughout the world. Here the basic issues and concepts invovled in computer controlof surface street traffic are discussed. With the current emphasis on ITS, computer control ofsystems is now classified as Advanced Transportation Management Systems (ATMS) and thecontrol centers themselves as Transportation Management Centers (TMCs).

Basic principles and flow of information

The basic system Originally, it was assumed that the power of the digital computer couldbe used to control many traffic signals from one location, allowing the development of controlplans. The basic concept can be summarized thus: the computer sends out signals along oneor more arterials. There is no feedback of information from detectors in the field, and thetraffic-signal plans are not responsive to actual traffic conditions. Earlier,the plans for such asystem are developed based on the engineers usage of data from field studies to generate planseither by hand, or by computer,using packages available at the time. The computer solutionswere then run on another machine, or in off hours on the control computer when it was notbeing used for control of the traffic signals. Though this “off-line” system of control plans givesan image of a deficient system, there are many advantages of this “limited” system. Theseinclude:

1. Ability to update signals from a Central Location: The ability to retime signals froma central location without having to send people along an entire arterial to retime thesignals individually at each intersection saves lot of time.

2. Ability to have multiple plans and special plans: In many localities a three-dial controlleris quite sufficient: if traffic is generally regular, three basic plans (A.M. peak, P.M. peak,off-peak) can meet the needs. The computer opens the possiblity to have an N-dialcontroller, with special plans stored for certain days. With appropriate plans stored foreach such event, the plans can be called up by time of day, or by operator intervention.

3. Information on equipment failures: The early systems simply took control of electrome-chanical controllers, driving the cam-shaft from the central computer and receiving a


CE415 Transportation Engineering II 25. Parking

2 3

Libraryof plans

Pattern Plan

X1 P1X2 P2. .. .Xn Pn

Detector

Controllerconfirm signal

Traffic data

4 1

y

computer pattern

Operator

Figure 25:1: Computer control system with detector information used

confirmation signal. Failure to receive this signal meant trouble. The information pro-vided by the control computer allowed such failures to be detected and repair crewsdispatched.

4. Performance data on contractor or service personnel: With a failure detected and noti-fication made, the system can log the arrival of the crew and/or the time at which theintersection is returned to active service.

Collection of traffic data The ability of a computer to receive great amount of data andprocess it is made use of by detectors in the field for sending information back to the centrallocation. If the information is not being used in an “online” setting and hence still does notinfluence the current plan selection. Typically, the computer is being used as the tool for thecollection of permanent or long-term count data.

Traffic data used for plan selection Fig. 25:1 shows a computer control system thatactually uses the traffic data to aid in plan selection. This can be done in one of three principalways: 1. Use library - Monitor deviations from expected pattern: This concept uses a time-of-day approach, looking up in a library both the expected traffic pattern and the preselectedplan matched to the pattern. The actual traffic pattern can be compared to the expected, andif a deviation occurs, the computer can then look through its library for a closer match and usethe appropriate plan. 2. Use library - Match plan to pattern: This is a variation on the firstconcept, with the observed pattern being matched to the most appropriate prestored patternand the coresponding plan veing used. 3. Develop plan on-line: This concept depends on theability to do the necessary computations within a deadline either as a background task or on acompanion computer dedicated to such a computations. This approach presumes an advantageto tailoring the control plan to specific traffic data. It is necessary to note that the time betweenplan updates is constrained by the speed with which the on-line plan computations can be done.The desire to have more frequent updates implicitly assumes that the real traffic situation canbe known precisely enough to differentiate between consecutive update periods.



Chapter 26

Parking

26.1 Overview

Parking is one of the major problems that is created by the increasing road traffic. It is animpact of transport development. The availability of less space in urban areas has increasedthe demand for parking space especially in areas like Central business district. This affects themode choice also. This has a great economical impact.

26.2 Parking studies

Before taking any measures for the betterment of conditions, data regarding availability ofparking space, extent of its usage and parking demand is essential. It is also required toestimate the parking fares also. Parking surveys are intended to provide all these information.Since the duration of parking varies with different vehicles, several statistics are used to accessthe parking need.

26.2.1 Parking statistics

Parking accumulation: It is defined as the number of vehicles parked at a giveninstant of time. Normally this is expressed by accumulation curve. Accumulation curveis the graph obtained by plotting the number of bays occupied with respect to time.

Parking volume: Parking volume is the total number of vehicles parked at a givenduration of time. This does not account for repetition of vehicles. The actual volume ofvehicles entered in the area is recorded.

Parking load : Parking load gives the area under the accumulation curve. It can alsobe obtained by simply multiplying the number of vehicles occupying the parking area ateach time interval with the time interval. It is expressed as vehicle hours.

Average parking duration: It is the ratio of total vehicle hours to the number ofvehicles parked.

parkingduration = parkingloadparkingvolume



Parking turnover: It is the ratio of number of vehicles parked in a duration to thenumber of parking bays available.

parkingturnover = parkingvolumeNo.ofbaysavailable

This can be expressed as number of vehicles per bay per time duration.

Parking index: Parking index is also called occupancy or efficiency. It is defined as theratio of number of bays occupied in a time duration to the total space available. It givesan aggregate measure of how effectively the parking space is utilized. Parking index canbe found out as follows

parking index =parking load

parking capacity× 100 (26.1)

To illustrate the various measures, consider a small example in figure 26:1, which showsthe duration for which each of the bays are occupied(shaded portion). Now the accumulationgraph can be plotted by simply noting the number of bays occupied at time interval of 15, 30,45 etc. minutes ias shown in the figure.

��

��

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3

2

1

1

2

3

30 45 7560 90 105 110150 Time

Parking accumulation curve

Bays and occupancy

No.

of v

ehic

les

Figure 26:1: Parking bays and accumulation curve

The various measures are calculated as shown below:Parking volume= 5 vehicles.Parking load = (1 + 2 + 1 + 0 + 1 + 2 + 3 + 1)15

60=11×15

60= 2.75 veh hour.

Average parking duration =2.75 veh hours5veh

= 33 minutes.

Parking turnover =5 veh/2 hours3bays

= 0.83 veh/hr/bay.

Parking index = 2.75 veh hour3×2 veh hours

× 100= 45.83%

26.3 Parking surveys

Parking surveys are conducted to collect the above said parking statistics. The most commonparking surveys conducted are in-out survey, fixed period sampling and license plate methodof survey.



1. In-out survey: In this survey, the occupancy count in the selected parking lot is takenat the beginning. Then the number of vehicles that enter the parking lot for a particulartime interval is counted. The number of vehicles that leave the parking lot is also taken.The final occupancy in the parking lot is also taken. Here the labor required is very less.Only one person may be enough. But we wont get any data regarding the time durationfor which a particular vehicle used that parking lot. Parking duration and turn over isnot obtained. Hence we cannot estimate the parking fare from this survey.

2. Fixed period sampling: This is almost similar to in-out survey. All vehicles arecounted at the beginning of the survey. Then after a fixed time interval that may varybetween 15 minutes to i hour, the count is again taken. Here there are chances of missingthe number of vehicles that were parked for a short duration.

3. License plate method of survey: This results in the most accurate and realisticdata. In this case of survey, every parking stall is monitored at a continuous interval of15 minutes or so and the license plate number is noted down. This will give the dataregarding the duration for which a particular vehicle was using the parking bay. This willhelp in calculating the fare because fare is estimated based on the duration for which thevehicle was parked. If the time interval is shorter, then there are less chances of missingshort-term parkers. But this method is very labor intensive.

26.4 Ill effects of parking

Parking has some ill-effects like congestion, accidents, pollution, obstruction to fire-fightingoperations etc.

Congestion: Parking takes considerable street space leading to the lowering of the roadcapacity. Hence, speed will be reduced, journey time and delay will also subsequentlyincrease. The operational cost of the vehicle increases leading to great economical loss tothe community.

Accidents: Careless maneuvering of parking and unparking leads to accidents which arereferred to as parking accidents. Common type of parking accidents occur while drivingout a car from the parking area, careless opening of the doors of parked cars, and whilebringing in the vehicle to the parking lot for parking.

Environmental pollution: They also cause pollution to the environment because stop-ping and starting of vehicles while parking and unparking results in noise and fumes. Theyalso affect the aesthetic beauty of the buildings because cars parked at every availablespace creates a feeling that building rises from a plinth of cars.

Obstruction to fire fighting operations: Parked vehicles may obstruct the movementof firefighting vehicles. Sometimes they block access to hydrants and access to buildings.



L

2.5

5.0

5.9

Figure 26:2: Illustration of parallel parking

26.5 Parking requirements

There are some minimum parking requirements for different types of building. For residentialplot area less than 300 sq.m require only community parking space. For residential plot areafrom 500 to 1000 sq.m, minimum one-fourth of the open area should be reserved for parking.Offices may require atleast one space for every 70 sq.m as parking area. One parking spaceis enough for 10 seats in a restaurant where as theatres and cinema halls need to keep only 1parking space for 20 seats. Thus, the parking requirements are different for different land usezones.

26.6 On street parking

On street parking means the vehicles are parked on the sides of the street itself. This will beusually controlled by government agencies itself. Common types of on-street parking are aslisted below. This classification is based on the angle in which the vehicles are parked withrespect to the road alignment. As per IRC the standard dimensions of a car is taken as 5× 2.5metres and that for a truck is 3.75× 7.5 metres.

Parallel parking: The vehicles are parked along the length of the road. Here there isno backward movement involved while parking or unparking the vehicle. Hence, it is themost safest parking from the accident perspective. However, it consumes the maximumcurb length and therefore only a minimum number of vehicles can be parked for a givenkerb length. This method of parking produces least obstruction to the on-going traffic onthe road since least road width is used. Parallel parking of cars is shown in figure 26:2.The length available to park N number of vehicles, L = N

5.9

30◦ parking: In thirty degree parking, the vehicles are parked at 30◦ with respect to theroad alignment. In this case, more vehicles can be parked compared to parallel parking.Also there is better maneuverability. Delay caused to the traffic is also minimum in thistype of parking. An example is shown in figure 26:3. From the figure,

AB = OBsin30◦ = 1.25,

BC = OPcos30◦ = 4.33,

BD = DQcos60◦ = 5,

CD = BD − BC = 5 − 4.33 = 0.67,



5 m

2.5

m

30

1.25 m

4.66 m1 2 n....

A B C D E

L1.254.33

O QP

Figure 26:3: Illustration of 30◦ parking

5.0 m

45

1.77

5.31 m

2.5 m


AB + BC = 1.25 + 4.33 = 5.58

For N vehicles, L = AC + (N-1)CE =5.58+(N-1)5 =0.58+5N

45◦ parking: As the angle of parking increases, more number of vehicles can be parked.Hence compared to parallel parking and thirty degree parking, more number of vehiclescan be accommodated in this type of parking. From figure 26:4, length of parking spaceavailable for parking N number of vehicles in a given kerb is L = 3.54 N+1.77

60◦ parking: The vehicles are parked at 60◦ to the direction of road. More number ofvehicles can be accommodated in this parking type. From the figure 26:5, length availablefor parking N vehicles =2.89N+2.16.

Right angle parking: In right angle parking or 90◦ parking, the vehicles are parkedperpendicular to the direction of the road. Although it consumes maximum width kerb

2.5m

60

L




L

2.5


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ENTRY

EXIT

Figure 26:7: Illustration of off-street parking

length required is very little. In this type of parking, the vehicles need complex maneu-vering and this may cause severe accidents. This arrangement causes obstruction to theroad traffic particularly if the road width is less. However, it can accommodate maximumnumber of vehicles for a given kerb length. An example is shown in figure 26:6. Lengthavailable for parking N number of vehicles is L = 2.5N.

26.7 Off street parking

In many urban centres, some areas are exclusively allotted for parking which will be at somedistance away from the main stream of traffic. Such a parking is referred to as off-streetparking. They may be operated by either public agencies or private firms. A typical layout ofan off-street parking is shown in figure 26:7.

Example 1

From an in-out survey conducted for a parking area consisting of 40 bays, the initial count wasfound to be 25. Table gives the result of the survey. The number of vehicles coming in andout of the parking lot for a time interval of 5 minutes is as shown in the table 26:1. Find theaccumulation, total parking load, average occupancy and efficiency of the parking lot.



Table 26:1: In-out survey dataTime In Out

5 3 210 2 415 4 220 5 425 7 330 8 235 2 740 4 245 6 450 4 155 3 360 2 5

Solution The solution is shown in table 26:2

Table 26:2: In-out parking survey solutionTime In Out Accumulation Occupancy Parking load(1) (2) (3) (4) (5) (6)5 3 2 26 65 13010 2 4 24 60 12015 4 2 26 65 13020 5 4 27 67.5 13525 7 3 31 77.5 15530 8 2 37 92.5 18535 2 7 32 80 16040 4 2 34 85 17045 6 4 36 90 18050 4 1 39 97.5 19555 3 3 39 97.5 19560 2 5 36 90 180

Total 1735

• Accumulation can be found out as initial count plus number of vehicles that entered theparking lot till that time minus the number of vehicles that just exited for that particulartime interval. For the first time interval of 5 minutes, accumulation can be found out as25+3-2 = 26. It is being tabulated in column 4.



• Occupancy or parking index is given by equation For the first time interval of five minutes,Parking index = 26

40×100 = 65%. The occupancy for the remaining time slot is similarly

calculated and is tabulated in column 5.

Average occupancy is the average of the occupancy values for each time interval. Thusit is the average of all values given in column 5 and the value is 80.63%.

• Parking load is tabulated in column 6. It is obtained by multiplying accumulation withthe time interval. For the first time interval, parking load = 26 × 5 = 130 vehicle minutes.

• Total parking load is the summation of all the values in column 5 which is equal to 1935vehicle minutes or 32.25 vehicle hours

Example 2

The parking survey data collected from a parking lot by license plate method is s shown inthe table 26:3 below. Find the average occupancy, average turn over, parking load, parkingcapacity and efficiency of the parking lot.

Table 26:3: Licence plate parking survey dataBay Time

0-15 15-30 30-45 45-601 1456 9813 - 56782 1945 1945 1945 19453 3473 5463 5463 54634 3741 3741 9758 48255 1884 1884 - 75946 - 7357 - 78937 - 4895 4895 48958 8932 8932 8932 -9 7653 7653 8998 482110 7321 - 2789 278911 1213 1213 3212 477812 5678 6678 7778 8888

Solution See the following table for solution 26:4. Columns 1 to 5 is the input data. Theparking status in every bay is coded first. If a vehicle occupies that bay for that time interval,then it has a code 1. This is shown in columns 6, 7, 8 and 9 of the table corresponding to thetime intervals 15, 30, 45 and 60 seconds.

• Turn over is computed as the number of vehicles present in that bay for that particularhour. For the first bay, it is counted as 3. Similarly, for the second bay, one vehicle is



Table 26:4: Licence plate parking survey solutionBay Time Time(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

15 30 45 60 15 30 45 60 Turn over1 1456 9813 - 5678 1 1 0 1 32 1945 1945 1945 1945 1 1 1 1 13 3473 5463 5463 5463 1 1 1 1 24 3741 3741 9758 4825 1 1 1 1 35 1884 1884 - 7594 1 1 0 1 26 - 7357 - 7893 0 1 0 1 27 - 4895 4895 4895 0 1 1 1 18 8932 8932 8932 - 1 1 1 0 19 7653 7653 8998 4821 1 1 1 1 310 7321 - 2789 2789 1 0 1 1 211 1213 1213 3212 4778 1 1 1 1 312 5678 6678 7778 8888 1 1 1 1 4

Accumulation 10 11 9 11Occupancy 0.83 0.92 0.75 0.92 2.25

present throughout that hour and hence turnout is 1 itself. This is being tabulated incolumn 10 of the table. Average turn over = Sum of turn−over

Total number of bays= 2.25

• Accumulation for a time interval is the total of number of vehicles in the bays 1 to 12 forthat time interval. Accumulation for first time interval of 15 minutes = 1+1+1+1+1+0+0+1+1+1+1+1= 10

• Parking volume = Sum of the turn over in all the bays = 27 vehicles

• Average duration is the average time for which the parking lot was used by the vehicles.It can be calculated as sum of the accumulation for each time interval × time intervaldivided by the parking volume = (10+11+9+11)×15

27= 22.78 minutes/vehicle.

• Occupancy for that time interval is accumulation in that particular interval divided bytotal number of bays. For first time interval of 15 minutes, occupancy = (10×100)/12 =83% Average occupancy is found out as the average of total number of vehicles occupyingthe bay for each time interval. It is expressed in percentage. Average occupancy =0.83+0.92+0.75+0.92

4× 100 = 85.42%.

• Parking capacity = number of bays × number of hours = 12× 1 = 12 vehicle hours

• Parking load = total number of vehicles accumulated at the end of each time interval ×time = (10+11+9+11)×15

60= 10.25 vehicle hours



• Efficiency = Parking loadTotal number of bays

= 10.2512

= 85.42%.

26.8 Summary

Providing suitable parking spaces is a challenge for traffic engineers and planners in the scenarioof ever increasing vehicle population. It is essential to conduct traffic surveys in order to designthe facilities or plan the fares. Different types of parking layout, surveys and statistics werediscussed in this chapter.

26.9 Problems

1. The parking survey data collected from a parking lot by license plate method is shownin table 26:5 below. Find the average occupancy, average turnover, parking load, parkingcapacity and efficiency of parking lot.

Table 26:5: Licence plate: problemBay Time

0-15 15-30 30-45 45-601 1501 1501 4021 -2 1255 1255 1255 12553 3215 3215 3215 32154 - - 3100 31005 1623 1623 1623 -6 2204 2204 - -

Solution Refer table 26:6. Column 1 to 5 is the input data. The parking status in every bayis coded first. If a vehicle occupies that bay for that time interval, then it has a code 1. This isshown in columns 6, 7, 8 and 9 of the tables corresponding to the time intervals 15,30,45 and60 seconds.

• Turn over is computed as the number of vehicles present in that bay for that particularhour. For the first bay, it is counted as 2. Similarly, for the second bay, one vehicleis present throughout that hour and hence turnout is 1 itself This is being tabulated incolumn 10 of the table. Total turn over in all the bays or parking volume= 2+1+1+1+1+1= 7 vehicles Average turn over = Sum of turn−over

Total number of bays= 7

6=1.17

• Average duration is the average time for which the parking lot was used by the vehicles.It can be calculated as sum of the accumulation for each time interval × time intervaldivided by the parking volume = (5+5+5+3)×15

7= 38.57 minutes/vehicle.


CE415 Transportation Engineering II 26. Congestion Studies

Table 26:6: License Plate Problem: SolutionBay Time Time(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

15 30 45 60 15 30 45 60 Turn over1 1501 1501 4021 - 1 1 1 0 22 1255 1255 1255 1255 1 1 1 1 13 3215 3215 3215 3215 1 1 1 1 14 - - 3100 3100 0 0 1 1 15 1623 1623 1623 - 1 1 1 0 16 2204 2204 - - 1 1 0 0 1

Accumulation 5 5 5 3Occupancy 0.83 0.83 0.83 0.5

• Average occupancy is found out as the average of total number of vehicles occupyingthe bay for each time interval. It is expressed in percentage. Average occupancy =0.83+0.83+0.83+0.5

4× 100 = 75%.

• Parking capacity = number of bays × number of hours = 6× 1 = 6 vehicle hours

• Parking load = total number of vehicles accumulated at the end of each time interval ×time = (5+5+5+3)×15

60= 4.5 vehicle hours

• Efficiency = Parking loadTotal number of bays

= 4.56

= 75%.



Chapter 27

Congestion Studies

27.1 Introduction

27.1.1 Challenges in transportation system

Transportation system consists of a group of activities as well as entities interacting with eachother to achieve the goal of transporting people or goods from one place to another. Hence,the system has to meet the perceived social and economical needs of the users. As these needschange, the transportation system itself evolves and problems occur as it becomes inadequateto serve the public interest. One of the negative impacts of any transportation system istraffic congestion. Traffic congestion occurs wherever demand exceeds the capacity of thetransportation system. This lecture gives an overview of how congestion is generated, howit can be measured or quantified, and also the various countermeasures to be taken in orderto counteract congestion. Adequate performance measures are needed in order to quantifycongestion in a transportation system. Quality of service measures indicates the degree oftraveller satisfaction with system performance and this is covered under traveller perception.Several measures have been taken in order to counteract congestion. They are basically classifiedinto supply and demand measures. An overview of all these aspects of congestion is dealt within this lecture.

27.1.2 Generation of traffic congestion

The flow chart in Fig. 27:1 shows how traffic congestion is generated in a transportation system.With the evolution of society, economy and technology, the household characteristics as wellas the transportation system gets affected. The change in transport system causes a change intransport behaviour and locational pattern of the system. The change in household character-istics, transport behaviour, locational pattern, and other growth effects result in the growth oftraffic. But the change or improvement in road capacity is only as the result of change in thetransportation system and hence finally a situation arises where the traffic demand is greaterthan the capacity of the roadway. This situation is called traffic congestion.

27.1.3 Effects of congestion

Congestion has a large number of ill effects which include:



Transport Behavior

Household Characteristics & Norms Evolution of Society Economy Technology

Transportation System

Location Patterns

Roadway Capacity

Traffic Congestion

Traffic Growth

Growth Effects

Figure 27:1: Generation of traffic congestion

1. Loss of productive time,

2. Increase in the fuel consumption,

3. Increase in pollutants (because of both the additional fuel burned and more toxic gasesproduced while internal combustion engines are in idle or in stop-and-go traffic),

4. Increase in wear and tear of automobile engines,

5. High potential for traffic accidents,

6. Negative impact on people’s psychological state, which may affect productivity at workand personal relationships, and

7. Slow and inefficient emergency response and delivery services.

The summation of all these effects yields a considerable loss for the society and the economyof an urban area

27.1.4 Traffic congestion

A system is said to be congested when the demand exceeds the capacity of the section. Trafficcongestion can be defined in the following two ways:

1. Congestion is the travel time or delay in excess of that normally incurred under light orfree flow traffic condition.

2. Unacceptable congestion is travel time or delay in excess of agreed norm which may varyby type of transport facility, travel mode, geographical location, and time of the day.

Fig. 27:2 shows the definition of congestion. The solid line represents the travel speed underfree-flow conditions and the dotted line represents the actual travel speed. During congestion,the vehicles will be travelling at a speed less than their free flow speed. The shaded area inbetween these two lines represents the amount of congestion. Traffic congestion may be of twotypes:



Street 1 Street 3 Street 4

Distance

Street 2

Travel SpeedFree flow

Spe

ed

Congestion

Amount of

ActualTravel Speed

Figure 27:2: Definition of congestion

1. Recurrent Congestion: Recurrent congestion generally occurs at the same place, atthe same time every weekday or weekend day.

2. Non-Recurrent congestion: Non-Recurrent congestion results from incidents such asaccidents or roadway maintenance.

27.2 Measurement of congestion

Congestion has to be measured or quantified in order to suggest suitable counter measuresand their evaluation. Congestion information can be used in a variety of policy, planningand operational situations. It may be used by public agencies in assessing facility or systemadequacy, identifying problems, calibrating models, developing and assessing improvements,formulating programs and policies and priorities. It may be used by private sector in makinglocational or investment decisions. It may be used by general public and media in assessingtraveller’s satisfaction.

27.2.1 System performance measurement

Performance measure of a congested roadway can be done using the following four components:

1. Duration,

2. Extent,

3. Intensity, and

4. Reliability.



27.2.2 Duration

Duration of congestion is the amount of time the congestion affects the travel system. Thepeak hour has now extended to peak period in many corridors. Measures that can quantifycongestion include:

• Amount of time during the day that the travel rate indicates congested travel on a systemelement or entire system.

• Amount of time during the day that traffic density measurement techniques (detectors,aerial surveillance, etc.) indicate congested travel.

Duration of congestion is the sum of length of each analysis sub period for which the demandexceeds capacity. The maximum duration on any link indicates the amount of time beforecongestion is completely cleared from the corridor. Duration of congestion can be computedfor a corridor using the following equation:

H = N × T (27.1)

where, H is the duration of congestion (hour), N is the number of analysis sub periods forwhich v/c > 1, and T is the duration of analysis sub-period (hour)The duration of congestionfor an area is given by:

Hi =T vi

ci(1 − r)

1 − r(vi

ci)

(27.2)

where, Hi is the duration of congestion for link i (hour), T is the duration of analysis period(hour), r is the ratio of peak demand to peak demand rate, vi is the vehicle demand on link i(veh/hour), and ci is the capacity of link i (veh/hour).

27.2.3 Extent

Extent of congestion is described by estimating the number of people or vehicles affected bycongestion and by the geographic distribution of congestion. These measures include:

1. Number or percentage of trips affected by congestion.

2. Number or percentage of person or vehicle meters affected by congestion.

3. Percentage of the system affected by congestion.

Performance measures of extent of congestion can be computed from sum of length of queuingon each segment. Segments in which queue overflows the capacity are also identified. Tocompute queue length, average density of vehicles in a queue need to be known. The defaultvalues suggested by HCM 2000 are given in Table 1.



Table 27:1: Queue density default valuesSubsystem Storage density(veh/km/lane) Spacing(m)Free-way 75 13.3

] Two lane highway 130 7.5Urban street 130 7.5

Queue length can be found out using the equation:

Qi =T (v − c)

N × ds(27.3)

where; Qi is the queue length (meter), v is the segment demand (veh/hour), c is the segmentcapacity (veh/hour), N is the number of lanes, ds is the storage density (veh/meter/lane), andT is the duration of analysis period (hour). If v < c, Qi=0 The equation for queue length issimilar for both corridor and area-wide analysis.

27.2.4 Intensity

Intensity of congestion marks the severity of congestion. It is used to differentiate between levelsof congestion on transport system and to define total amount of congestion. It is measured interms of:

• Delay in person hours or vehicle hours;

• Average speed of roadway, corridor, or network;

• Delay per capita or per vehicle travelling in the corridor, or per person or per vehicleaffected by congestion;

• Relative delay rate (relative rate of time lost for vehicles);

Intensity in terms of delay is given by,

DPH = TPH − T 0PH (27.4)

where, DPH is the person hours of delay, TPH is the person hours of travel under actualconditions, and T 0

PH is the person hours of travel under free flow conditions. The TPH is givenby:

TPH =OAV × v × l

S(27.5)

where, OAV is the average vehicle occupancy, v is the vehicle demand (veh), l is the length oflink (km), and S is the mean speed of link (km/h). The TPH is given by:

T 0PH =

OAV × v × l

S0(27.6)



where, OAV is the average vehicle occupancy, v is the vehicle demand (veh), l is the length oflink (km), and S0 is the free flow speed on the link

27.2.5 Numerical example

On a 2.8 km long link of road, it was found that the demand is 1000 Vehicles/hour mean speedof the link is 12 km/hr, and the free flow speed is 27 km/hr. Assuming that the average vehi-cle occupancy is 1.2 person/vehicle, calculate the congestion intensity in terms of total personhours of delay.

Solution: Person hours of delay is given as

DPH = TPH − T 0PH

Person hours of travel under actual conditions,

TPH =OAV × v × l

S

=1.2 × 1000 × 2.8

12= 280 person hours

Person hours of travel under free flow conditions,

T 0PH =

OAV × v × l

S0

=1.2 × 1000 × 2.8

27= 124.4 person hours

Therefore, person hours of delay,

DPH = = 280 − 124.4

= 155.6 person hours.

27.2.6 Relationship between duration, extent, and intensity of con-gestion

The relationship between duration,extent, and intensity of congestion can be show in a time-distance graph Fig. 27:3 The extent of congestion is seen on the x-axis, the duration on they-axis. The intensity is shown in the shading. Based on the extent and duration the congestioncan be classified into four types as shown in Fig.27:4



��

��

Tim

e

Distance

Duration

Extent

Figure 27:3: Intensity of congestion-relation between duration and distance

Ext

ent

Duration

Congestion

General

Broad Critical

System−Wide

Problems

Critical

Links or

Corridors

Limited

Problem

Figure 27:4: Intensity of congestion-Relation between extent and duration of delay

The variation in extent and duration of congestion indicates different problems requiring dif-ferent solutions. Small delay and extent indicates limited problem, small delay for large extentindicates general congestion, great delay for small extent indicates critical links and great delayfor large extent indicates critical system-wide problem. The product of extent and durationindicates the intensity, or magnitude of the congestion problem.

27.3 Congestion countermeasures

Various measures to address congestion are discussed here. These include supply side, demandside, and pricing.

27.3.1 Supply measures:

Congestion countermeasures on the supply side add capacity to the system or make the systemoperate more efficiently. They focus on the transportation system. Supply measures include



1. Development of new or expanded infrastructure. This includes civil projects (new free-ways, transit lines etc), road widening, bridge replacement, technology conversions(ITS),etc.

2. Small scale capacity and efficiency improvement. This includes signal system upgradeand coordination, freeway ramp metering, re-location of bus stops etc.

27.3.2 Demand measures:

Demand measures focuses on motorists and travelers and attempt to modify their trip makingbehaviour. Demand measures include:

1. Parking pricing: It discourages the use of private vehicles to specific areas, thereby re-ducing the demand on the transportation system.

2. Restrictions on vehicle ownership and use: It includes heavy import duties, separatelicensing requirement, heavy annual fees, expensive fuel prices, etc. to restrain privatevehicle acquisition and use.

27.3.3 Congestion pricing

Congestion pricing is a method of road user taxation, charging the users of congested roadsaccording to the time spent or distance travelled on them. The principle behind congestionpricing is that those who cause congestion or use road in congested period should be charged,thus giving the road user the choice to make a journey or not.

Economic principle behind congestion pricing

Journey costs are made of private journey cost, congestion cost, environmental cost, and roadmaintenance cost. The benefit a road user obtains from the journey is the price he preparedto pay in order to make the journey. As the price gradually increases, a point will be reachedwhen the trip maker considers it not worth performing or worth performing by other means.This is known as the critical price. At a cost less than this critical price, he enjoys a net benefitcalled as consumer surplus(es) and is given by:

s = x − y (27.7)

where, x is the amount the consumer is prepared to pay, and y is the amount he actuallypays. The basics of congestion pricing involves demand function, private cost function as wellas marginal cost function. These are explained below.



No. of trips

Cos

t of T

rips

O Q

R

PS

Figure 27:5: Demand Curve

Demand

Fig. 27:5 shows the general form of a demand curve. In the figure, area QOSP indicates theabsolute utility to trip maker and the area SRP indicates the net benefit.

Private cost

Total private cost of a trip, is given by:

c = a +b

v(27.8)

where, a is the component proportional to distance, b is the component proportional to speed,and v is the speed of the vehicle (km/h) which is given by:

v = d − eq (27.9)

where, q is the flow in veh/hour, d and e are constants.

Marginal cost

Marginal cost is the additional cost of adding one extra vehicle to the traffic stream. It reducesspeed and causes congestion and results in increase in cost of all journey. The total cost incurredby all vehicles in one hour(CT ) is given by:

CT = cq (27.10)



Flow(q)

Priv

ate

Cos

t of T

rips

Marginal Cost / Flow

Private Cost / Flow

Figure 27:6: Private cost/flow and cost and marginal curve

Marginal cost is obtained by differentiating the total cost with respect to the flow(q) as shownin the following equations.

d(cq)

dq= c + q

dc

dq(27.11)

dc

dq=

dc

dv× dv

dq(27.12)

= (−b)/v2 ×−e (27.13)

= be/v2 (27.14)

d(cq)

dq= c + q

dc

dq(27.15)

= a +b

v+

d − v

e× be

v2(27.16)

Note that c and q in the above derivation is obtained from Equations 27.8 and 27.9 respectively.Therefore the marginal cost is given as:

M = a +b

v+

(d − v)b

v2(27.17)

Fig. 27:6 shows the variation of marginal cost per flow as well as private cost per flow. It is seenthat the marginal cost will always be greater than the private cost, the increase representingthe congestion cost.

Equilibrium condition and Optimum condition

Superimposing the demand curve on the private cost/flow and marginal cost/flow curves, theposition as shown in Fig. 27:7 is obtained. The intersection of the demand curve and the privatecosts curve at point A represents the equilibrium condition, obtained when travel decisions are



Flow(q)

Marginal Cost / Flow

Private Cost / Flow

AX

Y

B

Z

Condition

Optimum

Equilibrium

Condition

Cos

t / B

enifi

t

Figure 27:7: Relation between material cost, private cost and demand curves.

based on private costs only. The intersection of the demand curve and the marginal costs curveat point B represents the optimum condition. The net benefit under the two positions A and Bare shown by the areas ACZ and BY CY Z respectively. If the conditions are shifted from pointA to B, the net benefit due to change will be given by area CCyY X minus AXB. If the areaCCyY X is greater than arc AXB, the net benefit will be positive. The shifting of conditionsfrom point A to B can be brought about by imposing a road pricing charge BY. Under thisscheme, the private vehicles continuing to use the roads will on an average be worse off in thefirst place because BY will always exceed the individual increase in benefits XY.

27.3.4 Numerical example

Vehicles are moving on a road at the rate of 500 vehicle/hour, at a velocity of 15 km/hr. Findthe equation for marginal cost.

Solution:

c =a + b

v

=a + b

15v = d − eq

= d − 500e

M = a +b

v+

(d − v)b

v2

= a +b

15+

(d − 15)b

225



27.3.5 Uses of congestion pricing

1. Diverts travelers to other modes

2. Causes cancellation of non essential trips during peak hours

3. Collects sufficient fund for major upgrades of highways

4. Cross-subsidizes public transport modes

27.3.6 Requirements of a good pricing system

1. Charges should be closely related to the amount of use made of roads

2. Price should be variable at different times of day/week/year or for different classes ofvehicles

3. It should be stable and ascertainable by road users before commencement of journey

4. Method should be simple for road users to understand and police to enforce

5. Should be accepted by public as fair to all

6. Payment in advance should be possible

7. Should be reliable

8. Should be free from fraud or evasion

9. Should be capable of being applied to the whole country

27.4 Conclusion

In this lecture, we have discussed about the causes and effects of congestion and how congestioncan be defined. We also discussed how congestion can be quantified by various performancemeasures such as duration, extent and intensity. The measures to be taken in order to coun-teract congestion were also discussed. The principle and process of congestion pricing was alsodiscussed.


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CE415 Transportation Engineering II

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