Effect of stop line detection in queue length estimation at traffic signals from probe vehicles data

European Journal of Operational Research 226 (2013) 67–76

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

Effect of stop line detection in queue length estimation at traffic signals from probe vehicles data Gurcan Comert ⇑ Department of Physics and Engineering, Benedict College, 1600 Harden St., Columbia, SC 29204, USA

a r t i c l e

i n f o

Article history: Received 16 December 2011 Accepted 28 October 2012 Available online 12 November 2012 Keywords: Traffic Queuing Traffic signals Stop-line detector Estimation Probe vehicles

a b s t r a c t Stop line detectors are one of the most deployed traffic data collection technologies at signalized intersections today. Newly emerging probe vehicles are increasingly receiving more attention as an alternative means of real-time monitoring for better system operations, however, high market penetration levels are not expected in the near future. This paper focuses on real-time estimation of queue lengths by combining these two data types, i.e., actuation from stop line detectors with location and time information from probe vehicles, at isolated and undersaturated intersections. Using basic principles of statistical point estimation, analytical models are developed for the expected total queue length and its variance at the end of red interval. The study addresses the evaluation of such estimators as a function of the market penetration of probe vehicles. Accuracy of the developed models is compared using a microscopic simulation environment-VISSIM. Various numerical examples are presented to show how estimation errors behave by the inclusion of stop line detection for different volume to capacity ratio and market penetration levels. Results indicate that the addition of stop line detection improves the estimation accuracy as much as 14% when overflow queue is ignored and 24% when overflow queue is included for less than 5% probe penetration level. Ó 2012 Elsevier B.V. All rights reserved.

1. Problem definition The vehicles equipped with tracking technologies (e.g., GPS, cell phones) or the so called probe vehicles (PVs) can disseminate both direct and indirect measurements as they traverse traffic network. As PV data get deployed more in the field, a possible problem with the parameter estimation models will be absence of a PV on the link especially for cycle-to-cycle applications unlike fixed detectors which are always available once installed except errors/failures. Since the most deployed detection technology at the intersections today is loop detectors (Ngo-Quoc and Zhu, 2003), this widely available detection can be incorporated into the models to improve the estimation performance at such scenarios. This study develops analytical models that use data from stop line detection (SLD) and PVs to estimate queue lengths at traffic signals and evaluates the accuracy of these estimators using a microscopic simulation environment-VISSIM 5.10, PTV 2008). In order to simply observe the impact of addition of SLD on queue length estimation from probe vehicles, the paper basically revises author’s previous work (Comert and Cetin, 2009, 2011) and presents numerical examples for fixed time controlled isolated and undersaturated intersections. From the technical point of view, the conditional expectation of N given the stop line detection (i.e., actuation with time stamp) and ⇑ Tel.: +1 803 705 4803; fax: +1 803 771 7015. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.10.035

the probe information (i.e., locations and the time instances at which join the back of the queue) is formulated. Then, the steady state error of this conditional expected value from the actual N is derived where N is the total number of vehicles accumulated in the queue at the end of red interval. Throughout the paper, assumptions broadly include that the arrivals follow a Poisson distribution, which is commonly used to describe arrivals at isolated intersections; vehicles accumulate in a vertical (point) queue; a stop line detector is able to provide the actuation and time of the actuation after red duration starts; and PVs are able to provide their location and queue joining time information. For SLD, similar applications can be found in the literature (Chang et al., 2000; Sharma et al., 2007). For PVs, differential GPS systems currently can provide less than a meter accuracy (Du and Barth, 2008; Qing et al., 2011). For the purpose of only locating vehicles on a link, this accuracy would be sufficient. When the position and the time of a vehicle waiting on the link are known, the number of vehicles ahead of the observed vehicle can be estimated by assuming an average length/spacing per vehicle. The concept of using an average or effective vehicle length is common in estimating density and speed based on occupancy (i.e., percentage of time the detector is occupied/activated) measured by inductive loop detectors (e.g., Dailey, 1999; Hellinga, 2002; Coifman and Kim, 2009). Although identical type is assumed, multiple vehicle classes can be incorporated to the models by selecting the average vehicle length to be representative of different vehicle types. Multiple vehicle classes

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Fig. 1. Snapshot of an intersection right before the red interval terminates.

can also be added through specifying arrival rates for each vehicle type. The paper does not discuss information flow architecture and the exact nature of data processing needed to obtain the location or the time of probes in the queue. Moreover, determination of whether a PV is in the queue or not (i.e., waiting or moving) which is usually decided by its speed, acceleration and selected thresholds (e.g., Quiroga and Bullock, 1998) are dealt with in VISSIM simulations (see Appendix C). Fig. 1 shows a snapshot of a signalized intersection approach at the end of a red1 phase in any given cycle. Stop line detector and the detected vehicle are shown as shaded. As an example, the value of the count information for stop line detector H = 1 and the time of actuation T⁄ = 3rd second after the light turns red. Solid vehicles in the figure represent PVs. For simplicity, the location of vehicles in queue is measured in terms of the number of vehicles (i.e., order of vehicle in the queue) both in analytical derivations and in VISSIM evaluations. In the example, there are three PVs and the location of the last probe, denoted by L, is 8. The time instance at which the last probe joins the back of the queue, denoted by T = 35 seconds, which means that the 8th vehicle joins the back of the queue 35 seconds after the red duration begins timing. The value of N is estimated in real-time at the end of red phase based on the location and the time information of PVs and the actuation of SLD. From the practical view, the estimators developed can be used when duration of red is fixed or variable as the probe vehicle and SLD data can be obtained with a snapshot of the intersection at the end of red or cycle durations. To be able to develop simple analytical models, it is assumed that the waiting lane has infinite capacity and vehicles can accelerate and decelerate instantaneously which are the basic assumptions of point queue models. However, the estimation errors of the models are compared using VISSIM which provides a realistic movement and queuing of vehicles. VISSIM estimation errors are calculated by recording vehicle data from VISSIM and processing in Excel VBA. Apart from obvious horizontal versus vertical queuing, some difference is expected between VISSIM and analytical evaluations: because of the car following in VISSIM, for a long approach lane vehicles tend to form platoons as they move downstream which makes the arrival profile not exactly Poisson which is also observed by Viti (2006), number of runs in VISSIM relatively low to show true steady state behavior, in queue definition depends on several factors which is not optimized here, and VISSIM estimation uses discrete time values in seconds for T and T⁄ where analytical formula takes continuous time random variables. 1.1. Literature review In the literature, traffic parameter (i.e., speed, travel time, delay, and queue length) studies typically use data from the loop detectors for estimation. Majority of them allocates dual loop detectors for estimation. For instance, Sharma et al. (2007) allocate SLD data to 1 For interpretation of color in Fig. 1, the reader is referred to the web version of this article.

estimate cycle-to-cycle approach delay and queue length by identifying the arrival flow profiles with phase change and advance passage detector data upstream. Some of the recent work develops algorithms to estimate the parameters from single loop detectors (Guo et al., 2009; Coifman and Kim, 2009) and combination of different data sources (Bhaskar et al., 2011; Qing et al., 2011). The models for queue length estimation are mainly developed in the context of ramp metering. Except, Sheu (2003) predicts the queues beyond the detection zone at signals and lately Liu et al. (2009) presents models for queue length estimation from a stop line and an upstream detector at traffic intersections based on shockwave theory. The research with vehicle probes focuses in general on reliability of the travel time (Cetin et al., 2005) or travel speed estimations and understanding the relationships between the accuracy of the estimations and the market penetrations (Kwella and Lehmann, 2000; Chen and Chien, 2000; Cheu et al., 2002; Ferman et al., 2005; Lin et al., 2008; Sohn and Hwang, 2008; Liu and Ma, 2009; Kianfar and Edara, 2010). Network coverage, which is also an important issue, is addressed (Srinivasan and Jovanis, 1996; Turner and Holdener, 1995; Boyce et al., 1991; Dion et al., 2011). These studies use the direct measurements from PVs (e.g., average travel time, speed) that do not require probabilistic inference for model development. Moreover, the studies are mostly simulation based and develop empirical analyses based on the data for numerous scenarios with different PV percentage levels. Typically, data from microscopic traffic simulation models are used since real-world data with a large number of probes to support such analyses are not available. Queuing at traffic signals is well-developed topic in transportation literature. One of the most fundamental work is done by Webster (1958) who generated relationships for the number of stops and delays by simulating traffic flow on a one-lane approach to an isolated signalized intersection. Related queuing literature can be broadly divided into; the studies involving simulation analysis and curve fitting; efforts to formulate expected value and variance of average queue length (Miller, 1963; Newell, 1965; McNeil, 1968) for different arrival distributions; derivations with and without the overflow queue (the leftover queue from a previous cycle); and approximations to average overflow queue (Adams, 1936; Miller, 1963; Newell, 1965; McNeil, 1968; Ohno, 1978; Tarko and Rouphail, 1994; Fu and Hellinga, 2000; Rouphail et al., 2001; Van den Broek et al., 2006; Viti and Van Zuylen, 2010). Recently with the help of computational power, there are several works on formulating the steady-state probability generating function of overflow queue, numerical evaluation of distribution parameters (Darroch, 1964; Abate and Whitt, 1992; Van Leeuwaarden, 2006), and Markov Chain formulations of total queue lengths (Olszewski, 1990; Viti, 2006). Extensive literature review on the queuing at traffic signals can be seen in (Rouphail et al., 2001; Van Leeuwaarden, 2006; Viti, 2006). A very similar idea to estimate system performance measures in computer communication networks, ‘‘probe packets’’ are sent from a source to one or more receiver nodes in the network in order to deduce the quality of service or performance (e.g., loss rate, delay) at the internal nodes or links (e.g., routers). In this method,

G. Comert / European Journal of Operational Research 226 (2013) 67–76

performance of the internal links/nodes is estimated by exploiting the correlation present in end-to-end (origin to destination) measurements obtained from probe packets (e.g., Bowei et al., 2006; Duffield, 2006; Lawrence et al., 2006). The paper is organized as follows: Section 2 presents an analytical queue length estimator and its errors with numerical examples while omitting the overflow queue. In Section 3, the formulation is presented including the overflow queue. Numerical examples are shown for fixed-cycle traffic light. Also, in this section an approximate model for estimation errors is given. In Sections 2 and 3, for all three models VISSIM comparisons are given. Section 4 summarizes the findings and results. Finally, the set up for simulations and the detailed step-by-step derivations are provided in appendices. 2. Effect of stop line detection for the case q = 0 In this section, the location L, the time instance T at which the last probe joins the back of the queue, and the actuation on the stop line loop detector (i.e., H = 1, T⁄ = t⁄ when there is an arrival, H = 0, T⁄ = 0 when no arrival during red phase) are utilized in the estimation of the total queue length N. Under the assumption that the arrivals follow Poisson distribution, the formulation is carried out for a simplified scenario where the overflow queue Q is assumed to be zero. Detailed step-by-step derivations are provided in Appendix A. A more general case with the overflow queue is investigated in the next section. The total queue length can be written as the sum of two queue lengths: the total number of vehicles up to the last probe (L P 1) or up to the vehicle on the stop line detector (L = 0, H = 1)N1; and the number of vehicles after the last probe (L P 1) or after the vehicle on the stop line detector (L = 0, H = 1)N2. Therefore, the total queue length given the SLD and the information from the last probe can be expressed as:

NjL; T; H; T  ¼ N1 jL; T; H; T  þ N2 jL; T; H; T 

ð1Þ

8 > < Iðl P 1Þ½l þ hDþ EðNjL ¼ l; T ¼ t; H ¼ h; T  ¼ t  Þ ¼ Iðh ¼ 1; l ¼ 0Þ½hD þ pþ > : Iðh ¼ 0; l ¼ 0Þ0:

ð2Þ

8 > < Iðl P 1Þ½hDþ VarðNjL ¼ l; T ¼ t; H ¼ h; T  ¼ t  Þ ¼ Iðh ¼ 1; l ¼ 0Þ½hD  ð1  pÞþ > : Iðh ¼ 0; l ¼ 0Þ0: ð3Þ where

8 > < R  t; l P 1 D ¼ R  t ; l ¼ 0; h ¼ 1 > : R; l ¼ 0; h ¼ 0

and h ¼ ð1  pÞk

The expectation in (2) is an estimator of the queue length given SLD and information of the last PV. From real-life implementation point of view, estimating the total number of vehicles based on SLD and the probe data can be used at the end of red duration. It can be applied either when R is known (i.e., fixed) or on the last second of red duration (i.e., varying R). In addition, the conditional variance of the total queue length given H, T⁄, T, and L can be written as in (3). However, to assess the overall performance, more specifically, how the accuracy changes with respect to the percentage of probes p, the steady state variance of the difference from the observed queue length D = N  E(NjL = l, T = t, H = h, T⁄ = t⁄) is derived.

VarðDÞ ¼ ½½kR  EðLÞ  ekR ½ð2  pÞðehR1 Þ þ hR

ð4Þ

69

Interestingly, the variance of D can be evaluated by a very simple form in Eq. (4). It can be observed that the first term is the variance of D when only probe information is available, whereas the second term is the gain from addition of SLD. 2.1. Comparison with vissim results for the case q = 0 In this subsection, performance of the estimator in (2) is presented using evaluations from Eq. (4) (i.e., independent Poisson arrivals and vertical queuing) and from VISSIM simulations. The set up for simulations is explained in Appendix C. The figures show the accuracy of the analytical model and the impact of SLD in the estimation errors changing by probe percentage (between p = 0 and p  1.0) at different volume-to-capacity ratios. Fig. 2 shows Var(D)s that are calculated directly from (4) and from VISSIM at three q levels. Volume-to-capacity ratios of q = 0.88 and q = 0.98 are not shown as overflow queue is significantly affects the estimation performance. Clearly from the figure, the error decreases as probe proportion increases and it increases as q level increases. In Fig. 2d, the comparison is made inpterms ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiof the percent coefficient of variations (%CV) (i.e.,100 VarðDÞ= EðNÞÞ, where the %CV of the values generated from the analytical model is subtracted from VISSIM. The analytical error formula pretty much follows the estimation errors of VISSIM with less than 3% in %CV for all three q 6 0.80 levels and at any given probe proportion. To see the impact of SLD in estimation errors, Fig. 3 is constructed which depicts the percent difference of VAR(D)s of the two models with and without SLD at three different arrival rates for various probe proportions using the analytical VAR(D) and the errors from VISSIM. Calculated from Eq. (4), the improvement in the accuracy gets as much as 14% at q = 0.60, p  0 and about 10% improvements for higher q values. The highest gain from VISSIM is calculated as 15% at q = 0.60 for p  0. The impact becomes ignorable after 50% probe penetration for both evaluations and for all three q values. 3. Effect of stop line detection the case with q Estimation model with SLD including overflow queue is presented in this section. Overflow queue is defined in this paper as the cycle to cycle residual queue or random queue. In the formulations, it is assumed that the arrival rate is less than the average capacity, i.e., undersaturated conditions. Let Q be the overflow queue length and A be the queue length that occurs due to new arrivals during the red period. The total queue length N is expressed as summation of Q and A (5). At any given cycle, PVs may or may not be present in one or both of Q and A, depending on the SLD five scenarios can be observed: (i) (ii) (iii) (iv) (v)

The last probe is present within the overflow queue. The last probe is present within new arrivals. Detector is actuated by an overflow queue vehicle Detector is actuated by a new arrival No arrival at all.

Using the methodology in the previous section, the queue length estimator in (6) and its error in (7) is derived. See detailed discussions in Appendix B.

NjL; T; H; T  ¼ QjL; T; H; T  þ AjL; T; H; T 

ð5Þ

8 Iðl 2 Q Þ½l þ hðC  t 0 Þ þ hRþ > > > > > > < Iðl 2 AÞ½l þ hDþ EðNjL;T;H;T  Þ ¼ Iðh ¼ 1;Q > 0Þ½1 þ ð1  pÞðEðQ Þ  1 þ kRÞþ > > > Iðh ¼ 1;Q ¼ 0Þ½hD þ pþ > > > : Iðh ¼ 0Þ½0:

ð6Þ

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(a)

(b)

ρ =0.80

ρ=0.70

(d)

(c)

ρ=0.60

ρ=0.60

ρ=0.70 ρ=0.80

Fig. 2. Comparison of Var(D)s and %CVs from Eq. (4) and VISSIM simulation as probe proportion increases at four different arrival rates.

ρ=0.60

ρ=0.60

ρ=0.70

ρ=0.70

ρ=0.80

ρ=0.80

Fig. 3. Effect of SLD in terms of percent difference of errors evaluated as probe proportion increases at various arrival rates by Eq. (4) and by VISSIM simulation.

where I() is the indicator function and T0 is the time information from previous cycle that measures the arrival time of the last probe relative to the beginning of red phase.

8 Pðl 2 Q Þ½hðC  EðT 0 ÞÞ þ hRþ > > > > > > < Pðl 2 AÞ½hðR  EðTÞÞþ VarðDÞ ¼ Pðh ¼ 1; Q > 0Þ½ð1  pÞVarðQ Þ þ hRþ > > > > Pðh ¼ 1; Q ¼ 0Þ½ð1  pÞðkðR  k1 ÞÞþ > > : Pðh ¼ 0Þð0Þ:

ð7Þ

In order to proceed further to obtain a closed-form solution for the Var(D), probability distributions for both Q and T0 (or T0 jl) need to be known. The distribution of Q can be obtained by Markov Chain formulation or by solving complex roots of the probability generating function (Haight, 1959; Heidemann, 1994; Ohno, 1978). Likewise, for the distribution of T0 jl, one needs to know the probability distribution of l (number of vehicles in front of

ρ Fig. 4. Average total queue lengths in number of vehicles for VISSIM, point queue simulation in C++, and approximation model Eq. (10).

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(a)

ρ =0.98

(c)

(b)

ρ=0.88

(d) ρ=0.70

ρ=0.80

(e) ρ=0.60

(f)

ρ=0.60 ρ=0.70 ρ=0.80 ρ=0.88 ρ=0.98 Fig. 5. Difference in Var(D) and %CV with SLD as probe proportion increases evaluated by Eq. (7) and VISSIM.

Fig. 6. Impact of SLD in terms of %difference of Var(D)s as probe proportion increases including the overflow queue for various arrival rates from Eq. (7) and from VISSIM simulation.

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the last probe), which also depends on P(Q = q) (see (B4) in Appendix B). Consequently, obtaining a relatively simple formulation for Var(D) is not straightforward. Instead an approximate method is presented in Section 3.2. Before, some numerical examples that are obtained using simulations are discussed in the next section. 3.1. Comparison with vissim results for the case with q Numerical examples are provided for the case with overflow queue to show how estimation errors as well as the impact of SLD on these estimation errors change as a function of probe proportion (0.0 6 p 6 1.0) and volume-to-capacity ratios q = (0.60, 0.70, 0.80, 0.88, 0.98). Results for various scenarios are obtained by a simulation application developed in C++ (i.e., Poisson arrivals

(a)

(c)

(e)

ρ =0 .98

ρ =0 .80

ρ =0 .60

and point queue) which is exact evaluation of Eq. (7) and by VISSIM as described in Appendix C. Fig. 4 gives average total queue lengths in number of vehicles at the end of red interval for point queue model, VISSIM, and approximation by Eq. (10). It can be seen from Fig. 4 that at these q levels, VISSIM and analytical models generate similar average total queue length. In Fig. 5 Var(D) of the model are given when the overflow queue and SLD are included under two simulation environments. The overflow queue starts to become significant part of the total queue when q gets larger than 0.88 e.g., E(Q)  1.80 vehicles per cycle) where Var(D) becomes especially large at low probe levels. Overall, at all volume-to-capacity ratios and probe proportions, the model in Eq. (7) evaluated by C++ simulation gives reasonable estimation errors that of VISSIM staying within ±3% in %CV.

(b)

(d)

(f)

ρ =0 .88

ρ =0 .70

ρ=0.60 ρ=0.70 ρ=0.80 ρ=0.88 ρ=0.98

Fig. 7. (a)–(e) Comparison of the three models and VISSIM errors: without Overflow Queue, with Overflow Queue, and Approximation in terms of Var(D), and (f) difference between the Approximation model and VISSIM in %CV.

G. Comert / European Journal of Operational Research 226 (2013) 67–76

Fig. 6 shows the percent difference of Var(D) between the estimators without and with SLD when overflow queue is included. Evaluation from VISSIM and the analytical model shows very similar impact of addition of SLD on estimation accuracy. In VISSIM, the estimation errors reduce by 40% for q = 0.98 at p = 20%. For low p 6 5% penetration, improvement reaches up to 27% for q = 0.88–0.98 and about 15% for lower demand levels. The analytical formula shows that the inclusion of SLD improves the accuracy as high as 41% at the same probe penetration for q = 0.98 and stays above 5% for p 6 40% and above 5% for p 6 20% at all q levels. The results are desirable as the effect is observed at low probe proportions and at higher q levels where overflow queue causes great deal of uncertainty. 3.2. Approximate model This subsection provides a simple approximation model for expected total queue length E(N) and Var(D) in (7). The model calculates Var(D) analytically with the addition of SLD while accounting for the overflow queue. In the literature, there are several models for the expected value of overflow queue (Miller, 1963; Newell, 1965; McNeil, 1968), and (Akcelik et al., 1980; Olszewski, 1990; Viti and Van Zuylen, 2010; Ohno, 1978; Tarko and Rouphail, 1994; Fu and Hellinga, 2000; Rouphail et al., 2001; Viti and Van Zuylen, 2010) and relatively fewer models for Var(Q) (Medhi, 1991). In the approximation models below the Pollaczek–Khintchine (K–P) formulas for steady-state M/G/1 queue are used (Ohno, 1978; Tarko and Rouphail, 1994; Fu and Hellinga, 2000; Rouphail et al., 2001). Adopted K–P formula below assumes constant queue discharge rate and a linear relation between overflow queue Q and overflow delay W (i.e., the Little’s law Q = kW) (Tarko and Rouphail, 1994; Fu and Hellinga, 2000). The expectation and variance for the overflow queue are given in (8) and (9). For detailed derivations see Medhi (1991).

EðQ Þ ¼ q2 =2ð1  qÞ

ð8Þ

VarðQ Þ ¼ ½4ð1  qÞq3 þ 3q4 =12ð1  qÞ2

ð9Þ

The Eqs. (8) and (9) are embedded into Eqs. (7) and (11) is derived. Scenario probabilities in (B4) are also approximated by simply using P(L = 0). After testing various weights, E(Q) in (12) is weighted by 0.5 to slowly reduce P(L = 0) for high q values when probe proportion increases.

VarðDÞ ffi

After p > 20%, Eq. (2) can be used for any q < 1.0 level to estimate the total queue lengths under similar assumptions in this study. 4. Conclusions This paper presents analytical models for real-time estimation of queue lengths at a signalized intersection by fusing the information from a stop line detector and probe vehicles. Analytical expressions are derived for estimators of the total queue length and their errors under steady-state conditions and known arrival rate and probe proportion. When the overflow queue can be ignored, a closed form solution for the mean and the variance of the estimator is presented for an isolated traffic signal where arrivals are assumed to follow a Poisson distribution. The estimation of queue length with overflow queue is also investigated. For the models, the effect of SLD especially at various PV penetration levels is illustrated. The impact of SLD reaches as much as 14% improvement in estimation errors at low probe penetration values for low volume-to-capacity ratios when overflow queue is ignored. Including the overflow queue, the impact of SLD shows different behavior having different optimum values at different probe proportions p and volume-to-capacity ratio q values. For less than 5% probe penetration, estimation accuracy improved by 24% for q = 0.98. The highest improvement of 41% in VAR (D) is shown at about 20% probe proportion for q = 0.98. In addition, estimation error of the developed models is compared with errors from VISSIM microscopic simulation. All three analytical estimation error models closely follow VISSIM errors staying within ±3% in %CV at all volume-to-capacity ratios and probe penetration levels. Acknowledgements This research has been partly sponsored by 2011-2012 National Nuclear Security Administration Summer Faculty Research Fellowship through Office of Research, Benedict College. The author also would like to thank the anonymous reviewers for their insightful comments which significantly improved the content of the paper. Appendix A. Derivations of estimators without q

Pðl ¼ 0Þ½ð1  pÞVarðQ Þð1  ðRkÞ1 Þ þ ð1  pÞðkR  1Þþ

Assuming Poisson arrivals with mean kR and the proportion of PVs p is known, estimator of the total queue length without overflow queue and its error at the end of red duration are derived as follows. First, the total queue length can be written as summation of two independent queues in (A1). Taking the expectation of both sides, the conditional expected value of total queue length can be written as in (A2),

ð1  Pðl ¼ 0ÞÞ½hðR  EðTÞÞ:

NjL; T; H; T  ¼ N1 jL; T; H; T  þ N2 jL; T; H; T 

EðNÞ ffi ½EðQ Þð1  ðRkÞ1 Þ þ kR  q (

73

ð10Þ

ðA1Þ

ð11Þ EðNjL ¼ l; T ¼ t; H ¼ h; T  ¼ t  Þ ¼ EðN1 jL ¼ l; T ¼ t; Pðl ¼ 0Þ  ð1  pÞ  ep½kRþ0:5EðQÞ ½1  eð1pÞ½kRþ0:5EðQÞ 

ð12Þ

In Fig. 7, approximate model and VISSIM errors are presented. Generally, the approximation method gives acceptable results staying within ±3% of the VISSIM errors in %CV at all probe proportion and q levels. Also in the figure, Var (D) values of under four different evaluations: without the overflow queue (i.e., analytical w SLD without Q); C++ simulation model with the overflow queue (i.e., analytical w SLD with Q); VISSIM simulation model with the overflow queue; and the approximate model are shown. For probe percentages p < 20% when average overflow queue is Q < 1 Eq. (2) seems very reasonable estimator to use. For Q > 1 and p < 20% the estimator in Eq. (6) can be used.

H ¼ h; T  ¼ t  Þ þ EðN2 jL ¼ l; T ¼ t; H ¼ h; T  ¼ t Þ

ðA2Þ

The conditional expectations are evaluated depending on the value of L and H. First conditional expectation is constant since L and H are given and E(N1jL = l, T = t, H = h, T⁄ = t⁄) changes according to the values of L and H is shown in (A3),

8 > < l; l P 1 EðN 1 jL ¼ l; T ¼ t; H ¼ h; T  ¼ t  Þ ¼ 1; h ¼ 1; l ¼ 0 > : 0; h ¼ 0; l ¼ 0

ðA3Þ

The expected value of N2 corresponds to the number of arrivals in a time interval that is equal to the difference between the red phase and the random arrival time of the last probe or the

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actuation time of the stop line detector if no probe exists as shown in (A4), where R denotes the length of red phase. Since no PV is present for the time period, the arrival rate during this period is given as in Eq. (A4) where k is the arrival rate for all vehicles and p is the proportion of probes in the traffic stream. To simplify notation, k(1  p) is denoted by h. Then, N2  Poisson(hD) or Poisson[hD  (1  p)], these observations are true for the cases and varies in formulation when L > 0, L = 0, H = 1, and H = 0.

Appendix B. Derivations of estimators with q

8 > < Iðl P 1Þ½hDþ EðN2 jL ¼ l; T ¼ t; H ¼ h; T  ¼ t  Þ ¼ Iðh ¼ 1; l ¼ 0Þ½hD  ð1  pÞþ > : Iðh ¼ 0; l ¼ 0Þ0:

Taking expectation of both sides, the conditional expectation of the total queue length becomes as follows.

ðA4Þ 8 > < Iðl P 1Þ½l þ hDþ EðNjL ¼ l; T ¼ t; H ¼ h; T  ¼ t  Þ ¼ Iðh ¼ 1; l ¼ 0Þ½hD þ pþ > : Iðh ¼ 0; l ¼ 0Þ0:

ðA5Þ

8 > < R  t; l P 1 D ¼ R  t ; l ¼ 0; h ¼ 1 > : R; l ¼ 0; h ¼ 0 Estimator of the total queue length (A5) can be obtained by adding two expectations (A3) and (A4). The error of this estimator can be calculated by using the scenario probabilities in (A6). Simply, probability of the scenario 1 is that the probability of having at least one PV. Probability of the third scenario is having no vehicle arriving to the approach. Hence, probability of the second scenario can be found as it is complement of scenarios 1 and 2.

ðA6Þ

Since the first term in (A1) is constant (N1 = l or 1), its variance is zero. The second term specifies a Poisson distribution with a parameter either hD or hD  (1  p), which gives the conditional variance in (A7) that represents the conditional variance when t or t⁄ is known.

8 Iðl P 1Þ½hDþ > <   VarðNjL ¼ l; T ¼ t; T ¼ t Þ ¼ Iðh ¼ 1; l ¼ 0Þ½hD  ð1  pÞþ > : Iðh ¼ 0; l ¼ 0Þ0: ðA7Þ where D is a random variable that measures the difference between the actual queue length and the estimated queue length. The variance of D can be found as shown below in (A8) by using the laws of expectation.



ð1  epkR Þh½R  EðTÞ=ð1  epkR Þþ 

ðePkR  ekR Þ½ð1  pÞ½kðR  EðT ÞÞ  1

H ¼ h; T  ¼ t  Þ þ EðAjL ¼ l; T ¼ t; H ¼ h; T  ¼ t Þ

ðB2Þ

8 Iðl 2 Q Þ½l þ hðC  t0 Þ þ hRþ > > > > > Iðl 2 AÞ½l þ hDþ > > > < Iðh ¼ 1; Q > 0Þ½1 þ ð1  pÞ EðNjL ¼ l; T ¼ t; H ¼ h; T  ¼ t Þ ¼ > ðEðQ Þ  1 þ kRÞþ > > > > > Iðh ¼ 1; Q ¼ 0Þ½hD þ pþ > > : Iðh ¼ 0Þ½0: ðB3Þ where I() is the indicator function. Eq. (B3) involves a new variable T0 that is the time information when the last probe is within the overflow queue. From the practical perspective, T0 is the time information from previous cycle that measures the arrival time of the last probe relative to the beginning of red phase. T0 is subtracted from cycle time C to determine the time interval during which no PV is arrived. Similarly, T⁄ is the time information from stop line detector, that is continuous actuation meaning a queued vehicle. The time of actuation for a vehicle in overflow queue is zero. Based on the actuation of stop line detector and possible locations of the last probe in the queue, there are five scenarios as mentioned above. In the first case, the last probe is present within the overflow queue. The probability of this scenario simply corresponds to having at least one PV in Q and no PV in A. The probability of the second scenario is the same as the probability of having at least one probe in A. The probability of the third scenario is having no PV both in Q and A at the same time the detector is actuated by an overflow queue vehicle. In the forth scenario while there is no PV in the queue, the detector is actuated by a new arrival. The probability of the last scenario is having no vehicle in the queue. These probabilities are given in (B4) below.

Pðl 2 Q Þ ¼ ½1 

X ð1  pÞq PðQ ¼ qÞekRp ; q¼0

Pðl 2 AÞ ¼ 1 

X ð1  pÞa PðA ¼ aÞ;

# X q ð1  pÞ PðQ ¼ qÞ ð1  ekRp Þ; Pðh ¼ 1; Q > 0Þ ¼ 1 



¼ l; T ¼ t; H ¼ h; T ¼ t Þ VarðDÞ ¼

ðB1Þ

EðNjL ¼ l; T ¼ t; H ¼ h; T  ¼ t  Þ ¼ EðQ jL ¼ l; T ¼ t;

a¼0

VarðDÞ ¼ Var½N  EðNjL ¼ l; T ¼ t; H ¼ h; T  ¼ t  Þ ¼ E½VarðNjL 

NjL; T; H; T  ¼ QjL; T; H; T  þ AjL; T; H; T 

Expressing (B2) as the sum of five scenarios, the estimator for the total queue length becomes,

where

8 pkR Þ; > < Pðl P 1Þ ¼ ð1  e Pðh ¼ 1; l ¼ 0Þ ¼ ðepkR  ekR Þ; > : Pðh ¼ 0; l ¼ 0Þ ¼ ðekR Þ;

Assuming Poisson arrivals with mean kR and known probe proportion p, estimator of the total queue length with overflow queue and its error at the end of red interval are derived. First, the total queue length given the data is expressed as follows.

ðA8Þ

Pðh ¼ 1; Q ¼ 0Þ ¼

"

X

ðB4Þ

q¼0

ð1  pÞq PðQ ¼ qÞð1  ekRp Þ;

q¼0

The expected value of T and T⁄ are needed to compute the variance given in (A8). Derivation of E(T) = [E(L)  pkR]/h can be found in (Comert and Cetin, 2011). E(T⁄) is just the expected value of arrival time of the first vehicle when arrival rate is k then E(T⁄) = 1/k. The variance of D can then be found by inserting E(T) and E(T⁄) into Eq. (A8). Finally, error of the estimator (A5) is:

Pðh ¼ 0Þ ¼

X ð1  pÞq PðQ ¼ qÞekR ; q¼0

The estimation error, D, can be found using the same reasoning as in (A7) taking the expected value of (B5), Var (D) is obtained as the total of the errors in five scenarios, and expressed in (B6).

VarðNjL ¼ l; T ¼ t; H ¼ h; T  ¼ t  Þ ¼ VarðQ jL ¼ l; T ¼ t; VarðDÞ ¼ ½½kR  EðLÞ  ekR ½ð2  pÞðehR  1Þ þ hR

ðA9Þ

H ¼ h; T  ¼ t  Þ þ VarðAjL ¼ l; T ¼ t; H ¼ h; T  ¼ t Þ

ðB5Þ

G. Comert / European Journal of Operational Research 226 (2013) 67–76

8 Pðl 2 QÞ½hðC  EðT 0 ÞÞ þ hRþ > > > > > > < Pðl 2 AÞ½hðR  EðTÞÞþ VarðDÞ ¼ Pðh ¼ 1; Q > 0Þ½ð1  pÞVarðQ Þ þ hRþ > > > > Pðh ¼ 1; Q ¼ 0Þ½ð1  pÞðkðR  k1 ÞÞþ > > : Pðh ¼ 0Þð0Þ:

ðB6Þ

Appendix C. Set up for simulations In this appendix, VISSIM and point queue simulation set ups are explained. VISSIM generates vehicles with exponential interarrival times at the origin that move and queue realistically. The arrival profile changes as vehicles move along the network based on the vehicle composition, vehicle characteristics, driving behavior, number of lanes, and other network settings. Similar to the complex traffic system with multiple parameters to control, a very detailed analysis can be done in order to make sure the fully accurate comparison under different scenarios which is beyond the scope of this paper. For a basic evaluation purpose, default parameters for vehicle characteristics (e.g., speed distribution (29.8 miles per hour, 36 miles per hour), acceleration/deceleration, type 100 car), car following characteristics, and driving behavior are adopted within a single lane fixed time controlled isolated intersection located 4921 feet downstream of a total 9797 feet link. The approach length is tested for having enough capacity to hold long queues. Fixed time signal with Red = 45 seconds, Green = 45 seconds, Cycle Length = 90 seconds, and no amber time is programmed. Given these parameters, two points are critical in order to compare the estimation errors of VISSIM with developed analytical models. First, saturation headway (i.e., reciprocal of saturation flow) is needed to obtain the volume-to-capacity ratios. Second, since the models estimate the total queue lengths at the end of the red interval using SLD and probe vehicles, a sensitive queue definition is needed to be able to accurately capture the information from stopped vehicles. Saturation flow rate is determined as follows. First, VISSIM 5.10 user manual is reviewed. The manual gives saturation headway 1.80 seconds per vehicle (spv) for Wiedemann 74 Car Following Model under very similar assumptions. Second, for the designed intersection it is checked with several simulation runs the saturation flow is around 1000 vehicles per hour (vph). Third, offline analysis under the special evaluation tool is used with multiple test runs which showed discharge rates changing 1.79–1.83 seconds per vehicle. Hence, the Saturation Headway of 1.80 seconds per vehicle is used throughout this study. It is recorded that approximately 24 vehicles could be discharged within a cycle yielding to Effective Green  1.8  24 = 43.2 seconds. Based on these observations, 600, 700, 800, 900, and 985 vehicles per vehicle volumes yielding close to q  0.60, 0.70, 0.80, 0.88, and 0.98 are used respectively in VISSIM and analytical model comparisons. VISSIM is run 1000 cycles for four different random seeds (121–124) and 12 probe percentages at each demand level. First 50 cycles of the runs are not included in estimation error calculations for warm up period. Queue length estimation in VISSIM from PVs requires tracking of the vehicles on the link and using the information when they stop. Therefore, the detection zone needs to be longer when the flow gets close to the saturation level as vehicles stop at well upstream of the intersection. Queue definition in terms of min, max speed, and max queue length is given as follows; for arrival rates 600, 700, 800, and 900 vehicles per hour (<6.2 miles per hour, >9.3 miles per hour, and 1640.5 feet); and for close to saturated flow 985 vehicles per hourgf (<9.3 miles per hour, >12.4 miles per hour, and 3281 feet) are used. Notice that queue is defined rather relaxed as vehicles stop and go at very low speeds near saturation flow level.

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