unreliability due to inherent properties of the traffic stream, especiallycongestion phenomena. In other words, unreliability due to acci-dents or other special occurrences is not addressed—although thatcould be incorporated in the overall framework for reliability char-acterization and information provision. Thus the main focus is toquantify the unreliability resulting from traffic interactions as thesystem approaches congested states, namely those associated withthe breakdown phenomenon.

Given that long delays are usually caused by traffic breakdownon one or more links along a driver’s path, the intention is to com-pute travel time reliability taking into account flow breakdownimpacts. Due to its stochastic nature, traffic breakdown does notnecessarily occur at a deterministic maximal flow rate, but occurswith some probability at various flow rates (8, 9). Traffic flow reli-ability, referred to in some of the literature as capacity reliability,describes a traffic facility’s ability to sustain service flow rates nearthe operational optimum, that is, at prebreakdown levels (4). Traf-fic flow reliability and travel time reliability have been studied in theliterature as two independent performance characteristics. However,empirical data reveal a high correlation between these two perfor-mance dimensions (which also conforms to theoretical prediction),that is, when traffic breaks down, travelers usually encounter longdelays, resulting in unreliable trip times. In this study the probabil-ity of traffic breakdown at different flow rates is examined, and thetraffic flow reliability is related to the added delay that a travelermay experience. The proposed methodology predicts the traveltime and its reliability based on the prevailing traffic conditions,which can be readily obtained from sensors. As such, it provides anapproach for predicting flow and travel time reliability in real time,for possible dissemination to the traveling public.

The rest of the paper is organized as follows. The probabilisticcharacter of traffic breakdown is examined next, followed by a sec-tion proposing the discrete time Markov chain approach to modelthe breakdown event. The next section discusses the travel time dis-tribution conditional on flow rate and computes travel time reliabil-ity reflecting breakdown probability. Conclusions and remarks onpossible directions for future investigation end the paper.

PROBABILISTIC DESCRIPTION OF FLOW BREAKDOWN

This study is concerned with predicting travel time reliability alongalternate paths to the destination. Specifically, given historical andreal-time traffic measurements from road sensors, the problem is tocharacterize the breakdown event and to quantify travel time relia-bility. In this section the probabilistic nature of flow breakdown isdiscussed.

Several empirical studies have documented that flow breakdowndoes not necessarily occur at the same location, nor at the same pre-vailing flow level or nominal maximal flow; rather it occurs at flows

Flow Breakdown and Travel Time Reliability

Jing Dong and Hani S. Mahmassani

203

This paper proposes a methodology for online prediction of travel timereliability based on real-world measurements in light of the probabilisticcharacter of traffic breakdown. A discrete time Markov chain is devel-oped to represent the flow breakdown event. Through the calibrated tran-sition probability matrix of the Markov chain, the probability of flowbreakdown or recovery along a given facility can be estimated. Traveltime reliability that captures the probability of flow breakdown and theconditional expected delay associated with occurrence of breakdown cantherefore be predicted. Because both the mean and the variance of traveltime are represented as functions of current traffic conditions and cali-brated for each road section on the basis of field data, travel time andits reliability could be obtained offline by analyzing historical data andcomputed online when real-time measurements are available.

Although most existing studies, as well as prevailing advanced trav-eler information system practice, have considered the mean traveltime as the sole measure of interest to drivers, travel behaviorresearchers have long recognized that various factors other than traveltime govern driver route choice (1). In particular, travel time reliabil-ity is recognized as an important attribute in travelers’ route anddeparture time choices (2). However, there is no single agreed-ontravel time reliability measure. Different researchers have used differ-ent definitions of transportation network travel time reliability (3–7).For example, Nicholson et al. defined it from the road user’s perspec-tive as the probability that a trip could reach its destination within agiven period, which might be applied at the path or origin–destination(O-D) level (3). Focusing on the unreliability due to capacity varia-tions arising from network degradation, Chen et al. defined the traveltime reliability for each O-D as the probability of maintaining a satis-factory level of service (LOS) despite the capacity degradation on cer-tain links in the network (4). Clark and Watling examined reliabilityat the network level and estimated the probability density function forthe total travel time (6). In this study travel time reliability refers tothe variability of travel times because it affects the probability that atrip can be successfully completed within a certain time. Reliabilitybecomes more critical under congested traffic conditions in whichtravel time is expected to be longer and more variable. On the basisof data collected in Los Angeles, California, Chen et al. confirmedthat under LOS A through E, the travel time was low and consistent,whereas under LOS F the travel time was three times longer and thestandard deviation was also much greater (7). Although travel timeunreliability could be due to a variety of factors, some endogenousand others exogenous to the traffic stream, the concern here is with

Transportation Center, Northwestern University, 600 Foster Street, Evanston IL 60208. Corresponding author: H. S. Mahmassani, masmah@northwestern.edu.

Transportation Research Record: Journal of the Transportation Research Board,No. 2124, Transportation Research Board of the National Academies, Washington,D.C., 2009, pp. 203–212.DOI: 10.3141/2124-20

204 Transportation Research Record 2124

FIGURE 1 Flow rate and speed time series for I-405N (Jeffrey section; observed May 30, 2008).

FIGURE 2 I-405N, Jeffrey section. (Source: Google map.)

that may be lower or higher than the traditionally accepted (engineer-ing) capacity (8–11). Although traffic scientists have long debatedthe mechanisms by which such breakdown occurs, and whether apurely deterministic explanation could be provided (e.g., transientor moving bottlenecks induced by specific driver maneuvers) ver-sus allowing for some inherent irreducible randomness, for practi-cal application purposes the phenomenon is naturally treated as astochastic process.

To model the probabilistic nature of traffic breakdown, previousstudies view the facility capacity as a random variable, with trafficbreakdown occurring when the traffic demand exceeds the randomcapacity (12). That results in a probability of breakdown occurring ata given flow (demand) level. Alternatively, to avoid terminologicalconfusion and remain consistent with the traditional (engineering)definition of highway capacity (as the maximum flow rate that canreasonably be expected to pass through a section of a roadway), the

flow rate observed immediately before traffic breaks down is viewedhere as a random variable.

Figure 1 depicts a time series of 5-min averages of flow and speedon the I-495N freeway at the Jeffrey section in California (13). Asshown in Figure 2, the detector is located upstream of the on-ramp.With reference to Figure 1, flow breakdown can be viewed as a two-phase event including (a) the prebreakdown phase when traffic is freeflowing and (b) the breakdown phase when the queue builds up.Other theories have been developed to model traffic jams from dataobserved at finer intervals (e.g., in seconds), such as Kerner’s three-phase theory (14) and the five different kinds of spatiotemporalcongestion patterns identified by Schönhof and Helbing (15). Thetwo-phase assumption is appropriate to model flow breakdown for thepurpose of this study because it is focused on the flow breakdown’seffect on travel time in an operational setting. The following quantitiesare defined in that context:

• Prebreakdown flow rate, q. The flow rate, expressed as a perlane equivalent hourly rate, observed immediately before the onsetof traffic breakdown.

• Breakdown flow rate, qb. The average flow rate, expressed as aper-lane equivalent hourly rate, observed after the onset of breakdownand before traffic recovers.

• Breakdown duration, tb. The time period between the onset ofbreakdown and the recovery of traffic flow.

The prebreakdown flow rate q and the breakdown flow rate qb

are viewed as random variables, with distribution functions FQ(q)and FQB(qb), respectively. By definition, the prebreakdown flowdistribution at flow rate q, FQ(q), expresses the probability thattraffic breaks down at flow rates less than or equal to q (in the next timeinterval, for a given time discretization). In other words, 1 − FQ(q) rep-resents the probability of flow rate sustaining at q, indicating thetravel reliability at flow rate q. The breakdown flow rate is a deter-minant of the added delay that may be experienced by a travelerwhen breakdown occurs. To estimate the distributions of theserandom variables, speed and flow data at 5-min intervals during a 1-year period (peak period on workdays) are collected along I-405 northbound (Irvine, California) from the PeMS website (13).Traffic breakdowns are detected when the speed drop between twoconsecutive time intervals exceeds a threshold (i.e., minimumspeed difference) and the low speed (i.e., threshold speed) is sus-tained for some time (i.e., minimum breakdown duration). Thethreshold speed is location dependent; for example, Brilon et al.found 70 km/h to be representative for German freeways based onvolume and speed data at a 5-min aggregation level (12). With anexamination of the 5-min averages of flow and speed, 55 mph isconsidered a proper threshold speed for I-405N to part the free-flowing state from the congested state. Because the free-flow speedis 65 mph, the minimum speed difference is defined as 10 mph, thatis, the difference of speed limit and threshold speed. The minimumbreakdown duration is used to eliminate false alarm because offlow fluctuation. Once breakdown occurs it takes a relatively longtime period to recover, so the low speed condition has to sustainfor at least 15 min to be recognized as a “true” breakdown inci-dent. Although identifying breakdown based on speed drops is anintuitive and commonly used criterion, other breakdown detectionmethods, for example, based on detector occupancy (as a surrogateto density), could also be used. To estimate the prebreakdown

Dong and Mahmassani 205

distribution, the maximum likelihood estimation (MLE) of a dis-tribution function with increasing failure rate, based on censoredobservations, is used (12). However, the distribution of averageflow rate during the breakdown can be estimated by MLE on thebasis of uncensored data. The likelihood function is defined as follows (16):

where

n = number of observations;δi = 1, if uncensored, 0, otherwise;

f(•) = probability density function; andF(•) = probability distribution function.

One of the plausible functional specifications for the distributionsFQ(q) and FQB

(qb), namely the Weibull distribution, is used to fit theobservations. The Weibull distribution has been calibrated and sug-gested in the literature for reliability analysis, such as in Al-Deekand Emam (5) and Brilon et al. (12).

where σ is a scale parameter and s is a shape parameter.Figure 3 shows the distribution curves of prebreakdown and break-

down flow rate, calibrated for the I-495N freeway (Jeffrey section)in California, based on field data obtained from the Freeway Per-formance Measurement System (PeMS) website, that is, 5-minaverages of flow rate and speed (13). The dotted portion of the plotis used to complete the curve but corresponds to unrealistic (unob-served) flow rates. The solid line instead represents the realisticrange of flow rates. The difference between two distributions indi-cates lower flow rate when traffic breaks down compared with theprebreakdown flow level, which has been referred to as “capacitydrop” in some studies of freeway bottlenecks (17, 18). The flowreduction is not an exogenous effect caused by a drop in demand orby a queue from another bottleneck downstream. Rather, it is viewedas an inherent or endogenous feature of the bottleneck (19). The phe-nomenon is rooted in the behavioral characteristics of drivers, includ-ing reaction times, risk perception, and driving maneuvers. The

F q eq

s

( ) = −−⎛

⎝⎜⎞⎠⎟1 2σ ( )

L f q F qi ii

ni i= ( ) − ( )[ ] −

=∏ δ δ

1 11

1

( )

FIGURE 3 Weibull distribution curves of flow rate.

flow reduction phenomenon has also been characterized by two-regime or dual-mode traffic theories (20), and more recently inKerner’s three-phase theory (14).

A MARKOV CHAIN APPROACH TO MODEL BREAKDOWN

A discrete time Markov chain is proposed to model the breakdownevent considering the stochasticity in traffic flow evolution. A Markovchain is a stochastic process with the Markov property, that is, thefuture evolution of the process is influenced only by the present stateand is independent of the past states (21). At each step, the system maychange its state from the current state to another one, or remain thesame, according to certain state transition probabilities.

Identification of States

The feasible flow rate range is discretized into several equal-width bins. Speed is categorized into two possible levels, high andlow. Each state i of the Markov chain is defined by a flow–speedpair, namely a flow range [qi, qi+1) and a speed level, as listed inTable 1. Note that the concept of state is different from the traf-fic phases discussed in the section on the probabilistic descriptionof flow breakdown although the speed level indicates whether astate is in the free-flow phase or in the breakdown phase. In fact,because the discretization of flow is arbitrary, the definition of statescould be different depending on the desired resolution. In line withthe speed level, the states are divided into two subsets, that is, SL andSH. The set SH includes the free-flowing states before traffic breaksdown and after it recovers. The set SL includes the congested statesduring breakdown. The transition from a high-speed state i ∈ SH to alow-speed state j ∈ SL represents the onset of breakdown. Con-versely, the transition from a low-speed state i ∈ SL to a high-speedstate j ∈ SH represents the recovery of traffic flow.

206 Transportation Research Record 2124

Transition Probability

The transition probability pertains to the probability of moving fromone state to another and is denoted by pij. The transition probabilitymatrix is therefore defined as follows:

The transition matrix can be partitioned into four blocks: (a) H →H, namely, i ∈ SH and j ∈ SH, traffic continues flowing; (b) H → L,namely, i ∈ SH and j ∈ SL, the onset of breakdown; (c) L → L, namely,i ∈ SL and j ∈ SL, traffic remains congested; and (d) L → H,namely, i ∈ SL and j ∈ SH, traffic starts to recover.

To calibrate the transition matrix speed and flow rate, data at5-min intervals during a 1-year period (peak period on workdays)are collected at the Jeffrey section of I-405 northbound, where recur-rent traffic breakdown occurs during the morning peak period. Themaximum likelihood estimation of the transition probability matrixis visualized in Figure 4. The upper-right quarter of the plot repre-sents the probability of sustaining free-flow conditions, that is, PHH.The lower-left quarter of the plot illustrates state transitions during traf-fic breakdown, that is, PLL. The lower-right quarter characterizes PHL,which corresponds to a flow breakdown event. As shown in Table 2,PHL is a lower triangular matrix, because of the flow reduction phenom-enon during breakdown. Each element in the matrix represents thetransition probability from a free-flowing state to a congested state. Forexample, if the traffic is currently flowing at 2,400 to 2,500 vehiclesper hour per lane (vphpl) (i.e., s15H), the probability of traffic break-

PP P

P P

LL LH

HL HH

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

( )4

P p i j S

p i j S

p

ij

ij

ijj S

= ⎡⎣ ⎤⎦ ∈

≥ ∈

=∈∑

,

,0

1

for all

ffor all i S∈ ( )3

TABLE 1 Definition of States, Set S

State State State StateNumber Name Flow (vphpl) Speed Number Name Flow (vphpl) Speed

1 s1L [1,000, 1,100) Low 16 s1H [1,000, 1,100) High

2 s2L [1,100, 1,200) Low 17 s2H [1,100, 1,200) High

3 s3L [1,200, 1,300) Low 18 s3H [1,200, 1,300) High

4 s4L [1,300, 1,400) Low 19 s4H [1,300, 1,400) High

5 s5L [1,400, 1,500) Low 20 s5H [1,400, 1,500) High

6 s6L [1,500, 1,600) Low 21 s6H [1,500, 1,600) High

7 s7L [1,600, 1,700) Low 22 s7H [1,600, 1,700) High

8 s8L [1,700, 1,800) Low 23 s8H [1,700, 1,800) High

9 s9L [1,800, 1,900) Low 24 s9H [1,800, 1,900) High

10 s10L [1,900, 2,000) Low 25 s10H [1,900, 2,000) High

11 s11L [2,000, 2,100) Low 26 s11H [2,000, 2,100) High

12 s12L [2,100, 2,200) Low 27 s12H [2,100, 2,200) High

13 s13L [2,200, 2,300) Low 28 s13H [2,200, 2,300) High

14 s14L [2,300, 2,400) Low 29 s14H [2,300, 2,400) High

15 s15L [2,400, 2,500) Low 30 s15H [2,400, 2,500) High

ing down and the flow rate dropping to 1,900–2,000 vphpl (i.e., s10L)is 0.4. Finally, the upper-left quarter depicts the transition matrix PLH

in line with the recovery of traffic flow, which is listed in Table 3.The recovery of traffic flow could be a result of congestion elim-ination or demand drop. In the latter case, the flow rate would notrecover to a higher level than the breakdown flow rate. In fact, thelower off-diagonal part of the matrix, corresponding to the transi-tions from a congested high flow rate state to an uncongested lowflow rate state, indicates that situation. Conversely, the upper off-diagonal part characterizes the commonly recognized flow recovery

Dong and Mahmassani 207

process, that is, transition from a congested low flow rate state to anuncongested high flow rate state.

From the probability transition matrix PHL the probability ofbreakdown can be estimated. Specifically, at a prebreakdown flowlevel q ∈ [qi, qi+1) and i ∈ SH, the probability of flow drops to qb ∈[qj, qj+1) and j ∈ SL due to breakdown is pij. The breakdown proba-bility at a given flow level can therefore be estimated using Equa-tion 5. The transition matrix PHL provides the basis for predictingtravel time reliability at a given flow level because the travel timecorrelates to the breakdown flow level.

FIGURE 4 Probability of transition.

TABLE 2 State Transition Probability: Transition Matrix PHL

s1L s2L s3L s4L s5L s6L s7L s8L s9L s10L s11L s12L s13L s14L s15L

s1H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s2H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s3H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s4H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s5H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s6H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s7H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s8H 0 0 0 0 0 0.011 0 0 0 0 0 0 0 0 0

s9H 0 0 0 0 0.0073 0 0 0.0146 0.0219 0 0 0 0 0 0

s10H 0 0 0 0 0.0046 0 0.0092 0.0322 0.0415 0 0 0 0 0 0

s11H 0 0 0 0 0.0031 0 0 0.0367 0.0795 0.0275 0 0 0 0 0

s12H 0 0 0 0 0 0.0024 0.0024 0.0219 0.1046 0.0341 0.0097 0.0024 0 0 0

s13H 0 0 0 0 0 0 0.0083 0.0292 0.0917 0.0583 0.0125 0.0042 0 0 0

s14H 0 0 0 0 0 0 0.012 0.0241 0.0843 0.1084 0.012 0.012 0 0 0

s15H 0 0 0 0 0 0 0 0 0 0.4 0 0 0 0 0

In addition, the duration of breakdown can be derived from thetransition matrix PLH. Because the time interval between two transi-tions is a fixed value ∆t, the breakdown duration is proportional tothe number of steps taken in set SL before shifting to SH. The expectedbreakdown duration is expressed in Equation 6.

TRAVEL TIME RELIABILITY

The travel time reliability measure is intended to convey additionalinformation to drivers for consideration in their route selection, inaddition to the mean travel time itself. Conventional travel timereliability measures are based on day-to-day travel time variabilityor theoretical distributions. In this section travel time reliability ismodeled as a function of the prevailing flow rate on the facility ofinterest.

Travel Time and Flow Rate

The Bureau of Public Roads (BPR) function has been used exten-sively in transportation planning studies and traffic assignmentapplications to capture the dependence of travel time on the averageflow rate (over a sufficiently long observation time interval) (22).However, the BPR function does not capture the dynamic nature oftraffic in the context of real-time traffic management and informationsystems, especially under congested traffic conditions. In a classic ref-

E t q t i tq t S q t ib

H

( ) =( ) =+( ) ∈ ( ) =( ) −

⎛⎝⎜

⎞⎠⎟

∆

1

11

Pr

== −⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

∀ ∈

∈∑

∆tp

i Sij

j S

L

H

11 6( )

Pr ( )s n S s n i p i SL ijj S

H

L

+( ) ∈ ( ) =( ) = ∀ ∈∈∑1 5

208 Transportation Research Record 2124

erence (Gerlough and Huber), it is shown that only when the aggrega-tion interval is long enough (e.g., 60 min) is the shape of the BPR curveappropriate to represent the variation of travel time with average flow(23). With smaller aggregation intervals (e.g., 6 min), longer traveltime delays are observed along with lower flow rate under highly con-gested conditions, and the backward-bending curve is obtained (24).The backward-bending curve can also be derived from common formsof the speed–density relation. For example, consider the two-regimemodified Greenshields model:

where

vf = free-flow speed,v0 = minimum speed,

vcut = speed intercept,cjam = jam density,cbp = breakpoint density, andω = power term.

The parameters of the speed–density equation can be calibrated onthe basis of field data obtained from the California PeMS website, thatis, 5-min averages of density and speed on I-495N in California (13).The calibrated speed–density curve and the scatter plot of the obser-vations are shown in Figure 5. Defining the travel time index as theratio of the travel time to the speed-limit travel time (i.e., the time totraverse the link if traveling at the speed limit), the flow–travel timeindex relation shown in Figure 6 can be derived. The lower part rep-resents the travel time under uncongested or moderately congestedconditions (corresponding to the upper part shown in Figure 5) and ismonotonically increasing with flow rate. The upper part correspondsto the synchronized and congested flow conditions (in line with thelower part shown in Figure 5). Under such conditions, lower flowrates reflect severe congestion, resulting in longer delay.

v v c c

v v v vc

c

f= ≤ ≤

− = −( ) −⎛⎝⎜

⎞⎠⎟

if bp

cutjam

0

10 0

ω

iif bp jamc c c< ≤ ( )7

TABLE 3 State Transition Probability: Transition Matrix PLH

s1H s2H s3H s4H s5H s6H s7H s8H s9H s10H s11H s12H s13H s14H s15H

s1L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s2L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s3L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

s4L 0.0081 0 0.0081 0.0081 0.0081 0 0 0 0 0 0 0 0 0 0

s5L 0 0 0.0064 0 0.0192 0.0128 0 0 0 0 0 0 0 0 0

s6L 0 0 0.0038 0 0.0114 0.0265 0.0265 0.0227 0 0 0 0 0 0 0

s7L 0 0 0 0.0018 0.0018 0.0071 0.0302 0.0266 0.0124 0 0 0 0 0 0

s8L 0 0 0 0.0009 0 0.0053 0.0222 0.0257 0.0035 0 0 0 0 0 0

s9L 0 0 0 0 0 0.0025 0.0056 0.0113 0.0031 0.0013 0.0006 0 0 0 0

s10L 0 0 0 0 0 0 0.004 0.002 0.005 0.004 0.002 0 0.001 0 0

s11L 0 0 0 0 0 0.0035 0 0.0017 0.0052 0.0017 0.007 0.0017 0 0 0

s12L 0 0 0 0 0 0 0 0 0.0075 0.0038 0 0.0038 0.0038 0 0

s13L 0 0 0 0 0 0 0 0 0 0 0 0 0.0256 0 0

s14L 0 0 0 0 0 0 0 0 0 0 0.25 0 0 0 0

s15L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Dong and Mahmassani 209

FIGURE 5 Two-regime modified Greenshields model.

FIGURE 6 Backward-bending flow–travel time index curve.

Travel Time Distribution

In light of its uncertain value, travel time has been viewed as a randomvariable in the literature (2, 25). Given the nonmonotone flow–traveltime relation, this study considers the joint distribution of flow rate andtravel time. Two types of travel time distributions can be inferred: theunconditional travel time distribution depicting the day-to-day vari-ability in travel time and the travel time distribution conditional on flowrate. Given the joint distribution of flow rate and travel time,

the marginal probability density function (pdf) of travel time, orunconditional distribution, can be derived by integrating the jointpdf over the flow rate.

Empirical studies have shown that the travel time distribution fT(t)was not symmetrical. Rather, the distribution is skewed with a flat and

f t f q t dq f t q f q dqT Q T T Q Q( ) = ( ) = ( ) ( )−∞

+∞

′−∞

+∞

∫ ∫, , (( )9

F q t f x y dxdyQ T Q T

qt

, ,, , ( )( ) = ( )−∞−∞∫∫ 8

long right tail. Li et al. suggest that a lognormal distribution best char-acterizes the distribution of travel time when a large (in excess of 1 h)time window is under consideration, especially in the presence of con-gestion (25). However, when the focus is on a small departure timewindow (e.g., on the order of minutes), a normal distribution appearsmore appropriate. The morning peak (7–9 a.m.) travel times collectedon a freeway section of I-405N are used to estimate the marginal dis-tribution of travel time. The travel time index data show that the mean(1.59) and the median (1.48) are to the right of the mode (0.96), whichsuggests a positive-skewed (right-skewed) distribution. In Figure 7 thehistogram is plotted as an approximate density estimator. In addition,the data are fitted to a lognormal distribution.

The conditional distribution of travel time at a given flow rate canbe expressed as follows:

Because the travel time and flow are correlated variables, devel-oping and calibrating continuous joint or conditional distribution arevery challenging. Alternatively, the flow rate can be discretized into

f t qf q t

f qT Q

Q T

Q

( ) =( )( )

, ,( )10

equal-width bins. Hence the conditional distribution can be approx-imated by several distributions of travel times at different flow lev-els. The discretization of flow rate could be, but not necessarily,consistent with the definition of the states in Table 1.

To illustrate the evolution of the travel time distribution at vary-ing flow levels, the peak period observations (a set of travel timeindex–flow pairs averaged over 5-min intervals) are categorized intogroups based on flow level. Figure 8 presents histograms of obser-vations from three flow-rate ranges: 1,400–1,500 vphpl (Figure 8a),1,700–1,800 vphpl (Figure 8b), and 1,900–2,000 vphpl (Figure 8c).The histograms show two distinct peaks, suggesting a bimodal dis-tribution. Therefore a Gaussian mixture model, typically used toaccount for heterogeneity in populations, is used to represent thebimodal feature of the travel time density function at a given flowlevel. The Gaussian mixture model adapted in this context is a con-vex combination of two Gaussian distributions, corresponding to thetravel times under uncongested and congested conditions respec-tively. The probability density function is expressed in Equation 11.

where

αH, αL = coefficients,μH = mean travel time under uncongested conditions (free

flowing),σH

2 = variance of travel times under uncongested conditions(free flowing),

μL = mean travel time under congested conditions (break-down), and

σL2 = variance of travel times under congested conditions

(breakdown).

As shown in Figure 8 the travel time index data at three flow rangesare fitted into three Gaussian mixture distributions. The lower (left)mode, corresponding to uncongested conditions, remains almost thesame with varying flow rate level. In contrast, the higher (right) modemoves toward the lower one as the flow rate increases. At each flowrange, the travel time distribution has two sets of means and variances,corresponding to the congested and uncongested traffic conditions,

f t q e eT Q H

H

t

L

L

tH

H( ) = +−

−( )−

ασ π

ασ π

µ

σ

1

2

1

2

2

22

−−( )µ

σL

L

2

22 11( )

210 Transportation Research Record 2124

respectively. In this fashion, each state defined in Table 1 is associ-ated with either the lower (left) or higher (right) portion of a specificconditional travel time distribution.

Travel Time Reliability

Assume the traffic is currently in state i, and moves to state j in thenext step with a probability of pij. Whether the flow will break downor not and the resulting flow rate at which vehicles move through thefacility in the next time interval largely determine the travel time andits reliability. As discussed previously, each state j corresponds toone of the two sets of the mean and variance that define the traveltime distribution conditional on the flow range (qj, qj+1). Therefore,given the current traffic state i, the mean and the variance of the traveltime in the next time interval can be predicted as follows.

The transition probability pij is a component of the transition prob-ability matrix defined in Equation 3. The μj’s and σj’s are obtainedfrom the calibrated travel time probability density function, definedin Equation 11, at the corresponding flow level. Because the modelhas been calibrated using data from a single detector, the mean andthe variance of travel time correspond to the freeway section inwhich the detector is located. To predict travel time and its variabil-ity along a path, the mean and variance of the constituent links cansimply be summed up assuming all the links are independent. Or thecovariance matrix could be calibrated so as to capture the interactionbetween the links.

CONCLUSION

This paper has proposed a methodology to predict travel time and itsreliability in real time based on real-world measurements in light ofthe probabilistic nature of flow breakdown. The discrete-time Markov

Var T i pijj S

j( ) =∈∑ 2 2 13 σ ( )

E T i pijj S

j( ) =∈∑ µ ( )12

FIGURE 7 Distribution of travel times during peak period (7–9 a.m.).

Dong and Mahmassani 211

(a)

(b)

(c)

FIGURE 8 Distributions of travel times at varying flow levels: (a) flow range of1,400–1,500 vphpl, (b) flow range of 1,700–1,800 vphpl, and (c) flow range of1,900–2,000 vphpl.

chain approach for modeling traffic flow evolution enables predictionof flow breakdown probability, as well as the resulting flow rate. Theexpected duration of breakdown can also be derived from the proba-bility transition matrix. In addition, a bimodal travel time distributionconditional on flow levels, consistent with the backward-bendingflow–travel time curve, reflects the difference in experienced traveltimes when flow breaks down versus when traffic is free-flowing.Therefore given the current traffic condition, the travel time and itsreliability can be predicted by integrating the transition probabilitiesof possible subsequent states and the corresponding travel time distri-butions. Because the flow rate and speed can be readily obtainedfrom sensors that capture prevailing traffic conditions, the proposedmethodology could be used to provide real-time traveler information.

The present paper has illustrated the probability of flow break-down and its effect on travel time. However, improvement is possi-ble in modeling the traffic breakdown phenomenon, as well as inproviding predictive traveler information. For instance, constrainedby data availability, 5-min averages of flow and speed data were usedto calibrate a two-phase traffic flow model that captures prebreak-down and breakdown regimes. With finer observation intervals,intermediate traffic regimes might be observed and represented bymore detailed models. The memoryless assumption that the futurestate depends only on the present state can also be relaxed by includ-ing past states, which calls for a Markov chain of order m (or aMarkov chain with memory m). In addition, based on current trafficconditions, the prediction of flow, travel time, and its reliability ispossible in one-step look ahead, namely predicting for the next 5-mininterval. To better assist travelers planning a trip, prediction of time-dependent travel time and reliability for the near future, that is, mul-tiple steps ahead, is desirable. This can be accomplished through atraveler information provision procedure operated in tandem with areal-time traffic estimation and prediction system (26). Finally, thereliability measure considered in this study is limited to sources asso-ciated with the intrinsic flow breakdown phenomenon. Other sourcesof unreliability, for example, the occurrence of accidents, may alsobe included by extending the framework presented in this paper.

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The Traffic Flow Theory and Characteristics Committee sponsored publication ofthis paper.