stochastic nature of freeway capacity and its estimation

Stochastic nature of freeway capacity and itsestimation

Abishai Polus and Moshe A. Pollatschek

Abstract: The main purpose of this study was to investigate the meaning of freeway capacity, in particular to exploreits stochastic nature and to estimate its distribution. The evaluation is based on traffic data from three busy urban free-way sections over 3 full days. Momentary capacity is defined as the intersection of the best-fit regression lines calcu-lated for the dense- and unstable-flow regimes close to the maximum flow. An algorithm was developed for theselection of the relevant pairs for each regression line and is discussed. It is argued that momentary capacity values arestochastic in nature and distributed according to the shifted gamma distribution. Estimation of the parameters of thisdistribution for the three urban freeway sections is studied. It is proposed that the 5th percentile of the distribution,found to be approximately 2330 vehicles per hour per lane for the prevailing conditions of the basic section, could beadopted as the representative design value of capacity. Another conclusion is that the average distribution of capacitiesfor all three through lanes is relatively close to the distribution of capacities in the middle lane.

Key words: freeway flow, capacity, flow breakdown, dense flow, unstable flow.

Résumé : L’objectif principal de cette étude était d’effectuer une recherche sur la signification de la capacité autorou-tière, et en particulier d’explorer sa nature stochastique et d’estimer sa distribution. L’évaluation est basée sur des sta-tistiques de circulation provenant de trois sections d’autoroutes urbaines chargées. Ces statistiques portent sur troisjours complets. La capacité momentanée est définie comme l’intersection des droites de régression calculées pour ledébit dense et les régimes de débits instables approchant le débit maximal. Un algorithme permettant de sélectionnerles couples significatifs de chaque droite de régression a été développé, et est discuté. Il est démontré que les valeursde capacité momentanée sont de nature stochastique et distribuées selon la distribution gamma décalée. L’estimationdes paramètres de cette distribution pour les trois sections d’autoroutes urbaines est étudiée. Il est suggéré que les cinqcentiles de la distribution, qui se sont avérés correspondre approximativement à 2330 véhicules par heure et par couloirde circulation pour les conditions dominantes de la section de base, pourraient être adoptés comme valeur de concep-tion représentative de la capacité. Une autre conclusion est que la distribution moyenne des capacités pour chacune destrois voies de circulation est relativement proche de la distribution des capacités de la voie du milieu.

Mots clés : débit autoroutier, capacité, interruption du débit, débit dense, débit instable.

[Traduit par la Rédaction] Polus and Pollatschek 852

Introduction

In most large metropolitan areas, freeways are very likelyto be congested during peak periods, which commonly lastfor several hours, first on weekday mornings and then in theevenings. Frequent breakdowns, particularly when the vol-ume of traffic approaches capacity during peak periods, cre-ate unstable-flow conditions, resulting in random andrecurrent congestion. In a congestion state, the flow is denseand susceptible to breakdowns, because of incidents, acci-

dents, and repeated interruptions associated with heavy vol-umes. When breakdowns are frequent, it becomes necessaryto consider the addition of through-lanes to certain basicfreeway sections between interchanges as warranted by thetraffic-volume increase, the disturbing traffic congestion,and the insufficient capacity of the facility.

In queue-discharge situations, congestion can be mea-sured; however, in other dense-flow conditions congestion isnot easily defined and the observation of freeway trafficleads to the frequent question of whether the traffic flow isapproaching “congestion” or is already “congested.” Notethat congestion happens initially in dense flow, not necessar-ily in unstable flow (“Congestion … can be thought of as arestriction or interference to normal free flow;” AASHTO2001, p. 76). This problem, although it may seem abstract,has an impact on several practical issues, such as travelsafety, travel times, and proper terminology on variable-message signs. It also influences a decision on when to addlanes to an existing freeway or tollway. The congestion levelis sometimes characterized as a specific volume to capacityratio. It is then necessary, however, to know the capacity ofthe section, which is typically assumed to be a deterministicvalue. This paper advances reasons why it is not.

Can. J. Civ. Eng. 29: 842–852 (2002) DOI: 10.1139/L02-093 © 2002 NRC Canada

842

Received 11 June 2002. Revision accepted 7 October 2002.Published on the NRC Research Press Web site athttp://cjce.nrc.ca on 6 December 2002.

A. Polus. Department of Civil Engineering, Technion – IsraelInstitute of Technology, Haifa 32000, Israel.M.A. Pollatschek.1 Department of Industrial Engineering andManagement, Technion – Israel Institute of Technology, Haifa32000, Israel.

Written discussion of this article is welcomed and will bereceived by the Editor until 30 April 2003.

1Corresponding author (e-mail: [email protected]).

I:\cjce\cjce2906\L02-093.vpTuesday, December 03, 2002 1:37:57 PM

Color profile: Generic CMYK printer profileComposite Default screen

The stability of the flow and the reduction of congestionare essential to planners, designers, decision-makers, andmotorists because of the importance of free flow and travelsafety. Higher speeds on freeway sections have significanteconomic implications for millions of commuters. Moresteady flow, characterized by stable speeds, flows, and den-sities, is vital to reducing crash rates and improving safety.Other flow characteristics, such as travel comfort, ease ofmanoeuver, and the ability to arrive on time as a result ofhigh operational reliability, are also important to a goodlevel of service.

This study originated from a need to explore the meaningof freeway capacity and, consequently, to develop capacityvalues for freeways. The paper develops an algorithm toidentify the flow rates and speeds to be adopted as thresh-olds for an analysis of the relevant data from which capacitycould be determined. The evaluation is based on observa-tions of flow and speed parameters during 5 min time inter-vals and on simulations that consider the variability ofregression lines fitted to the data. The variability of the traf-fic characteristics also results in a stochastic property of mo-mentary capacity. This stochastic implication can be foundeven in the basic capacity concept presented by the Highwaycapacity manual (Transportation Research Board 2000),which suggests that “capacity is the maximum hourly rate atwhich persons or vehicles reasonably can be expected to tra-verse a point or a uniform section.” The manual furtherstates that “reasonable expectancy is the basis for definingcapacity,” yet no quantitative terms are given to this addeddimension of the term capacity. The random characteristicsof the traffic flow make it necessary to add the random com-ponent and stochastic meaning to the capacity term andthereby improve the prevalent deterministic concept.

Literature reviewThe capacity of a freeway facility has been treated mostly

as a deterministic parameter, although some recent studieshave proposed a probabilistic approach. According to theTransportation Research Board (2000), the maximum sus-tainable rate of flow depends primarily on the geometry; asurrogate variable used is the free-flow speed, which is theoperating speed at low volume. In a general definition of ca-pacity, the Transportation Research Board adds the expec-tancy element for the term. An earlier consideration for theexpectancy element was presented by the AASHTO (2001,p. 74), which considered the maximum rate of flow that “canreasonably be expected.” These two definitions, by theTransportation Research Board and the AASHTO, providean applicable tool for traffic analyses and planning; however,the expectancy component is not quantified.

The deterministic concept of capacity is also accepted bymost textbooks (McShane et al. 1998), which provide an im-portant clarification: operation at capacity is “quite poor”and difficult, although possible, to sustain capacity condi-tions for long periods because of breakdowns. Commonly,therefore, capacity is the rate of flow of the “worst” 15 minof the peak period and does not represent the hourly volume.

Edie (1961), who conducted an early and important study,suggested that traffic flow behaved differently at high densi-ties than at low densities and introduced the concept of twodifferent speed–density models. Several prominent studies of

the interrelationship of speed, flow, and density arediscussed by Gerlough and Huber (1975). An important ex-ample is a study by Drake et al. (1967), who conducted astatistical analysis of speed–density hypotheses and pre-sented a three-segment linear model for the speed–densityregime.

Persaud and Hurdle (1988) found that a drop in speed oc-curs because of a bottleneck upstream and that the speed ofvehicles discharged from the queue varies with their locationin the bottleneck. They demonstrated a vertical drop inspeeds near approximately 1800 vehicles per hour per lane,which clearly explains the phenomenon of traffic breakdownwhen a bottleneck exists. In a later work, Persaud and Hur-dle (1991) discussed several different capacity definitionsand examined several approaches, such as maximum flowand a specified percentile of the flow; they concluded thatthe mean queue-discharge flow was the most suitable. Halland Agyemang-Duah (1991) suggested that capacity must bemeasured in a bottleneck, not in a queue.

The time interval for observations of capacity is of impor-tance and impacts the resulting hourly capacity values. Thetime interval used for aggregation influences the variabilityof the data and its representative traffic-flow parameters,thus affecting capacity. Sachse (1993) recommended com-puting capacity during 5 min time intervals or at identifiedstationary periods in the traffic flow. The time interval rec-ommended by the Transportation Research Board (2000) forcapacity determination is 15 min. Other researchers, likeAgyemang-Duah and Hall (1991), determined capacity (de-fined as queue-discharge flow) on the basis of 15 min timeintervals. Their analysis of 30 s flow data, however, indi-cated that capacity in a bottleneck under free-flow condi-tions was greater than capacity when vehicles dischargedfrom a queue.

Persaud et al. (2001) conducted a study on the breakdownphenomenon and on ways to increase capacity by reducingbreakdowns. They demonstrated the use of the probability ofbreakdown concept as a basis for ramp-metering and in-creasing the throughput of the section studied. Elefteriadouet al. (1995) conducted a study on the probabilistic nature ofbreakdowns at freeway merge junctions and showed that abreakdown may occur at lower flows than the maximum ob-served flow (i.e., capacity). Elefteriadou et al. (1995) con-cluded that breakdown was a probabilistic event thatdepended on ramp vehicle clusters entering the main streamand disrupting the traffic flow.

Hunt and Yousif (1994) explored whether catastrophe the-ory could account for the discontinuous behavior of freewayoperations. They examined short time periods (30 s) andconcluded that the traffic data for freeway operation did notmatch a catastrophe surface but a continuous foldless sur-face.

Data for the analysisData for the analysis were obtained for a busy, modern,

urban freeway facility in Tel Aviv, named the Ayalon Free-way; some of its sections carry well in excess of 100 000 ve-hicles/day. The data were obtained from the control center ofthe Ayalon Authority, which manages and maintains thefreeway, in the form of computer files of densities, volumes,and speeds, by vehicle class, during 5 min intervals for 24 h

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844 Can. J. Civ. Eng. Vol. 29, 2002

over a period of 3 days at three sites. One site (KerenKayemet) is a three-lane “basic” section, as defined by theTransportation Research Board (2000), and has a tangent ge-ometry, distant from an interchange; the nearest decelerationlane is approximately 2000 m away. The second site(Rockach) is a three-lane transition section just prior to aweaving section and has a moderate curvature. The third site(Halacha) is in the middle of a four-lane weaving section,700 m in length, between two busy interchanges, about 1 kmsouth of the second site.

A scatterplot of typical data (Keren Kayemet site, middlelane) is presented in Fig. 1. The data can be divided intothree regimes: free flow, dense flow, and unstable flow, asshown in the figure. Traffic breakdown is defined as thechange of state from dense flow to unstable flow.

It can be observed that the “meeting” zone where thedense points meet the unstable points does not show a verti-cal drop in speeds, as in the case of a queue-dischargemodel, but rather a fuzzy sharp angle between the dense andthe unstable-flow zones. These data, which are repeated inall the sites, lead to the belief that traffic breakdown can oc-cur spontaneously, as a result of high densities combinedwith lane changes and speed changes. A classic example ofwhen speed reduction and variability bring about a spontane-ous breakdown would be when “gapers” at an accident sitecause a traffic jam.

The meaning of capacity and its stochasticnature

The use of the term capacity in transportation engineering

and planning is quite ordinary; it is used primarily inanalyses of level of service for estimations of throughputperformance and for certain traffic-assignment algorithms.The starting point of this study on capacity is the Transpor-tation Research Board (2000) definition as “the maximumhourly rate at which persons or vehicles reasonably can beexpected to traverse a point or a uniform section of a lane orroadway during a given time period ….” This definition,however, lacks three important aspects that need to be inves-tigated: (i) how to deal with capacity as a stochastic quan-tity; (ii) what is the meaning of “during a given timeperiod;” and (iii) how to quantitatively define the maximum.A recommended common method of estimating capacity hasnot been provided, although it is typically assumed that ca-pacity is reached under dense stable flow conditions, closeto when the flow breaks down and is transformed into an un-stable flow, with considerable fluctuations in speed, flow,and density.

After observation of data in a plane of any two of thethree primary parameters of the flow (e.g., flow and speed),it is possible to fit a best-fit parabolic curve to the entire dataset, both stable and unstable flows. This approach is notused, however, especially for queue-discharge situations(Persaud and Hurdle 1988). The capacity can then be esti-mated as the extreme point of this parabola; however, thiswould typically result in an underestimation of capacity be-cause many observations of higher flows would be left out-side the fitted curve.

Another approach is to fit a parabola to the entire free butdense (though stable) flow portion of the data set and an-other separate parabola to the entire unstable-flow portion of

Fig. 1. Scatterplot of speed–volume data.



the data. The intersection point of these two parabolas mayprovide a rational unbiased estimation of the maximum rea-sonable flow while excluding some nontypical outlyingpoints.

An improved estimation could be obtained by the inter-section of two fitted linear regression models to the data, butusing only relevant pairs of flow and speed near capacity,one for the dense flow and one for the unstable flow. Thisapproach has the advantage of simplicity and more conve-nient statistical properties; furthermore, its estimation isbased only on the relevant points in the vicinity of the maxi-mum flow. This intersection is termed momentary capacityin this analysis because it changes continuously with chang-ing flow conditions.

These three capacity estimation methods are presented inFig. 2. The linear approximation for the best-fit regressionlines near maximum flow is the approach adopted in thisstudy.

The estimated linear relationship between speed (s) andflow ( )f for the dense-flow region is denoted as

[1] s b b f= +0 1d d

and for the unstable-flow region as

[2] s b b f= +0 1u u

where s is the average speed for a 5 min interval; f is theflow in the same interval; b0 and b1 are coefficients resultingfrom a regression analysis; and the superscripts d and u rep-resent the dense and unstable flows, respectively.

The standard theory of regression analysis assumes thatthese coefficients are distributed by a bivariate normal distri-bution with a dispersion matrix, D, as

σ σσ σ

00 01

10 11

where σ00 is the variance of b0; σ11 is the variance of b1; andσ01 and σ10 are the covariances of b0 and b1, respectively.

The dispersion matrix measures the deviations of b0 andb1 from their “true” values, β0 and β1; the last two parame-ters are the expected values of b0 and b1. The coefficients b0and b1 are the best-fit estimators of β0 and β1 for the data athand. Because of fluctuations in the data, however, b0 and b1could have assumed other values with the same expectedvalues and dispersion matrix.

Hence, the two best-fit lines are not unique, but are ran-dom, behaving according to the dispersion matrix. There-fore, it follows that their intersection, which is an estimateof momentary capacity, is also not a constant value, but ran-dom, thus providing the stochastic nature of capacity. Thisreasoning is presented in Fig. 3. The fluctuations depend onthe stochastic nature of the traffic flow (namely, the con-stantly changing traffic conditions and driver behavior), traf-fic composition, and obviously “external” parameters suchas the geometry and environmental conditions of the section.

Determination of stochastic momentarycapacity

A prerequisite for fitting the two straight lines to the datanear capacity is the selection of relevant observations for the

© 2002 NRC Canada


Fig. 2. Three capacity estimation methods. �, unstable flow data points; �, dense flow data points.



regression analysis. The experimental 5 min values of flowand speed pairs may be denoted as fi, si. The algorithm forthe selection of relevant points for the respective lines ispresented in Fig. 4 as a flow chart and in Fig. 5 on the flow–speed plane. The call-outs on the arrows in Fig. 4 mean se-lection rules for relevant pairs. The superscripts d and u de-note dense and unstable flows, respectively. Consequently,the set of data pairs belonging to the unstable- and dense-flow regimes are denoted by and , respectively.

Preliminary observations indicated that maximum flow onall these urban freeway sections occurred in the vicinity of80 km/h. Therefore, it was decided to adopt this speed as aninitial threshold between dense and unstable flows. The un-stable points were then considered: to select only those ob-servations with relatively higher flows, it was decided toeliminate the unstable points for which the flow was lessthan 20 vehicles in 5 min. The dense data set was selected insuch a way that the initial sample size of both sets was equaland only those points closest to momentary capacity wereselected. As the pairs were selected for 24 h, the number ofpairs in the unstable set was relatively small in comparisonto all the pairs. This means that the data in the dense setwere concentrated on near momentary capacity. The processand selection rules are further elaborated in Figs. 4 and 5.

After the selection of the relevant pairs in sets and ,regressions were performed to estimate the linear relation-ships given by eqs. [1] and [2]. The results were significant.The p values were generally small; most results were betterthan 0.1%. For a given value of the flow, f, it is possible tocompute the 90% confidence interval of the predicted valueof s. The intervals are shown in Fig. 5 as parabolic lines

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846 Can. J. Civ. Eng. Vol. 29, 2002

Fig. 3. Random intersections of fitted regression lines. �, unstable flow data points; �, dense flow data points.

Fig. 4. Selection rules for the two data sets.



around the regression line. If for a pair (fi, si), si falls outsidethe interval for this fi, then it is called an outlier; otherwise itis called a fitting pair. The data sets and are adjusted bydiscarding the respective outliers and adding the fitting pairsfrom the other sets. In these analyses, only a few pairs werediscarded or added to each set.

The goal of this process of adjusting the data sets was torefine the determination of the intersection of the two linearbest-fit lines defining momentary capacity. It is imperativethat the intersection point is found as accurately as possible;this is dependent on the number of relevant pairs of best-fitlines. To reduce the error of each line, additional points wereadded to each line in the vicinity of the point of intersection.At this stage, it was necessary to conduct a new regressionanalysis (referred to in Fig. 4 as Re-Regress) for the adjustedsets. This re-regression yielded refined estimates of the pa-rameters in eqs. [1] and [2] and the information needed toestimate the dispersion matrices. The results of re-regressions were highly significant. The p values were verysmall; only 5% of the regressions had a value greater than1%, and the median p value of all the regression was 10–8.

The regression analysis used (obtained through MicrosoftExcel®) did not provide the estimate of the covariance, σ01;this, however, was easily reconstructed as

[3] σ σ01 00= − ∑ f

Ni

where fi is the flow value of the observations included in there-regression analysis, and N is the number of speed andflow observation pairs that entered the re-regression.

Momentary capacity (C) is defined by the intersection ofthe two lines in eqs. [1] and [2], given as

[4] Cb b

b b=

−−

0 0

1 1

u d

d u

Since all parameters b are estimated from the data and,therefore, are stochastic, it follows that C is also stochastic.To estimate the confidence intervals of momentary capacity,it is necessary to determine the distribution of the C values.The pair of parameters b for each line (unstable and dense)is normally distributed and statistically dependent; however,the parameters of the two lines are assumed to be independ-ent of each other. There is a slight dependency between thebu and bd parameters because of the “fitting pairs” addedduring the adjustment process, shown in Fig. 4. This de-pendency is ignored, however, because the number of mutualpairs that belong to both sets is small compared with the sizeof the sets.

Note, also, that the parameter estimate of each line has itsown distribution function that can be estimated. Because ofthe division in eq. [4] and the statistical interdependence of b0and b1 on the same line, however, it is not easy to develop thedistribution of C theoretically. Therefore, it is necessary to es-timate this distribution through Monte Carlo simulations.

Simulation process

The momentary capacity of the freeway section is definedas the intersection of the two lines that represent the relevant

© 2002 NRC Canada


Fig. 5. Confidence intervals for the unstable flow and its use in determining fitting pairs and outliers. �, unstable flow data points;�, dense flow data points.



dense and unstable flow–speed pairs, in the general vicinityof maximum flow, which were selected as previously dis-cussed. The simulation process, repeated n times, yields tworandom lines, with parameters b0

ui, b1ui and b0

di, b1di (i = 1,

2, …, n) (note Fig. 3), in such a way that the respective sta-tistical properties of the parameters (i.e., the expected valuesand dispersion matrices) are the same as those found in theregression analyses. To simplify the presentation, the ran-dom-line selection is discussed for only one data set, , butit refers to both sets, and , and was performed on eachset separately. Furthermore, it is convenient to introduce thefollowing vector notation:

[5] b b bu u u T= ( , )0 1

[6] b b bi i iu u u T= ( , )0 1

The estimate of the expected values of bui is bu, its disper-sion matrix is Du, and it is distributed by the bivariate nor-mal distribution. Denote by Z i

1 and Z i2 two independent

standard normal variables selected randomly with expectedvalues zero and the dispersion matrix I (unit matrix) and let

[7] Z Z Zi i iu T= ( , )1 2

It can be shown (see Rubinstein and Melamed 1998, pp. 30–32) that a matrix Au exists, such that

[8] b b Zi iu u u= + Au

Similar computations were performed for the dense regime,, resulting in bdi. Note that for bdi another pair of random

standard normal variables was selected. Now it is possible toestimate momentary capacity for the ith simulation bymeans of eq. [9]:

[9] Cb bb b

ii i

qi i

= −−

0 0

1

u d

d u

Sample values C i are based on a two-stage process: (i) esti-mation of the expected value and dispersion matrix of the re-gression line parameters from the flow–speed points in thegeneral vicinity of the maximum flow; (ii) n randomsamplings of the two lines describing the data pairs in thetwo flow regimes, based on the estimations in the first stage.In the second stage of the process, C i was sampled n = 1000times, resulting in a sizable sample of simulated capacities.The first stage is summarized in Fig. 4 and the second stagein Fig. 6.

Distribution of momentary capacity

Pairs of flow–speed data were obtained for three sectionsof the Ayalon Freeway, for three lanes on each section for3 days each. Each combination of section, lane, and day wasanalyzed separately by the two-stage process discussed inthe previous section. Several theoretical distributions weretested by goodness of fit tests. It was found that the shiftedgamma distribution fitted all the cases with a 5% level ofconfidence. For example, for the data given in Fig. 1, the pvalues for the χ2 test were less than 0.005, and for theKolmogorov–Smirnov test the p value was 0.0428.

The theoretical distribution that best fit most of the 27data sets (for three sections, three lanes, and 3 days) was theshifted gamma distribution, given by

[10] f C C m e C mC m( )( )

( ) ( )/= − ≥− − −1 1

Γ α β αα β

where C is the momentary capacity (number of vehicles in5 min); m is a shift parameter (number of vehicles in 5 min);α is a shape parameter (unitless); and β is a scale parameter(number of vehicles in 5 min). Since there is no apparentreason why any of the 3 days should behave differently (allwere mid-week, clear-weather days), they were combined inthe distribution-fitting process. The resulting shifted gammadistribution parameters for the three lanes at all three sitesare summarized in Table 1. The distribution parameters foran average lane, its flow and speed being the average of allthree lanes, are presented in the last row of Table 1.

The resulting probability density functions for a specificsite, Keren Kayemet (a “basic” section without weaves andnot near an interchange), are shown in Fig. 7. The capacityof the first (right-most) lane is considerably lower than thatof the third lane (left-most), which is also higher than that ofthe middle lane. Figure 7 also shows that some high capaci-ties, close to and more than 3000 vehicles/h, can be obtainedfor the third lane, but with very low probabilities.

Figure 8 presents the distribution of the average momen-tary capacity of all three lanes at the Keren Kayemet site.The mode of distribution (i.e., the most probable capacity) is2400 vehicles/h. It should be noted that this momentary ca-pacity distribution, and all other capacities studied, is basedon actual counts during the busiest periods because the lessbusy periods were eliminated, as discussed previously.Moreover, the flow values that represent capacities includetrucks (i.e., the capacities are for vehicles per hour, not justpassenger cars per hour).

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848 Can. J. Civ. Eng. Vol. 29, 2002

Fig. 6. The simulation process.



Difference between capacities based on 5and 15 min

The determination of capacity in traffic-engineering appli-cations is commonly based on 15 min rates of flow. In thisanalysis, we used 5 min observations of speed–flow pairsbecause it was desirable to increase the statistical power ofthe regression analysis, which is dependent on the number ofobservations. The analyses should also be adjusted to the15 min basis, however.

Without assuming independence between the flow obser-vations, it can be shown (note the Appendix) that the factorby which it is necessary to multiply the 5 min hourly capac-ity to calculate the 15 min based hourly capacity is given by

[11] Faev

aev= +

+3 33 3

where F is the correction factor to convert the 5 min basedcapacity to 15 min based capacity; a is the number of stan-dard deviations between the average flow and the observedmaximum flow; v is the coefficient of variation of flow dur-ing 5 min intervals, defined as the standard deviation dividedby the mean; and e is the correction factor for statistical de-pendence between flow observations.

When the independence assumption is not satisfied, thecovariances are positive because a higher flow in any 5 mininterval tends to be followed by a similarly higher flow inthe next interval, and then e > 1.

Figure 9 shows the correction factor F for three values ofa (i.e., 1.0, 1.5, and 2.0) for the relevant range of product v ×e. The figure shows that F decreases when the product v × eincreases; note that for more dependent flows, e increases.Analysis of the data from the three sites showed that the av-erage value of the coefficient of variation for 5 min intervalsnear the maximum flow is 0.07 and e = 1.3. Therefore, their

© 2002 NRC Canada


Keren Kayemet Rockach Halacha

Lane

m(vehicles/5 min)

β(vehicles/5 min) α

m(vehicles/5 min)


m(vehicles/5 min)


Right 1927 32 6 1596 15 8 1881 16 8Middle 2189 35 6 2120 14 8 2177 22 8Left 2599 42 6 2508 24 8 2348 16 8Average 2234 38 6 2075 19 8 2132 18 8

Table 1. Parameters of the distribution of momentary capacities for the three lanes at all three sites.

Fig. 7. Fitted probability density functions of the capacity for three through lanes.



product is 0.091, which results in a conversion factor, F(from 5 min based capacity to 15 min based capacity), of0.95. This is shown in Fig. 9. The hourly capacity values inFigs. 7 and 8 and Table 1 are computed according to theseconversion factors to make the resulting values commensu-rate with capacity values in the literature.

Conclusions

The main purpose of this study was to explore the defini-tion of capacity and investigate its meaning and explore itsstochastic nature. The evaluation is based on observations oftraffic parameters of a busy urban freeway during 5 min time

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850 Can. J. Civ. Eng. Vol. 29, 2002

Fig. 8. Fitted probability density function of the average capacity of all lanes.

Fig. 9. Correction factor to convert 5 min based capacity to 15 min based capacity.



intervals. The data are composed of pairs of flow and speedover a period of 3 full days. The observed variabilities offlow–speed pairs, particularly during the peak periods and inthe approximate vicinity of the maximum flows, representfluctuations in momentary capacity, i.e., the stochastic na-ture of capacity.

Momentary capacity is defined as the intersection of thebest-fit regression lines for the dense- and unstable-flow re-gimes. This estimation of momentary capacity is based on atwo-stage process: (i) the estimation of the expected valueand dispersion matrix of the regression-line parameters;(ii) a random sampling of the two lines, based on the estima-tions in the first stage. The second stage involved 1000 sim-ulations, resulting in a sizable sample of simulatedmomentary capacities.

Momentary capacity values were found to be random vari-ables, distributed according to the shifted gamma distribu-tion. An estimation of the parameters of the distribution forthree urban freeway sections studied is provided. Accordingto the Transportation Research Board (2000), the capacityshould be defined “during a given time period,” and there-fore the time period issue is important because of the sto-chastic nature of flow. It is not recommended to adopt themode or mean as a representative value for planning pur-poses. Rather, one has to adopt a threshold that provides aconfidence that a selected flow can be sustainable for a longperiod. Although central tendencies of random variables aremeasured by the expected value or mode, for the reasoninggiven it is recommended that the lower 5th percentile of thedistribution be adopted and not any values that measure acentral tendency.

The 5th percentiles of the momentary capacity distribu-tions were 2330, 2140, and 2195, and the modes were ap-proximately 2425, 2200, and 2250 vehicles/h for the basic,transition, and weaving sections, respectively. These 5th per-centile values could be adopted as the representative designvalues of capacity in practical applications. Another conclu-sion is that the average distribution of momentary capacityat all three sites was relatively close to the distribution of ca-pacities in the middle lanes; this may be noticed by compar-ing the parameter values in Table 1 for the middle lane withthe values in the last row in the table.

Further research is suggested on three topics: (i) an estima-tion of the parameters of the shifted gamma distribution ofmomentary capacity from the geometry of the alignment,truck percentages, while considering the pavement conditions;(ii) an estimation of the distribution of capacities for differentfreeway conditions, such as rural locations; and (iii) the devel-opment of models based on the proposed method and as-signed confidence levels for a definition of congestionboundaries and for the addition of lanes on freeways.

AcknowledgementsThis research was supported in part by the Israel Public

Works Department and by the Fund for Promotion of Re-search at the Technion-Israel Institute of Technology.

ReferencesAASHTO. 2001. A policy on geometric design of highways and

streets. American Association of State Highway and Transporta-tion Officials, Washington, D.C.

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Appendix

Consider three consecutive flow observations f1, f2, and f3in units of vehicles per 5 min. The 15 min base observation is

[A1] f = f1 + f2 + f3

Denote by µi and µ the expected values of fi and f, respec-tively, and by V fi( ) and Cov( fi, fj) the variance of fi and thecovariance of fi and fj, respectively. The relations amongthem are

© 2002 NRC Canada




© 2002 NRC Canada

852 Can. J. Civ. Eng. Vol. 29, 2002

[A2] V f V f V f V f f f( ) ( ) ( ) ( ) ( , )= + + +1 2 3 1 22 Cov

+ +2 21 3 2 3Cov Cov( , ) ( , )f f f f

[A3] µ = µ1 + µ2 + µ3

In general, Cov( fi, fj) > 0 because consecutive flows tendto be correlated. Also, for such a relatively short period,such as 5 min, it can be safely assumed that V fi( ) = V fj( ) andµi = µj. Therefore, eqs. [A2] and [A3] can be stated morecompactly as

[A4] V f e V f( ) ( )= 3 21

[A5] µ = 3µ1

when e2 is a factor taking into account the correlations. Ingeneral, 3 ≤ e2 ≤ 1 (e2 = 1 for uncorrelated flows and e2 = 3for identical flows). Typically, however, 3 < e2 < 1.

Multiplying the 5 min based flow observations by 3 isalso an estimate of the 15 min based observation, denoted byf. Estimate f variance and expected value as follows:

[A6] V f V f( ) ( )= 9 1

[A7] µ µ= 3 1

Since capacity is the maximal sustainable flow, deviationslarger than the expected value are considered, for example,µ + a V f( ) or µ + a V f( ) for the two cases for some a > 0.

The ratio between the two is denoted by F and is given as

[A8]µµ

++

a V f

a V f

( )

( )

Substituting eqs. [A4], [A5], [A6], and [A7] into [A8] anddividing by µ1, both the numerator and the denominator,yields

[A9]3 3

3 31 1

1 1

++

� ��

ae V f

ae V f

( ) /

( ) / ]

µµ

The coefficient of variation, v, is V f( ) /1 1µ , which, when

substituting into [A9] yields eq. [11].



stochastic nature of freeway capacity and its estimation

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