Stochastics and StatisticsQueue length estimation from probe vehicle location and the impacts of sample size
Introduction
Queue lengths and delays are two interrelated and fundamental performance measures of signalized intersections. Accurate estimates of these measures in real-time enable optimal control through efficiently allocating the available capacity (i.e., green time) such that a defined performance metric is optimized (e.g., minimize total delays or minimize the maximum queue length). To estimate these performance measures in real-time, various surveillance technologies are being employed today (e.g., inductive loops, video) to measure traffic flow parameters (e.g., volume, density) which are subsequently utilized in models for delay estimation/prediction. Such systems are called real-time traffic-responsive or traffic-adaptive control systems (Gartner et al., 2002). One fundamental component of any traffic-adaptive signal control system is the estimation of delays and/or queue lengths at signalized intersection approaches. This paper focuses on queue length estimation at an isolated intersection in real-time based on data from probe vehicles (i.e., vehicles equipped with GPS and wireless communication technologies). Even though the use of probe vehicles as a traffic data source has been investigated before, for example in the context of travel time estimation (e.g., Cetin et al., 2005), there has not been any significant research effort on investigating the use of probe data for queue length estimation. It is hoped that as the vehicle-based information collection technologies gain momentum, there will be great interest on capitalizing on the probe data for traffic signal timing and other Intelligent Transportation Systems (ITS) applications. In order to support the development of these applications, research is needed to understand how probe vehicle technology could potentially improve the estimation of desired parameters. One key issue that pertains to the vehicle-based data collection systems is understanding the relationship between the market penetration (or the percentage of probe vehicle population) and the accuracy of the estimated parameters. The relationship between the accuracy of the queue length estimates and the probe percentage is explored in this paper.
A number of studies on vehicle probes deals with understanding the relationships between the market penetration and the reliability of the travel time estimates or travel speed (Chen and Chien, 2000, Cheu et al., 2002, Ferman et al., 2005, Lin et al., 2008). Network coverage is also an important issue that is addressed in the literature (Srinivasan and Jovanis, 1996, Turner and Holdener, 1995, Boyce et al., 1991). Due to the complexity of the problem, none of these studies develop analytical models or closed form solutions that relate the number of probes to the reliability of the estimates (except Ferman et al., 2005). Instead, empirical analyses are performed in these studies based on data of numerous scenarios with different probe vehicle percentages. 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.
In this paper, analytical models are developed to assess how queue length estimation is influenced by the percentage of probe vehicles in the traffic stream. These models require the marginal probability distribution of queue length to be known. Even though this distribution may not be readily available, the proposed analytical approach is better than the simple empirical approaches cited above, which require data to be generated for numerous scenarios. The application of the proposed approach is illustrated through two examples for an isolated intersection with fixed signal timing. The arrivals are assumed to follow a discrete distribution such as Poisson whereas the vehicles are assumed to queue vertically for simplicity.
It should be noted that there is a vast body of literature on queues at signalized intersections. Since fixed cycle traffic light allows a detailed analytical analysis, it has been studied by many researchers. One of the earliest studies is 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. In particular, the curve he fitted to the simulation results has been fundamental to traffic signal setting procedures since its development. Miller (1963) found an approximation to the average overflow queue for any arrival and departure distributions. Later, Newell (1965) derived an analytical approximation to the mean queue length for general arrivals. McNeil (1968) derived a formula for the expected delay and approximate mean of the overflow queue length for general arrival process and constant departure rate.
Other than the studies on the estimation of mean, there have been some attempts to obtain the probability function for the queue length. Some researchers obtained the conditional probability distribution of the overflow queue at the end of one period given the queue length at the preceding period assuming homogenous Poisson arrival process (Haight, 1959, Heidemann, 1994, Mung et al., 1996). Yet some others derived the probability generating function (pgf) of the stationary overflow queue (Darroch, 1964) in the hope of obtaining the probability functions for delays and queue lengths. Obtaining a probability function from a pgf involves inverting the pgf function (Abate and Whitt, 1992), and this inversion process is quite complicated and entails finding complex roots and numerical evaluations of parameters (Van Leeuwaarden, 2006). Therefore, the applicability of this procedure is very limited and complicated. Following similar methods presented in Haight, 1959, Olszewski, 1990 used Markov chains to obtain the probability distribution of overflow queue, and developed a computer program that estimates the mean queue length and its variance under different conditions such as stationary and non-stationary arrival processes, and variable service rates. In a recent paper, Van Zuylen and Viti (2007) also used Markov chains to model the dynamics of the probability distributions of the total queue and overflow queue.
Lastly, it is interesting to note that the general idea of using probes to estimate the system state or system performance is not unique to vehicular traffic on roadways. 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 technique, 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., Xi et al., 2006, Duffield, 2006).
This article is organized as follows: Section 2 introduces the problem statement and the notation. Section 3 describes the analytical formulation for the queue length estimation problem. The application of the formulation is presented first for a fixed-cycle traffic light without considering the overflow queue in Section 4. Section 5 addresses the application of the formulation when overflow queue is also considered. Lastly, Section 6 summarizes the findings and results.
Section snippets
Problem statement and notation
Fig. 1 shows a snapshot of traffic at a signalized intersection approach at the end of a red phase. Solid rectangles represent probe 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 N given the probe information, and to evaluate the error when this conditional expected value is used as the predictor of the actual N, where N is the total number
Analytical formulation
Assuming that the proportion of probe vehicles p is known, the relationship between the total number of probe vehicles Np and the total number of all vehicles in the queue N can be written as follows:The equation above asserts that every vehicle has equal probability of being a probe vehicle. It is assumed that the random variables y1, y2, … are independent and independent of N. Both N and Np are two discrete random variables. The formulation here is for a
Analysis of arrivals on red (without overflow queue)
At signalized intersections, if the overflow queue is ignored and the number of arrivals per unit time is assumed to be Poisson with parameter λ then, N is also a Poisson with parameter λR, where R is the duration of the red interval. It should be noted that at isolated intersections the use of Poisson distribution is common for describing the arrival process (Rouphail et al., 2001). Without loss of generality it is assumed that R = 1, then the probability function for N can be written as
Analysis with the overflow queue
In the previous section, the overflow queue (cycle to cycle overflow) or the residual queue at the end of green period is ignored. In this section, the total queue length estimation that also includes the overflow queue due to randomness is presented. It is still assumed that average demand is less than the capacity – undersaturated conditions. Let No be the overflow queue length and Nr be the queue length that occurs from new arrivals during the red period. The total queue length to be
Conclusions
This paper presents a statistical method for real-time estimation of queue length at a signalized intersection approach from probe vehicle data under the assumption that the marginal probability distribution of the queue length P(N = n) is known. Analytical expressions are derived for queue length estimation and for error analysis where the queue length is estimated based on the position or location of probe vehicles in the queue. It is found that, in terms of real-time data, only location of the
Acknowledgements
The authors wish to thank the University of South Carolina for supporting this research. The authors are also grateful to the anonymous reviewers who provided very detailed and insightful comments that helped us improve the quality of this manuscript.
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