Application of a Markov chain traffic model to the Greater Philadelphia RegionJoseph Reiter, Villanova University

MAT 8435, Fall 2013

Traffic models – levels of focus

• Microscopic

• Focus on individual interactions

between vehicles

• Mesoscopic

• Examines vehicle movements

as part of a larger scale

mechanism

• Macroscopic

• Describes average values of

overall traffic variables

Representing a highway system as a matrix

• Exits are represented by vertices

and highway segments by edges

• The adjacency matrix for this

graph:

Representing a highway system as a matrix

• If the distance between exits is

inserted into the matrix, and the

unconnected exits take on the

value infinity, a distance matrix

can be defined:

• The number of lanes between

exits, where unconnected exits

are represented by -1:

These two matrices describe all the important information about the geometry

of the highway system

Determining Population

• The number of vehicles in the area of an exit can be represented by a

row vector:

q = (q1, q2, …, qm),

• A matrix P containing the probabilities that a vehicle travels to a

particular exit can be defined:

• A function representing the relative volume of traffic on the highway

system at a time t in needed to determine the probability that a

vehicle travels on the highway during a time interval:

v(t)

Determining Population

• When determining the new population around an exit after a time

interval, there are two considerations:

• a) The vehicles that traveled to another exit

• b) The vehicles that did not travel

• The population after a time interval is given by:

• This can be written using matrices and vectors:

Traffic Density

• In order to calculate the density of traffic on a segment of highway, we

must determine how many vehicles are traveling between exits during

a time interval. A route matrix describing the number of vehicles

going from one exit to another is defined:

• A matrix Q can be made from the population vector:

Traffic Density

• The route matrix can be rewritten using matrices:

• The number of vehicles passing through a segment of highway is the

sum of the vehicles traveling on all the routes that pass through this

segment:

• One way to determine which routes pass through a segment is to use

Dijkstra’s algorithm, which finds the shortest path between exits.

Traffic Density

• The density is then determined using the element ci,j , the average

speed of traffic, and the corresponding element of the matrix L:

• This is the predicted density for the segment from i to j

Application of the model

• Assumptions for this application:

• Only 46 exits are used in this model

• The initial number of vehicles at time 1AM is proportional to the number

of households in that area

• 3 transition matrices are used:

• From 5AM to 10AM – probability is proportional to the number of workers

• From 3PM to 4AM – probability is proportional to the number of households

• From 11AM to 2PM – the average of these two probabilities

• Average speed of vehicles is 65 mph

• False exits added to ends of highways that travel away from the network

in order to provide a buffer to the system

Application of the model

Application of the model

• Positive Predictive Value

• Shows how likely a highway segment has heavy traffic given that

the model predicts heavy traffic:

• Four versions of the model are evaluated using PPV

1 miles population radius 2 mile population radius

1 mile workers radius Model A Model B

2 mile workers radius Model C Model D

Application of the model

0

0.1

0.2

0.3

0.4

0.5

0.6

A B C D

PP

V

Model

Overall Positive Predictive Value

Application of the model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Monday Tuesday Wednesday Thursday Friday

PP

V

Day

Positive Predictive Value By Day

Model A

Model B

Model C

Model D

Application of the model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

7 8 9 10 11 12 13 14 15 16 17 18 19

PP

V

Hour

Positive Predictive Value By Hour

Model A

Model B

Model C

Model D

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Data

Google Maps. (2013, October 20). Philadelphia, PA Historic Traffic Data

Map for Monday. Retrieved October 20, 2013, from Google Maps:

https://maps.google.com/maps

ITO Map. (2013, October 1). Highway Lanes. Retrieved October 1, 2013,

from ITO Map: http://www.itoworld.com/map/179

PennDOT. (2011). Factoring Process, Hourly Percent Total Vehicles.

Bureau of Planning and Research. PennDOT.

US Census Bureau. (2013, October 20). Demographic Snapshot

Summary, Total Households. Retrieved October 20, 2013, from Free

Demographics: http://www.freedemographics.com

US Census Bureau. (2013, Oct 20). Longitudinal-Employer Household

Dynamics Program. Retrieved Oct 20, 2013, from OnTheMap Application:

http://onthemap.ces.census.gov/

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