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Estimating Dynamic Queue Distribution in a Signalized Network Through a Probability Generating Model Yang Lu and Xianfeng Yang
Abstract—Most existing discussions regarding the timedependent distribution of queue length was undertaken in the context of isolated intersections. However, computing queue length distributions for a signalized network with generic topology is very challenging because such process involves convolution and nonlinear transformation of random variables, which is analytically intractable. To address such issue, this study proposes a stochastic queue model considering the strong interdependence relations between adjacent intersections using the probability generating function as a mathematical tool. Various traffic flow phenomena, including queue formation and dissipation, platoon dispersion, flow merging and diverging, queue spillover, and downstream blockage, are formulated as stochastic events, and their distributions are iteratively computed through a stochastic network loading procedure. Both theoretical derivation and numerical investigations are presented to demonstrate the effectiveness of the proposed approach in analyzing the delay and queues of signalized networks under different congestion levels. Index Terms—Probability generating function (pgf), queue distribution, queue spillover, signalized transport network, stochastic network loading, stochastic queue model.
I. I NTRODUCTION
A
CCORDING to Urban Mobility Report 2012 [1], congestion causes urban Americans to travel 5.5 billion hours more and to purchase an extra 2.9 billion gallons of fuel per year. Congestion, as a social problem, has very negatively impacted the service quality of the urban transportation system. As key junctions controlling the traffic flow movements inside the urban networks, researchers have devoted a tremendous amount of efforts to developing reliable and effective queue estimation models for signalized intersections over the past decades. Existing literatures in this field can be categorized into two groups, i.e., offline evaluation models and real-time estimation models. Moreover, offline evaluation models can further be divided into exact analytical models, computational models, and simulation models.
Manuscript received February 1, 2013; revised May 10, 2013 and July 14, 2013; accepted August 13, 2013. Date of publication September 16, 2013; date of current version January 31, 2014. The Associate Editor for this paper was S. Sun. Y. Lu is with the MIT Alliance for Research and Technology (SMART) Laboratory, Singapore 138602 (e-mail:
[email protected]). X. Yang is with the University of Maryland, College Park, MD 20742 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TITS.2013.2279333
Exact analytical models mainly focus on the delay evaluation at isolated intersections with pretimed signal lights. Most early works in this category derive the distribution of queue length under a predetermined arrival process using stochastic queuing theory and equilibrium condition. Representative works in this direction include those by Webster [2], Miller [3], McNeil [4], Darroch [5], Newell [6], Ohno [7], Cowan [8], and Heidemann [9]. Although the above studies have established reasonable mathematical expressions of queue and delay, their dependence on the equilibrium condition limited their application to undersaturated traffic conditions. To overcome such issues, a time-dependent delay formula was introduced later by several studies to evaluate delay of intersections undergoing an oversaturated traffic condition. Studies conducted by May and Keller [10], Kimber and Hollis [11], Akçelik [12], and Akçelik and Rouphail [13] expanded the analytical model into the oversaturated domain. Exact analytical queue models are mathematically concise and computationally efficient. However, rigid mathematical assumptions and less consideration of traffic flow dynamics have rendered them less effective when analyzing signalized networks. Computational models often adopt a more realistic representation of traffic flow. Among all the factors, two phenomena relating to the arterial traffic flow dynamics are usually of the most concern, i.e., platoon dispersion and queue spillover. The concept of platoon dispersion is originally proposed and discussed by Pacey [14] where the discharging vehicle platoon from the upstream intersection gradually spreads as they travel toward the downstream junction. The correlation of statistical properties between the departure and arrival pattern was studied in many literatures, including those by Van As [15], Tarko and Rouphail [16], Geroliminis and Skabardonis [17], and Jain and Smith [18]. On the other hand, queue spillover occurs when the physical size of the queue exceeds the capacity of the link. Since queue spillover blocks entering flows from upstream links, proper treatment of queue spillover is very crucial for oversaturated traffic conditions. Recently, Osorio and Bierlaire [19], Osorio [20], and Osorio et al. [21] have published a series of studies regarding the dynamic demand loading of a signalized network with finite link capacity. In addition, Liu and Chang [22] discussed the signal optimization of arterial corridor experiencing queue overflow and spillback. Compared with exact analytical models, the computational model offers more accurate estimation of queue and delay at the expense of additional computation time.
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LU AND YANG: QUEUE DISTRIBUTION IN SIGNALIZED NETWORK THROUGH A PROBABILITY GENERATING MODEL
Instead of working with macroscopic flow parameters, simulation models attempt to explain the traffic flow phenomena from a more microscopic prospective. The behavior of each individual vehicle is reproduced and aggregated to generate macroscopic network performance measurements such as delay and number of stops. A complete review of simulation models is out of the scope of this paper, and one can refer to Brackstone and McDonald [23] and Hidas [24] for a more detailed introduction on microsimulations. Simulation models are highly faithful once calibrated; however, a huge initial investment makes them a wise choice only for serious and large-scale preconstruction planning decisions. Real-time queue estimation methods take full advantage of flow information captured by traffic sensors, such as flow rate, concentration, and speed. By utilizing real-time measurements, one can significantly enhance the computational accuracy of the model. Studies along this line include those by Sharma et al. [25], Bhaskar et al. [26], Liu et al. [27], Liu and Ma [28], and Ban [29]. Real-time queue estimation models are very useful in ATIS applications such as travel time prediction or adaptive signal control; however, they are not applicable for preconstruction evaluation or design purposes. To summarize, a more systematical method is to be developed to estimate the time-dependent distributions of queues inside a signalized network with generic topology. From a technical perspective, it is not easy to formulate a numerical representation of the distributions of arbitrary random variables, and it is even harder to compute their convolution or nonlinear transformations. However, vehicular queues are nonnegative integer random variables with finite upper bounds (determined by link capacity); therefore, the underlying idea of this study is to convert various operations regarding the queue distributions into vector operations that can be handled more effectively by modern computing machines. Using the probability generating function (pgf) as a mathematical tool, the proposed model recursively updates the queue distributions based on the transition equations developed in this study. Then, the distributions of queues are used to compute other relevant variables such as delay or blockage probabilities. Compared with existing methods, the proposed model embraces the following new features. 1) The entire distribution of queue is iteratively computed at high time resolution (every 2 s); therefore, the periodical accumulation and discharge of queue is stochastically modeled. Compared with batch service queue models, the proposed method can better capture the complex interaction between demand, signal timing, and geometric scale of the infrastructure. 2) Transition of queue distribution between consecutive time slots is first derived based on a combination of queuing theory, traffic flow theory, and basic properties of the pgf. These transition equations serve as the theoretical foundation that supports the recursive updating algorithm. 3) Transition equations are formulated to replicate the stochastic property of the following four basic phenomena of arterial traffic flows: platoon dispersion, queue formation and discharge before signal, flow merging and diverging, and link queue spillover.
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Fig. 1. Queuing network under the (a) vertical-queue concept and the (b) horizontal-queue concept.
The remaining part of this paper is organized as follows. Section II provides basic concepts and assumptions of the proposed model, and Section III presents the formulation of transition of queue distributions using the pgf concept considering both platoon dispersion and downstream blockage impact. In Section IV, the performance of the proposed model is tested and compared against other existing methods and simulation results. Conclusions and future works are summarized in Section V. II. P RELIMINARIES Arterial road networks with signal devices can be modeled as queuing networks composed of links and nodes, where a link represents a section of a homogeneous roadway, and a node represents control points. Traffic flows may stop due to red lights, diverge into several substreams, or merge with other streams when passing nodes. Since signal lights alternate between green and red, queues will accumulate and periodically discharge at the end of each link. Two modeling approaches regarding the queue were previously suggested, i.e., vertical queue and horizontal queue. In the vertical-queue concept, all vehicles stack before the node vertically and, hence, occupy negligible space on the link, whereas in the horizontal-queue concept, the actual physical size of the queue is considered. Fig. 1 illustrates the concept of a queuing network under the vertical- and horizontal-queue concept, respectively. To establish a solid theoretical foundation for later discussion, the following assumptions are introduced. Discrete-time assumption: The study period is decomposed into a sequence of time intervals, each of which is called a slot. The length of each slot is equal to the minimum headway during the effective green time. As such, the traffic flow variables are updated at the beginning of each slot. In addition, green time, red time, and cycle of all intersections are assumed to be some integer multiple of the slot. Hereafter, subscript index t represents the time index, and Δ represents the duration of each time slot. Vertical-queue assumption: All vehicles waiting before the signal stack vertically at the end of each link.
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TABLE I K EY N OTATIONS IN T HIS S TUDY
Fig. 2. Blockage and queue spillover with (a) only one downstream link and (b) multiple downstream links.
Static link travel time assumption: The total travel time of each link is decomposed into on-link travel time (the travel time from the upstream intersection to the stop line of the downstream intersection) and waiting time (the time spent in the queue). The distribution of on-link travel time is related only to the length and design speed of the link. Homogeneous vehicle composition assumption: All vehicles inside the network are identical in terms of both physical characteristics and driving behavior. Note that according to the vertical-queue assumption, the physical size of the queue has no impact on the operation of the network. Such assumption will render the model unrealistic under a congested condition considering the potential impact of queue spillover. Therefore, the proposed model computes the probability of queue spillover according to estimated queue distribution. The modeling of queue spillover and downstream blockage is elaborated in later sections. Moreover, to facilitate our future presentation, some key notations relating to the queuing network are summarized in Table I. III. M ODELING N ETWORK Q UEUE DYNAMICS A. Transition of Queue Distribution Given the time-dependent random arrival Ai,t , the transition of queue on link i between two consecutive time slots can be formulated as ⎧ ⎨ Qi,t + Ai,t − 1, if Qi,t > 0 and t ∈ [1, gi ] if Qt = 0 and t ∈ [1, gi ] (1) Qi,t+1 = 0, ⎩ if t ∈ [gi , gi + ri ]. Qi,t + Ai,t , According to (1), the random variable series {Qi,t } forms a heterogeneous Markov chain. Nevertheless, it is very difficult to solve this heterogeneous Markov chain analytically due to
two facts: 1) the service rate (i.e., discharging rate) of the signal periodically alternates between 0 and 1 and 2) the distribution of arrival Ai,t is time dependent. Thereafter, it would be more practical to compute the Markov chain defined by (1) numerically given nowadays high computation power. Since the transition of queue involves only addition and subtraction of nonnegative integer random variables, the concept of pgf can be utilized. The basic concept of pgf and its application in computing the convolution of integer random variables is explained in Appendix A. The advantage of implementing pgf demonstrates in the following two aspects: From the theoretical viewpoint, pgf can formulate the convolution of integer random variables in analytically tractable form and help us understand the transition mechanism of queues; from the computational viewpoint, pgf converts the transition of distribution into the convolution of vectors, and the fast Fourier transformation (FFT) technique can be applied to significantly reduce computation time. Using the definition of pgf, (1) can be reformulated into the following functional form: Qi,t+1 (z) Ai,t (z)z −1 [Qi,t (z)−Qi,t (0)]+Qi,t (0); = Ai,t (z)Qi,t (z);
t ∈ [1, gi ] t ∈ [gi , gi +ri ]. (2)
Note that in this study, a capital letter followed by a parenthesis represents the pgf associated with the random variable denoted by the letter. For instance, Qi,t (z) represents the pgf of Qi,t . The derivation of (2) can be found in Appendix B. B. Modeling Downstream Queue Spillback Queue spillover occurs when stopped vehicles occupied the entire space of a link and blocked entering traffic from its upstream link. Blockage is another concept associated with queue spillover, which refers to the situation in which the departure rate of a link is reduced to zero due to the queue spillover condition of its downstream links. Hence, blockage will occur on a link when at least one of its downstream links is experiencing queue spillover, as shown in Fig. 2. The recursive transition equation described by (1) should be reconstructed considering the impact of queue spillover, and a set of binary random variable, Bi,t , i.e., 1 if spillover occurs on downstream link Bi,t = (3) 0 o.w.
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where Bi,t is considered as a random variable following a binomial distribution with certain probability. Then, the evolution of queue can be also reformulated accordingly into the following new form: Fig. 3. Platoon dispersion on an arterial link.
Qi,t+1 ⎧ ⎨ Qi,t +Ai,t −1, = 0, ⎩ Qi,t +Ai,t ,
if Qi,t > 0 and Bi,t = 0 and t ∈ [1, gi ] if Qi,t = 0 and Bi,t = 0 and t ∈ [1, gi ] if Bi,t = 1 or t ∈ [gi , gi +ri ]. (4)
In addition, the transition of pgf corresponding to (4) is also derived as in (5), shown at the bottom of the page. It takes a considerable amount of mathematical induction before one can obtain (5). The proof of the above condition is provided in Appendix B. Therefore, there are two kinds of probabilities associated with each link, namely, the blockage probability denoted by βi,t is defined as the probability of Bi,t being equal to one, whereas the queue spillover probability of a link denoted by ϕi,t is defined as the likelihood of physical queue length exceeding its geometric capacity. Thus, based on Fig. 2, βi,t and ϕi,t can be estimated with the following expressions: ϕi,t = P (Hi,t ≥ Ci ) =
βi,t = 1 −
(k) Ci Hi,t (0) k=1
k!
(1 − ϕj,t )
(6) (7)
j∈Γ+ (i)
where Ci is the capacity of link i defined as the maximum number of vehicles storable on the link, and Γ+ (i) is the downstream link set of link i. C. Computing Link Discharging Flow Let Di,t represent the number of vehicles departed during time slot t then, the distribution of Di,t depends on Qi,t , Ai,t , and Bi,t . Thus ⎧ if Qi,t > 0 and Bi,t = 0 and t ∈ [1, gi ] ⎨ 1, Di,t = Ai,t , if Qi,t = 0 and Bi,t = 0 and t ∈ [1, gi ] (8) ⎩ 0, if Bi,t = 1 or t ∈ [gi , gi + ri ]. The transition of pgf of Di,t is given by the following equation: ⎧ ⎨ (1 − βi,t ) {[1 − Qi,t (0)] z +Qi,t (0)Ai,t (z)} , if t ∈ [1, gi ] Di,t (z) = ⎩ 1, if t ∈ [gi , gi +ri ]. (9) The proof of the above equation is given in Appendix B.
Qi,t+1 (z) =
D. Computing Time-Dependent Arrivals Platoon dispersion is an important phenomenon in signalized networks. Pacey [14] first established an analytical model for the platoon dispersion effect, assuming stochastic travel time on the link and deterministic upstream departure and downstream arrival flow rates. Fig. 3 illustrates the basic idea of platoon dispersion. Let τ denote the stochastic travel time from A to B measured in time slots, and in this study, we want to develop a stochastic version of platoon dispersion using the pgf concept. More specifically, given the departure history of the upstream intersection up to time slot t, {Di,k ; k = 1, 2, . . . , t}, how to compute the distribution of downstream arrival Ai+1,t given the on-link travel time distribution fi (τ ). Here, fi (τ ) is the probability density function of on-link travel time, i.e., fτ (t) = p(t ≤ τ ≤ t + 1).
(10)
Note that according to the independent link travel time assumption defined earlier, fi (τ ) is only related to the design features of the link, such as length, design speed, and curvature, but not correlated to the traffic flow volume running on the link. Given (10), downstream arrival Ai+1,t is formulated as a stochastic transformation of {Di,k ; k = 1, 2, . . . , t} based on Lemma 1 and Proposition 1. Lemma 1: Let X be a discrete and nonnegative random variable with pgf denoted by X(z) and R be a Bernoulli random variable independent from X with pgf of R(z) = pz + 1 − p. Then, the pgf of Y defined by (11) is given by (12). Thus Y =
X
R
(11)
i=1
Y (z) = X (R(z)) .
(12)
The following proposition is derived based on Lemma 1 and unravels the important correlation between the upstream departure and downstream arrival in terms of their distribution. Proposition 1: Let Ai,t (z) and Di,t (z) be the pgfs of Ai,t and Di,t , respectively, and T is maximum travel time on the link measured by time slots, then Ai,t (z) =
t
Di−1,k (fτ (t − k)z + 1 − fτ (t − k)) . (13)
k=t−T
The proof of both Lemma 1 and Proposition 1 can be also found in Appendix C.
Ai,t (z)z −1 [Qi,t (z) − Qi,t (0)] + Qi,t (0) (1 − βi,t ) + βi,t Qi,t (z)Ai,t (z), Qi,t (z)Ai,t (z),
if t ∈ [1, gi ] if t ∈ [gi , gi + ri ]
(5)
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Fig. 4. Flow merging and diverging.
E. Flow Merging and Diverging Flow merging and diverging occurs whenever a link is connected with multiple upstream/downstream links. In the merging case, traffic streams discharged from several upstream links are merged to form the inflow of a common downstream link [see Fig. 4(a)], whereas in the diverging case, one discharging flow is splat into several substreams feeding into several downstream links [see Fig. 4(b)]. The pgf of merged flow Fi,t (z) can be computed by directly applying the convolution theorem of pgf (see Appendix A), i.e., Fi,t (z) = Dj,t (z). (14)
Fig. 5.
Arterial network for numerical experiment. TABLE II L ANE C ONFIGURATIONS OF THE E XPERIMENTAL N ETWORK
j∈Γ− (i)
As for diverging, an external route choice model is required to determine the turning fraction to each downstream link δi,j,t , j ∈ Γ+ (i), and Lemma 1 is applied to compute Fj,t (z), i.e., Fj,t (z) = Di,t (δi,j,t z + 1 − δi,j,t )
j ∈ Γ+ (i).
(15)
Note that a direct division of Di,t according to turning fraction ratio will not generate valid results because both departure and arrival flows are restricted to integer random variables. Therefore, Fj,t is not always equal to δi,j,t Di,t . Moreover, the summation of all turning fractions of all downstream links sharing a common upstream link adds up to one for all time t, i.e., δi,j,t = 1 for all i, t. (16) j∈Γ+ (i)
F. Computation Procedure The computation procedure of the proposed model is summarized as follows. Step 1) Initialization. Set current time step t = 0, total period of analysis to T , initialize the input demand volume distribution of all demand generating nodes during study period, Vi,t (z), i ∈ Gn , t ∈ [0, T ]. Set all pgfs of traffic state variables to 1; thus, Qi,t (z) = Ai,t (z) = Di,t (z) = Fi,t (z) = 1 for all i and t, then entering the main loop where procedure steps (2)–(6) are repeated until t > T . Step 2) Compute total inflow of each link. First of all, for each link i ∈ Gn , compute its discharging flow based on Ai,t−1 (z) and Qi,t−1 (z) using (9), then compute the flow running from link i to all its downstream links Di,j,t (z) using (15) as well as the total traffic flow from upstream links Fi,t (z) using (14). Then, the total inflow of each link is equal to the summation of Vi,t and Fi,t thus, Vi,t (z)Fi,t (z) is the pgf of the total inflow of link i at time t.
Step 3) Estimate arrivals observed at stop line Ai,t . Using the stochastic platoon dispersion equation (13), one can estimate Ai,t (z) based on the total link inflow until the previous time step, Fi,k (z), k = 1, 2, . . . , t − 1, and the on-link travel time pgf function fi (τ ). Step 4) Compute downstream blockage probabilities. For each link, compute its queue spillover probability ϕi,t−1 using (6). Then, for each link, compute the downstream queue blockage probability βi,t using (7). Step 5) Update distribution of queue. For each link, compute Qi,t (z) based on Ai,t (z), Qi,t−1 (z), and ϕi,t−1 using recursive equation (5). Step 6) Advance to the next time step. If t > T , then the computation terminates; otherwise, let t = t + 1 and go to step (2) and repeat the above computation process. IV. N UMERICAL E XAMPLES A. Experiment Setup The numerical experiment presented here intends to 1) demonstrate the properties of the proposed model through numerical analysis and 2) evaluate the accuracy and applicability of the
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TABLE III D EMAND VOLUMES AT E ACH O RIGIN W ITHIN E ACH P ERIOD
model with VISSIM, which is a well-established microscopic simulator. First of all, a brief description of the procedures contained in the numerical analysis is presented as follows. 1) A hypothetical arterial network, consisting of two intersections, is used in the experiment (see Fig. 5). Key geometric parameters are summarized in Table II. 2) The entire period of analysis is divided into nine intervals, and the traffic demand is assumed to be constant within each interval. The average demand volumes between all origin/destination node pairs are summarized in Table III. All demands are generated according to Poisson distribution. The duration of each period is set to 200 s. 3) The network performance under all demand scenarios is simulated by VISSIM calibrated with field data. A total of 100 simulation replications have been executed under different initial random seeds to achieve output stability. 4) The queue length of each link is collected from the VISSIM simulator using its embedded queue counter and is later used for model validation. Moreover, the total network delay is also reported by VISSIM as an important network performance indicator. B. Analysis of Average Queues Here, we are going to present the comparison between simulated and estimated average queue length. We start from the discussion of average queue length for at least two reasons: First, the average queue length is directly related with network delay, which is one primary indicator of service quality of the network; second, compared with distributions, the average queue length is more straightforward and easy to understand. Due to space constraints, only some links are selectively displayed. Links are categorized into three groups according to their location and topological characteristics. Outer rim links (such as links 1 and 4) are responsible for loading the demands, intermediate links (such as links 3 and 6) connect adjacent intersections, and turning pocket links (such as links 7, 8, 9, and 10) provide accommodation space for different move-
ment groups to wait for their signal before intersections. Among these three groups of links, intermediate links are of our highest concerns since the transition of queues on those links is collectively determined by signal, upstream filtering, and downstream blockage effects. Therefore, six intermediate links 3, 9, 10 and 6, 7, 8 are selected. Fig. 6(a)–(f) presents the estimated and simulated average queue length of selected links. There are several noticeable points regarding the average queue length shown in Fig. 6. First of all, the distributions of queues are not stationary during the analysis period. Multiple queue plots showed obvious peaks rising from around a 900-s time mark. Therefore, it is certain that cycle failures occurred at both intersections. In addition, the model provides fairly accurate average queue estimation on outer rim links 1 and 4 [shown in Fig. 6(a) and (d)]. The mean absolute error (MAE) of the queue estimation on those two links is 22.5 and 12.8 m, respectively. Given such degree of absolute error, the relative estimation error measured by the mean absolute percentage error (MAPE) is reported as 21.0% and 20.8%, respectively. Meanwhile, the estimated queues on intermediate links and turning pockets demonstrated different degrees of accuracy depending on whether blockage occurred or not. For eastbound traffic, blockage is caused by the heavy through traffic; therefore, queue spillover occurred on link 10 and further propagated toward its upstream link 3; for westbound traffic, blockage is caused by heavy left-turn traffic traveling from link 6 to link 7, which is a left-turn bay pocket. The consequences above two downstream blockages can be observed from high average queues shown in Fig. 6(b) and (f). The MAEs corresponding to the above two links are 56.2 and 28.8 m, respectively. Table IV provides a summary of model performance on different links treating VISSIM outputs as ground truth values. The discrepancy between the model estimation and simulation results may stem from multiple sources. Here, an important concept is the difference between queue size and queue length. Queue size refers to the number of vehicles waiting in the queue, whereas queue length is defined as the distance between the stop bar and the rear position of the
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Comparison of estimated and simulated average queue length of several major links. (a) Link 1. (b) Link 10. (c) Link 9. (d) Link 4. (e) Link 8. (f) Link 7. TABLE IV E XAMINATION OF E STIMATION ACCURACY
queue. The actual physical queue length is affected by many factors, including queue size, average vehicle length, parking headway, distribution of vehicles among different lanes, etc. Microscopic simulations such as VISSIM can incorporate all of the above factors; however, the proposed model only focuses on the estimation of queue size. The analytical model will become too sophisticated if all those factors are considered. Therefore, a tradeoff between complexity and accuracy needs to be balanced. Since the primary intention of the proposed model is for offline evaluation, the reported estimation accuracy is still acceptable taking into account the limited time and human efforts required by the model. In Fig. 7, the time-dependent queue distributions of all six selected links are presented. The time-varying distributions of queue are plotted as 3-D cumulative density functions. In each figure, the x-axis represents time, the y-axis represents the percentiles of the queue distribution at that time point, and the z-axis represents the queue length. Using the distribution of queue length presented in Fig. 7, one can compute the probability of queue spillover on each link according to (6). In Figs. 6 and 7, one can conclude that the proposed model is a powerful tool in predicting the stochastic
property of the queues within the signalized network using only static input information, including network topology, signal timing, and distribution of inflows. Hence, the model is a viable analytical tool for delay estimation, evaluation of geometric design of intersections, and signal optimization. C. Analysis of Network Delay Based on the queue output, the network delay can be computed with simple mathematical manipulation. This section presents the validation of estimated network delay, treating the simulation result as ground truth. Moreover, in order to investigate the impact of simulation duration on the network performance, the period of analysis is set to 30, 45, 60, 75, and 90 min, respectively. For each scenario, the duration of each period is scaled accordingly, and the demand volumes presented in Table III are used. In each scenario, VISSIM simulation was run 100 replications to explore the variability of the delay. The comparison between the simulated delay, estimated delay from the proposed model, and estimated delay from the HCM 2010 (Highway Capacity Manual) procedure is plotted in Fig. 8. The detail delay data corresponding to Fig. 8 are summarized in Table V together with the MAPE of two models treating VISSIM simulated delay as ground truth values. Based on Table V, the relative estimation error of delay of the proposed model stayed with 10% under all the scenarios. Such observation confirmed that the proposed model is capable of providing reliable delay estimation even under extremely
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Fig. 7.
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Time-dependent distribution of queue length of selected links. (a) Link 9. (b) Link 10. (c) Link 3. (d) Link 7. (e) Link 8. (f) Link 6.
congested traffic conditions. On the other hand, the HCM procedure failed to provide accurate delay estimation, which is observed from the table.
V. S UMMARY This paper has presented a new stochastic network loading model for urban network where signalized junctions are often
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of the stochastic properties of the transport system in different types of applications. A PPENDIX A F UNDAMENTAL C ONCEPT OF PGF
Fig. 8. Estimated and simulated network delay of the network.
The pgf of a nonnegative discrete random variable is a power series representation of the probability mass function of that random variable. Let X be a discrete random variable taking nonnegative integer values, then the pgf of X is a polynomial function of another variable z, which is denoted by X(z), i.e., X(z) = E(z X ) =
TABLE V C OMPARISON OF E STIMATION ACCURACY
∞
p(X = k)z k
(a-1)
k=0
where p(k) is the probability density (mass) function of X. The pgf is a very powerful tool in handling the convolution of two nonnegative integer random variables. Convolution Theorem: If X1 , X2 , . . . , Xn is a sequence of independent random variables and Sn is the summation of this random series defined by distributed at high density. The primary power and uniqueness of the proposed model stems from the utilization of pgf through which various traffic flow phenomena, including queue evolution, platoon dispersion, and queue spillover and blockage, are modeled as stochastic events with different distributions. Mathematical formulations related to the proposed model are obtained through a rigorous derivation process. A comprehensive numerical investigation is also presented to verify the model performance. Compared with classical analytical models, the proposed model is built on a more realistic description of traffic flows in signalized networks, and compared with microsimulation techniques, the proposed model is less demanding in terms of computational power. Therefore, it can be employed in offline traffic impact analysis, evaluation of service quality of transportation supplies, or signal optimization. Nevertheless, the model also suffers from several deficiencies to be improved by subsequent studies. First of all, the uneven distribution of flow and queue among different lanes is not considered by the current formulation. However, such uneven distribution caused by drivers’ lane choice preference definitely has a huge impact on the flow dynamics. One example is the queue spillover and partial lane blockage (of through traffic) caused by heavy left-turn volume. In future, one may integrate drivers’ lane changing behavior into the model. Another shortcoming of the model involves the time-dependent turning ratios. Currently, the time-varying turning fractions are treated as exogenous variables. However, in future studies, one may want to seek a possible way to enhance the proposed model to predict the turning fractions. Finally, the tradeoff between model complexity and computational efficiency is always an important issue for stochastic modeling approaches. The convolution and transformation of random variables consumes a large amount of computational power; therefore, researchers ought to discuss the additional benefits introduced by the investigation
Sn =
n
Xk
(a-2)
k=1
then the pgf of Sn , which is denoted by Sn (z), is the product of Xk (z) k = 1, 2, . . . , n, i.e., Sn (z) =
n
Xk (z).
(a-3)
k=1
Proof: Sn (z) = E(z Sn ) = E(z X1 +X2 +···Xn ) = E(z X1 z X2 · · · z Xn ) =
n
E(z Xk ).
(a-4)
k=1
The convolution theorem indicates that the pgf of the summation of independent random variables can be written as the product of the pgfs of each individual random variable. Discrete Convolution and FFT: The discrete convolution of two vectors can be effectively computed using the FFT technique. Let V1 and V2 be two vectors, then the following discrete convolution theorem is applied in this study in order to compute the discrete convolution of V1 and V2 , i.e., V1 ⊗ V2 = F −1 (F (V1 ) ∗ F (V2 ))
(a-5)
where F and F −1 ⊗ ∗
Fourier and inverse Fourier transformation; discrete convolution operator; pointwise multiplication operator.
The computational complexity of (a-5) through naïve method is O(m1 ∗ m2 ), where m1 and m2 are dimensions of V1 and V2 . By applying FFT, the computational complexity can be reduced to O(m1∗ log(m2 )).
LU AND YANG: QUEUE DISTRIBUTION IN SIGNALIZED NETWORK THROUGH A PROBABILITY GENERATING MODEL
A PPENDIX B P ROOF OF R ECURSIVE PGF T RANSITION E QUATIONS Proof of (5): According to the definition of pgf, Qt (z) can be expanded conditioning on the values of Bt and Qt , i.e., Qt+1 (z) = E(z Qt+1 ) =
∞
p(Qt+1 = k)z k
(b-1)
Equation (b-9) describes the transition of pgf of Qt during the green phase. As for the red phase, the transition is modeled as the convolution of two nonnegative random variables Qt and At therefore, the convolution theorem of the pgf can be applied. Hence, the proof of (5) is finished. Proof of (8): During the green phase Dt (z) = E(z Dt ) =
∞
p(Qt + At − 1 = k|Qt > 0)z k
+ p(Bt = 0)p(Qt = 0) + p(Bt = 1) ×
(b-2)
= p(Bt = 0)p(Qt > 0)z −1
∞
p (Qt + At = k + 1) z k+1
k=0
p(Qt + At = k)z
∞
p(Dt = k|Qt = 0, Bt = 0)z k
∞
p(At = k)z k
m>0 m = 0.
(b-3)
= (1 − βt ) {[1 − Qt (0)] z + Qt (0)At (z)} .
(b-4)
A PPENDIX C P ROOF OF L EMMA 1 AND P ROPOSITION 2
∞ p (Qt = k) z k E z Qt = ∞ k=1
Y (z) = E(z Y ) =
∞
Qt (z) − Qt (0) ) − p(Qt = 0) = . (b-5) p(Qt > 0) 1 − Qt (0)
= p(X = 0) +
By inserting (b-5) into (b-3), Qt+1 (z) can be further rewritten into the following form:
= p(X = 0) +
Qt
∞
P (Qt + At = j) z j
∞
∞
k R E z j=1 p(X = k)
∞
E(z R )
k
p(X = k)
(c-2)
k=1 ∞
[R(z)]k p(X = k) = X (R(z)) .
(c-3)
k=0
p(Qt + At = k)z k + (1 − βt )Qt (0)
(b-6)
k=0
= (1 − βt ) (1 − Qt (0)) z −1 E z Qt E(z At ) + βt E(z Qt )E(z At ) + (1 − βt )Qt (0) = (1 − βt ) (1 − Qt (0)) z −1 At (z)
(c-1)
k=1
=
j=1
+ βt
E(z Y |X = k)p(X = k)
k=0
p(Qt = k) k z p(Qt > 0)
= (1 − βt ) (1 − Qt (0)) z −1
(b-13)
Proof of Lemma 1: The pgf of Y can be obtained by conditioning on the value of X, thus
The pgf of Qt can be computed as follows:
E(z
(b-12)
k=0
The distribution of Qt is defined by p(Qt =m) p (Qt = m) = p(Qt >0) 0
=
(b-11)
= [1 − Qt (0)] [1 − βt ]z + Qt (0)[1 − βt ] ×
k
+ p(Bt = 0)p(Qt = 0).
= p (Qt = 0) +
p(Dt = k|Qt > 0, Bt = 0)z k
k=0
k=0
k=0
∞
+ p(Qt = 0)p(Bt = 0) ×
k=0
+ p(Bt = 1)
(b-10)
k=0
p(Qt + At = k)z k
∞
p(Dt = k)z k
= p(Qt > 0)p(Bt = 0) ×
k=0
∞
∞ k=0
k=0
= p(Bt = 0)p(Qt > 0)
343
(b-7)
Qt (z) − Qt (0) 1 − Qt (0)
+ βt Qt (z)At (z) + (1 − βt )Qt (0)
(b-8)
= (1 − βt )At (z)z −1 [Qt (z) − Qt (0)] + βt Qt (z)At (z) + (1 − βt )Qt (0).
The proof is finished. Proof of Proposition 2: Let Ak,t be the number of vehicles departed from the upstream intersection at time slot k and reached at the stop line of the downstream intersection at time slot t. Then, one can decompose At into vehicle groups according to their departure time slot measured at upstream of the link (A point in Fig. 4). Moreover, Ak,t can further be decomposed into the sum of Dk Bernoulli random variables, i.e., At =
(b-9)
t k=1
Ak,t =
Dk t k=1 i=1
Rk .
(c-4)
344
IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 15, NO. 1, FEBRUARY 2014
In the above equation, the event of arrival of each vehicle departed at time slot k is modeled by a Bernoulli random variable Rk . The success rate of Rk is then determined by the distribution of on-link travel time fτ (t − k), i.e., 1 fτ (t − k) Rk = (c-5) 0 1 − fτ (t − k). Using Lemma 1, the pgf of Ak,t can be written as a compound function of Dk (z) and Rk (z). (Note that Dk (z) is the upstream departure rate.) Thus Ak,t (z) = E(z Ak,t ) = Dk (fτ (t − k)z + 1 − fτ (t − k)) . (c-6) Then, according to the convolution theory, the pgf of At is the product of Ak,t (z) for all k prior to current time slot t, i.e., t At (z) = E z k=1 Ak,t =
t
Dk (fτ (t − k)z + 1 − fτ (t − k)) . (c-7)
k=1
For practical reasons, the travel time of the link should have certain upper bound T . Then, the number of time slots to be included in the above convolution operation can be reduced for faster computation, i.e., At (z) =
t
Dk (fτ (t − k)z + 1 − fτ (t − k)) .
(c-8)
k=t−T
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Yang Lu received the B.S. degree in civil engineering from Tsinghua University, Beijing, China, in 2006; the M.S. degree in civil engineering from Tokyo University, Tokyo, Japan, in 2008; and the Ph.D. degree in civil engineering from the University of Maryland, College Park, MD, USA, in 2013. He is a Research Scientist with the Massachusetts Institute of Technology Alliance for Research and Technology (SMART) Laboratory, Singapore. His research interests include traffic flow modeling, traffic control theory, real-time traffic information integration, intelligent transportation systems, and traffic simulation.
Xianfeng Yang received the B.S. degree in civil engineering from Tsinghua University, Beijing, China, in 2009 and the M.S. degree in civil engineering from the University of Maryland, College Park, MD, USA, in 2012, where he is currently working toward the Ph.D. degree. He is a Research Assistant with the Traffic Safety and Operation Laboratory, University of Maryland. His research interests include intelligent transportation systems, freeway traffic control, arterial signal optimization, transit signal priority, traffic simulation, and unconventional intersection design.
