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ARTICLE IN PRESS2 G. Comert, M. Cetin / European Journal of Operational Research xxx (2008) xxx–xxxtwo examples for an isolated intersection with fixed signal timing.The arrivals are assumed to follow a discrete distribution such asPoisson whereas the vehicles are assumed to queue vertically forsimplicity.It should be noted that there is a vast body of literature onqueues at signalized intersections. Since fixed cycle traffic light allowsa detailed analytical analysis, it has been studied by manyresearchers. One of the earliest studies is by Webster (1958) whogenerated relationships for the number of stops and delays by simulatingtraffic flow on a one-lane approach to an isolated signalizedintersection. In particular, the curve he fitted to the simulation resultshas been fundamental to traffic signal setting proceduressince its development. Miller (1963) found an approximation tothe average overflow queue for any arrival and departure distributions.Later, Newell (1965) derived an analytical approximation tothe mean queue length for general arrivals. McNeil (1968) deriveda formula for the expected delay and approximate mean of theoverflow queue length for general arrival process and constantdeparture rate.Other than the studies on the estimation of mean, there havebeen some attempts to obtain the probability function for thequeue length. Some researchers obtained the conditional probabilitydistribution of the overflow queue at the end of one periodgiven the queue length at the preceding period assuming homogenousPoisson arrival process (Haight, 1959; Heidemann, 1994;Mung et al., 1996). Yet some others derived the probability generatingfunction (pgf) of the stationary overflow queue (Darroch,1964) in the hope of obtaining the probability functions for delaysand queue lengths. Obtaining a probability function from a pgf involvesinverting the pgf function (Abate and Whitt, 1992), and thisinversion process is quite complicated and entails finding complexroots and numerical evaluations of parameters (Van Leeuwaarden,2006). Therefore, the applicability of this procedure is very limitedand complicated. Following similar methods presented in Haight(1959); Olszewski (1990) used Markov chains to obtain the probabilitydistribution of overflow queue, and developed a computerprogram that estimates the mean queue length and its varianceunder different conditions such as stationary and non-stationaryarrival processes, and variable service rates. In a recent paper,Van Zuylen and Viti (2007) also used Markov chains to modelthe dynamics of the probability distributions of the total queueand overflow queue.Lastly, it is interesting to note that the general idea of usingprobes to estimate the system state or system performance is notunique to vehicular traffic on roadways. In computer communicationnetworks, ‘‘probe packets” are sent from a source to one ormore receiver nodes in the network in order to deduce the qualityof service or performance (e.g., loss rate, delay) at the internalnodes or links (e.g., routers). In this technique, performance ofthe internal links/nodes is estimated by exploiting the correlationpresent in end-to-end (origin to destination) measurements obtainedfrom probe packets (e.g., Xi et al., 2006; Duffield, 2006).This article is organized as follows: Section 2 introduces theproblem statement and the notation. Section 3 describes the analyticalformulation for the queue length estimation problem. Theapplication of the formulation is presented first for a fixed-cycletraffic light without considering the overflow queue in Section 4.Section 5 addresses the application of the formulation when overflowqueue is also considered. Lastly, Section 6 summarizes thefindings and results.vehicles. The main objective is to estimate the total queue length,N, if the locations of probe vehicles in the queue are known. Technically,the goal is to determine the conditional expected value of Ngiven the probe information, and to evaluate the error when thisconditional expected value is used as the predictor of the actualN, where N is the total number of vehicles accumulated in thequeue at the end of red interval.It is assumed that the distance of probe vehicles from the stopbar can be measured by location tracking technologies. For example,today’s differential GPS systems can provide one to three meteraccuracy (USDOT, 2008). For the purpose of locating vehicles ona link, this accuracy would be sufficient. Once the position of avehicle waiting on the link is known (i.e., distance from the stopbar), the number of vehicles ahead of the subject vehicle can beestimated by assuming an average length/spacing per vehicle. Eventhough all vehicles are assumed to be identical in this paper, it ispossible to capture the impact of multiple vehicle classes by selectingthe average vehicle length to be representative of differentvehicle types. (The concept of using an average or effective vehiclelength is common in estimating density and speeds based on occupancy(i.e., percentage of time the detector is occupied/activated)measured by inductive loop detectors (e.g., Hellinga, 2002)). Anotherpossibility to model multiple vehicle classes is through specifyingarrival rates for each vehicle type. Even though incorporatingmultiple vehicle classes (e.g., busses, trucks) into the formulationpresented below is possible, this is avoided to simplify notationand presentation of the main ideas. Furthermore, this paper doesnot discuss information flow architecture and the exact nature ofdata processing needed to obtain the location of probes in thequeue. Albeit important, such details as determining whether aprobe vehicle is in the queue or not (i.e., waiting or moving), whichcan be determined based on its speed, acceleration and selectedthresholds, are not dealt with in this paper.For simplicity, the location of vehicles in queue will be measuredin terms of the number of vehicles. For example, in Fig. 1,there are three probe vehicles and the location of the last probe,L p , is 6. The value of N is estimated in real-time at the end of eachcycle based on the location of probe vehicles. Even though themodels are developed for vertical queues, this figure is includedto clarify the notation. In vertical queuing models, it is assumedthat vehicles do not occupy space and can accelerate and decelerateinstantaneously.The models developed in this paper are based on the assumptionthat only probe vehicle locations are observable. From a technologicalpoint of view, a system that can collect and use othertypes of data (e.g., arrival times of probes or joining back-of-queueevents) might also be envisioned. However, the scope of this paperis limited to those situations where only location data are used. Asit is demonstrated in the next section, the location of the last probein the queue L p is sufficient for estimating the queue length;neither locations of other probe vehicles nor the total number ofprobes vehicles in the queue N p are needed. It is assumedthat the percentage of probe vehicles in the traffic stream p isknown.N=72. Problem statement and notationFig. 1 shows a snapshot of traffic at a signalized intersection approachat the end of a red phase. Solid rectangles represent probeL p= 6Fig. 1. Snapshot of an intersection right before the red interval terminates.Please cite this article in press as: Comert, G., Cetin, M., Queue length estimation from probe vehicle location and the impacts of samplesize, European Journal of Operational Research (2008), doi:10.1016/j.ejor.2008.06.024
ARTICLE IN PRESSG. Comert, M. Cetin / European Journal of Operational Research xxx (2008) xxx–xxx 3In order to estimate N, the conditional probability distributionof N given the probe locations is needed. The next section presentsthe general formulation whereas the following sections providetwo examples and some analyses. Even though the formulationin the next section can be generalized, the two example applicationsgiven in this paper are for a signal with fixed timing. Theapplication to actuated signals is a topic of future research.3. Analytical formulationAssuming that the proportion of probe vehicles p is known, therelationship between the total number of probe vehicles N p and thetotal number of all vehicles in the queue N can be written asfollows:N p ¼ XNi¼1y i where y i 2f0; 1g and Pðy i ¼ 1Þ ¼p: ð1ÞThe equation above asserts that every vehicle has equal probabilityof being a probe vehicle. It is assumed that the random variablesy 1 ,y 2 ,... are independent and independent of N. Both N and N pare two discrete random variables. The formulation here is for ageneral probability mass function (pmf) for N, which is denotedby P(N = n). Given N = n, it follows that the number of probe vehiclesin the queue has the following binomial distribution.PðN p ¼ n p jN ¼ nÞ ¼n p np ð1 pÞ n np ; 0 6 p 6 1;n pn p ¼ 0; 1; 2; ...; n:Given N = n and N p = n p , the probability distribution for the locationof last probe (L p ) can be derived by considering the number of possiblecombinations. It can be verified that the conditional probabilityof L p given n and n p is l p 1n p 1PðL p ¼ l p jN ¼ n; N p ¼ n p Þ¼ ;nn pfor l p ¼ n p ; n p þ 1; ...; n; n p P 1: ð3ÞThis equation can be explained through an example case illustratedin Fig. 1. From Fig. 1, N =7,N p = 3, and L p = 6. The sample space forthe ‘‘experiment” is the possible combinations of choosing n p probe7vehicles from n vehicles, which is equal to . The number of elementsin the event space is equal to the number of possible place-3ments of the remaining probe vehicles, other than the one atposition l p , into the previous slots since l p is fixed. In this example,5 slots are available to place 2 probes. Thus, the event probability isthe number of elements in the event space divided by the numberof elementsin the sample space, which is PðL p ¼ 6jn ¼ 7;n p ¼ 3Þ ¼ 5 7.2 3As mentioned before, the main purpose is to find the conditionaldistribution of the queue length given the location of lastprobe and the total number of probes, i.e., P(N = njL p = l p , N p = n p ).This conditional distribution can be written by using the Bayes’rule as follows:PðN ¼ njL p ¼ l p ; N p ¼ n p Þ¼ PðL p ¼ l p jN p ¼ n p ; N ¼ nÞPðN p ¼ n p jN ¼ nÞPðN ¼ nÞ: ð4ÞPðL p ¼ l p ; N p ¼ n p ÞBy the Total Probability LawPðL p ¼ l p ; N p ¼ n p Þ¼ X1k¼l pPðL p ¼ l p jN ¼ k; N p ¼ n p ÞPðN p¼ n p jN ¼ kÞPðN ¼ kÞ:ð2Þð5ÞThen (4) becomesPðN ¼ njL p ¼ l p ; N p ¼ n p Þ¼PðL p ¼ l p jN p ¼ n p ; N ¼ nÞPðN p ¼ n p jN ¼ nÞPðN ¼ nÞP 1k¼l pPðL p ¼ l p jN p ¼ n p ; N ¼ kÞPðN p ¼ n p jN ¼ kÞPðN ¼ kÞ : ð6ÞSubstituting the respective equations and after some simplifications,the following conditional distribution function for the queuelength is obtainedPðN ¼ njL p ¼ l p ; N p ¼ n p Þ¼ ð1 pÞn PðN ¼ nÞP 1k¼l pð1 pÞ k PðN ¼ kÞ ;for l p P 1:As it turns out, this probability only depends on the location of lastprobe l p and does not depend on the total number of probes in thequeue n p . It should be noted that Eq. (7) is valid for any probabilitydistribution of N. Even though the numerical examples presented inthe next sections are based on Poisson and Negative Binomial distributions,the formulation can be applied for any given P(N = n).Based on the above conditional probability distribution, the expectedqueue length can be easily computed as follows:EðNjL p ¼ l p Þ¼ X1¼ X1n¼l pnn¼l pnPðN ¼ njL p ¼ l p Þð7Þ½ð1 pÞŠ n PðN ¼ nÞP 1k¼l p½ð1 pÞŠ k PðN ¼ kÞ ; for l p P 1: ð8ÞThe equation above gives the mean of the random variable NjLp = lpwhich can be used as a predictor of the queue length given theprobe location information. For real-time applications, this equationcan be used to estimate the queue length, provided that P(N = n) isknown.In order to assess how accuracy of this estimated queue lengthchanges by the proportion of probe vehicles p, some additional resultsare obtained. First, the conditional variance of the queuelength (given l p ) can be written as follows:VARðNjL p ¼ l p Þ¼ X1n 2 ½ð1 pÞŠ n PðN ¼ nÞP 1n¼l p k¼l p½ð1 pÞŠ k PðN ¼ kÞ½EðNjL p ¼ l p ÞŠ 2 ; for l p P 1: ð9ÞSince the above variance depends on l p , a more general measurethat does not depend on l p is needed to determine the effects of pon accuracy. The error in the estimates, denoted by D, can be treatedas a random variable that is the difference between the actualqueue length N and its estimate:D ¼ N EðNjL p ¼ l p Þ: ð10ÞIf the expectation of both sides of this equation is taken, it can beseen that the expected error is zero. (Since E[E(NjL p = l p )] = E(N).)On the other hand, the variance of error is as follows:VarðDÞ ¼Var½N EðNjL p ¼ l p ÞŠ ¼ E½VarðNjL p ¼ l p ÞŠ: ð11ÞThe expected value on the right hand of this equation can be calculatedas follows:EðVarðNjL p ¼ l p ÞÞ ¼ X1l p¼0PðL p ¼ l p ÞVARðNjL p ¼ l p Þ:ð12ÞThis expected value in essence represents the weighted varianceover all possible values of l p . To calculate this weighted variance,the marginal probability distribution of L p is needed. This marginaldistribution can be readily written in terms of the conditional distributionsobtained thus farPlease cite this article in press as: Comert, G., Cetin, M., Queue length estimation from probe vehicle location and the impacts of samplesize, European Journal of Operational Research (2008), doi:10.1016/j.ejor.2008.06.024
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