Stochastic programming model for oversaturated intersection signal timing

Transportation Research Part C xxx (2015) xxx–xxx

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

Stochastic programming model for oversaturated intersection signal timing Yue Tong a, Lei Zhao a, Li Li b,c,⇑, Yi Zhang b,c a

Department of Industrial Engineering, Tsinghua University, Beijing 100084, China Department of Automation, Tsinghua University, Beijing 100084, China c Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies, SiPaiLou #2, Nanjing, China b

a r t i c l e

i n f o

Article history: Received 4 September 2013 Received in revised form 2 October 2014 Accepted 22 January 2015 Available online xxxx Keywords: Traffic signals Oversaturated intersection Stochastic programming

a b s t r a c t In large cities, signalized intersections often become oversaturated in rush hours due to growing traffic demand. If not controlled properly, they may collectively result in serious congestion. How to schedule traffic signals for oversaturated intersections has thus received increasing interests in recent years. Among various factors that may influence control performance, uncertainty in traffic demand remains as an important one that needs to be further studied. In some recent works, e.g., Yin (2008) and Li (2011), robust optimization models have been utilized to address uncertainty in traffic demand and to design fixed-timed signal control for oversaturated intersections. In this paper, we propose a stochastic programming (SP) model to schedule adaptive signal timing plans that minimize the expected vehicle delay. Our numerical experiments show that the proposed SP model better describes the fluctuations of traffic flows and outperforms the deterministic linear programming (LP) model in total vehicle delay, total throughput, and vehicle queue lengths. Moreover, we compare the proposed SP model with the adaptive signal control model proposed in Lin et al. (2011) to provide insights on such improvements from green time utilization and queue balancing perspectives. Furthermore, we study the feasibility of the proposed SP model in practice, with an emphasis on the required sample sizes. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction In large cities, signalized intersections often become oversaturated in rush hours due to growing traffic demand. In such situations, the vehicle inflow rate exceeds the intersection outflow rate at least in one branch (Green, 1967; Carstens, 1971). A number of oversaturated intersections then form oversaturated arterials (Lieberman et al., 2000) and oversaturated networks (Girianna and Benekohal, 2002). As a result, delays of vehicles increase dramatically and lead to congestions, environmental problems, and economic loss (Federal Highway Administration, 2008). One major cause of oversaturated intersections is improper signal timing plans. Among various factors that may result in improper signal timing plans, failure to appropriately address uncertainty in traffic demand remains as an important one that needs to be further studied. Heydecker (1987) showed that the signal timing plan dedicated to average inflows often fails to yield satisfactory performance, if the fluctuation degree of traffic flows is significant. Yin (2008) used the traffic data collected at Gainesville, Florida to show that the traffic flow varies significantly during 9:00–11:00 am at the studied intersection. ⇑ Corresponding author. Tel.: +86 (10) 62782071. E-mail address: [email protected] (L. Li). http://dx.doi.org/10.1016/j.trc.2015.01.019 0968-090X/Ó 2015 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Tong, Y., et al. Stochastic programming model for oversaturated intersection signal timing. Transport. Res. Part C (2015), http://dx.doi.org/10.1016/j.trc.2015.01.019

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There were several approaches to handle uncertainties and improve the reliability of traffic control systems (Calvert et al., 2012). We can roughly categorized these approaches into two kinds of schemes. The first kind of schemes is robust fixed-time signal control, which aims to maintain stable performance under fluctuating traffic conditions. Fixed-time signal control scheme assigns a timing plan with pre-determined cycle length, phase sequences, and phase lengths under certain traffic demands. This scheme mainly uses historical traffic measurements to set the timing plan (Denney et al., 2008). It is world-widely used, especially in developing countries, because of its simplicity and easy maintenance. Robust fixed-time signal control scheme adopts risk-averse type planning strategy so as to reduce the possible maximum delay under extreme situations. For example, Yin (2008) studied three robust fixed-time signal timing models: mean– variance model, conditional value-at-risk (CVaR) model, and min–max model. It was shown that the performance of an isolated intersection can be improved, if the fluctuations of inflow rates were considered appropriately via these models. The optimization problems for the first two models are easy to solve; while no globally optimal solution can be easily found for the min–max model. In Li (2011), a discretization approach was proposed to solve the min–max model using binary integer programs. The second kind of schemes is adaptive signal control, which aims to dynamically respond to the fluctuating traffic demands. Adaptive signal control scheme changes the timing plan in every cycle using real-time traffic measurements and usually also real-time prediction of future traffic demands to set the timing plan (Burger et al., 2013; Li et al., 2014). It gains substantially increasing interests nowadays, because of its flexibility and capability to further improve traffic control performance. For oversaturated intersection, many existing adaptive signal control models adopt greedy algorithms so as to fully utilize the green time (Liu et al., 2008; Zhao et al., 2011; Li et al., 2013). However, some initial tests show that, when the uncertainty of traffic demand cannot be neglected, greedy algorithms may not always be a good choice (Park and Kamarajugadda, 2007; Park et al., 2001). Recently, Lin et al. (2011) proposed a control model to dynamically update green time allocation so that the proportions of queue lengths in all directions can be well balanced. They showed that the performance of this control model is robust to random fluctuation in traffic demands. More importantly, this model significantly reduces the total vehicle delay compared with the signal timing plan optimized for average inflows, although it does not attempt to minimize total vehicle delay directly. However, their model selects the best alternative from a limited set of predetermined timing plans and thus offers limited flexibility. In this paper, we consider adaptive signal control against uncertainty in traffic flows from a different viewpoint. Our major contributions can be summarized as follows. First, we propose a two-stage stochastic programming (SP) model to incorporate uncertainty in traffic flows and to generate efficient signal timing plans. This SP model takes the risk-neutral planning strategy that minimizes the expected total vehicle delay of the intersection, subject to the intersection capacity and other operational constraints. Instead of solving a complex optimization problem of signal timing plans of several consecutive cycles like Han (1996), we recursively solve a series of two-stage stochastic programs, each for the signal timing plan of two consecutive cycles. We choose to use recursive two-stage stochastic programs rather than multistage stochastic programs for several consecutive cycles based on historical traffic statistics. Comparing with multistage stochastic programs, two-stage stochastic programs are significantly less demanding on both computational capability of signal control device and solution time. This makes the proposed signal timing plan strategy easy to implement in practice. Furthermore, by recursively solving two-stage stochastic programs, at the beginning of each cycle, we are able to update the estimates on the traffic demands of the next cycle based on real-time traffic information (Vlahogianni et al., 2014). This strategy not only significantly reduces computational cost but also makes better use of the updated traffic information. Second, besides considering the inflow rate uncertainty, we also consider uncertainty in the outflows of intersections in the proposed SP model. In most of the aforementioned studies, the outflow rates were assumed to be deterministic and the saturation flow rate is used as the outflow rate for any through direction. However, recent studies reveal that vehicle outflows may also fluctuate because of the uncertainty in driver behaviors (Michael et al., 2000), pedestrian intrusion (see, e.g., Fig. 1), and environmental disturbance. The most frequently-used measure of stochastic outflows is the departure headway distribution models (Luttinen, 1992; Jin et al., 2009). Departure headway is defined as the elapsed time between two consecutive vehicles departing from the intersection, when the light turns green (Niittymäki and Pursula, 1996). During the last four decades, departure headway distributions have been examined with the aid of the fast developing video analysis methods in many cities worldwide (Al-Ghamdi, 1999; Bonneson, 1994; Hung et al., 2002; Lee and Chen, 1986; Moussavi and Tarawneh, 1990; Zhang et al., 2007). Notice that the average vehicle departure headway is the reciprocal of the average outflow rate, the outflow rate should therefore be modeled as stochastic rather than deterministic. Our recent paper (Tan et al., 2013) illustrated how to calculate the distributions of outflow rate based on the measured departure headways. It was further revealed that, given a certain green time and assumed no pedestrian/bicycle intrusions, the effective outflow rate roughly follows a certain normal distribution due to stochastic vehicle departure headways. Since pedestrian and bicycle intrusions frequently happen in many developing countries, see Fig. 1 as an example, the variance of empirical outflow rate is often doubled or tripled.

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Fig. 1. An example of pedestrian intrusion and the resulted decrease of outflow rate.

Third, our test results show that, although our SP model is designed to minimize the expected total delay, the corresponding signal plan leads to more balanced queues as well. This is in agreement with the finding of Lin et al. (2011) that more balanced queues lead to short total delay. Lin et al. (2011) and our study provide different perspectives on the interconnection between the expected total delay and queue balancing at an oversaturated intersection. The rest of this paper is arranged as follows: Section 2 introduces the stochastic programming model for signal timing. Section 3 numerically demonstrates the effectiveness and properties of the proposed SP model. Section 4 compares the proposed SP model and the queue balancing model in Lin et al. (2011). Finally, we conclude the paper in Section 5.

2. Signal timing models for an oversaturated intersection In this section, we first review the deterministic signal timing model proposed by Liu et al. (2008). Then, we propose a stochastic programming model to describe the uncertainty of traffic demands and traffic flow dynamics. For the ease of comparison of the relevant work, we adopt the notation of Zhao et al. (2011) and Li et al., 2013, as in Table 1. 2.1. The deterministic linear programming model The most commonly used objective of signal timing model is to minimize the total vehicle delay, which can be achieved through the minimization of the area between the vehicle departure curves and arrival curves. Let AðtÞ represent the cumulative number of arrived vehicles by time t and DðtÞ represent the cumulative number of discharged vehicles by time t. The R integral ½AðtÞ  DðtÞdt represents the total delay during a certain time period. As shown in Chang and Lin (2000) and Chang and Sun (2004), this integral is usually a nonconvex function with respect to time t. Since our objective is to reduce the delay, the non-convexity of delay function adds great difficulties to the solution process. One attacking method is to apply meta-heuristic searching algorithm to seek for a sub-optimal solution (Park et al., 1999). However, meta-heuristic algorithms do not guarantee optimality. To address this issue, Liu et al. (2008) proposed a reverse causal-effect model to derive a simple yet sufficiently accurate approximation formula to calculate the total vehicle delay. This model depicts both traffic arrivals and departures in one signal cycle with continuous smooth functions of time as if there were no interruptions to the traffic flows with the signal cycle. The total delay can then be estimated by the area between the arrival curve and the smoothed departure curve. This allows us to apply linear programming (LP) to find the signal timing plan that minimizes the total vehicle delay.

Table 1 Nomenclature list. M P Mp g pmin C spm

g gp ðkÞ km ðkÞ

lm ðkÞ X m ðkÞ

The set of traffic streams, indexed by m The set of signal phases, indexed by p The set of allowable traffic streams in phase p The minimum allowable green time in phase p (h) The length of a signal cycle (h) The average saturation outflow rate for traffic stream m in phase p (veh/h) The total allowable green time ratio due to the lost time Green time ratio in phase p in signal cycle k; gp ðkÞ ¼ g p ðkÞ=C The inflow rate for traffic stream m in signal cycle k (veh/h) The smoothed outflow rate in signal cycle k in traffic stream m, lm ðkÞ is affected by spm (veh/h) The vehicle queue length of traffic stream m at the beginning of signal cycle k (veh)

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Let AðtÞ be the cumulative number of vehicles arriving at the intersection over time, and DðtÞ be the cumulative number of vehicles departing from the intersection over time. Liu et al. (2008) introduced the concept of ‘‘smoothed’’ traffic flow and assumed uniform arrivals/departures of vehicles within each cycle. That is, in each cycle k; AðtÞ and DðtÞ increase linearly with constant rates kðkÞ and lðkÞ. Therefore, the total vehicle delay in the first n cycles of an oversaturated period can be written as

Total Delay ¼

Z

nC

½AðtÞ  DðtÞdt ¼

Z

0

¼

n Z X k¼1

¼

nC

dt

0

Z

t

½kðsÞ  lðsÞds ¼

0

kC

ðnC  sÞ½kðkÞ  lðkÞds ¼ ðk1ÞC

n Z X k¼1

ðnC  sÞ½kðsÞ  lðsÞds

ðk1ÞC

Z n X ½kðkÞ  lðkÞ k¼1

kC

kC

ðnC  sÞds

ðk1ÞC

n C2 X ð2n  2k þ 1ÞðkðkÞ  lðkÞÞ: 2 k¼1

ð1Þ

Furthermore, Liu et al. (2008) proposed a decomposition method to sequentially solve the signal timing plan of each signal cycle. At the beginning of the kth cycle, green time is allocated to each phase (C  gp ðkÞ) to minimize the total vehicle delay. Omitting the constant scalars, we can see that minimizing the total vehicle delay in cycle k is equivalent to

XX min ðkm ðkÞ  lm ðkÞÞ: p g ðkÞ

ð2Þ

p2P m2M p

Due to the effect of lost green time, in any given cycle, the sum of green time ratios of all phases does not exceed the total allowable green time ratio. We have

X

gp ðkÞ 6 g:

ð3Þ

p2P

The green time in each phase should be no less than the minimum green time, that is,

gp ðkÞC P g pmin ; 8p 2 P:

ð4Þ

Meanwhile, the vehicle outflow is bounded by the saturation outflow rate as in Eq. (5), and by the number of vehicles arrived in the current cycle plus the vehicle queue length at the end of the previous cycle as in Eq. (6).

lm ðkÞ 6 gp ðkÞspm ; 8m 2 M; lm ðkÞ 6 X m ðkÞ=C þ km ðkÞ; 8m 2 M:

ð5Þ ð6Þ

Moreover, all the decision variables should be nonnegative, i.e., lm ðkÞ P 0; 8m 2 M; gp ðkÞ P 0; 8p 2 P. Thus, we can formulate a deterministic linear program (LP) for cycle k as

min p g ðkÞ

s:t:

XX

ðkm ðkÞ  lm ðkÞÞ

p2P m2Mp

X

gp ðkÞ 6 g

p2P

gp ðkÞC P g pmin ; 8p 2 P lm ðkÞ 6 gp ðkÞspm ; 8m 2 M lm ðkÞ 6 X m ðkÞ=C þ km ðkÞ; 8m 2 M lm ðkÞ P 0; 8m 2 M gp ðkÞ P 0; 8p 2 P:

ð7Þ

After we solve the LP model (7) and obtain the signal timing plan in cycle k, we can calculate the queue lengths at the beginning of cycle k þ 1 as in Eq. (8) and continue to solve for the signal timing plan for cycle k þ 1.

X m ðk þ 1Þ ¼ X m ðkÞ þ km ðkÞC  lm ðkÞC;

8m 2 M:

ð8Þ

Following this pattern, we can recursively solve a series of LP models, one for each cycle, until we reach the pre-selected end of the planning horizon. The detailed discussions on the property of this deterministic model can be found in Zhao et al. (2011). It was proven in Li et al. (2013) that the obtained signal timing plan is global optimal, if the order of the vehicle clean-up times for different traffic streams remain unchanged during the planning horizon (this condition can usually be satisfied in practice). However, when the inflow and outflow rates are uncertain when making the signal timing plan at the beginning of each cycle, we need to develop an adaptive policy to adjust signal timing plans based on the realized traffic conditions over the planning horizon, and at the same time, considering the uncertain fluctuation in the inflow and outflow rates. Please cite this article in press as: Tong, Y., et al. Stochastic programming model for oversaturated intersection signal timing. Transport. Res. Part C (2015), http://dx.doi.org/10.1016/j.trc.2015.01.019

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2.2. The two-stage stochastic linear programming model To better describe the uncertain traffic flows, we assume inflow and outflow rates are random variables whose probability distributions are known based on statistical analysis of empirical observations. It should be pointed out that the so called demand/flow rate usually refers to the mean value of vehicle arrival rates during a certain time period. It provides a simple sketch for the traffic dynamics and is thus widely used in transportation engineering (especially in traffic control field). However, from the viewpoint of real-time traffic control, we need to address the fluctuations of traffic demand/flow rate. Notice that conventional traffic control models usually used the term demand/flow rate and seldom used the term vehicle arrival rate. In comply with this tradition, we clung to the term demand/flow rate to depict the vehicle arrival features but allow them to be random. Indeed, in several papers discussing the influence of fluctuating vehicle arrival rate, the authors all used the term demand/flow rate and meanwhile took them to be random (Yin, 2008; Li, 2011). Since stochastic inflow and outflow rates are incorporated, we need to redefine some notation in Table 1. In the remaining p;x part of this paper, we assume km ðkÞ and spm are random variables with known distributions, and kx m ðkÞ and sm are their realp p izations under scenario x. Furthermore, we define n ¼ ðkm ðkÞ; sm ðkÞ; km ðk þ 1Þ; sm ðk þ 1ÞÞ as the random vector, and nx as its x p;x realization under scenario x. Note that in Eq. (10), sp; m ðkÞ in (10b) and sm ðk þ 1Þ in (10g) are two realizations of the same p random variable sm , one for cycle k and the other for cycle k þ 1. Then, we can extend the deterministic linear programming model (7) into a stochastic programming model. Similar to Liu et al. (2008), we take a sequential decision planning strategy that can be applied online to find the best signal timing plan (green time allocation), that is, gp ðkÞ for each cycle k. However, in the stochastic programming model, we not only consider the effect of the signal timing plan in the current cycle (cycle k), but also the impact of this signal timing plan on the next cycle (cycle k þ 1). We formulate each step of this sequential decision process as a two-stage stochastic program (SP) (Kall and Wallace, 1994; Birge and Louveaux, 1997; Higle, 2005). At the beginning of cycle k, we do not know the exact values of the inflow and outflow rates in both cycles k and k þ 1 but we have a priori knowledge of their distributions. Therefore, in the first stage master problem, we determine the signal timing plan gp ðkÞ for cycle k, subject to the deterministic constraints on the allowable green time ratio (9b) and minimum green time (9c). The second stage subproblem evaluates the total vehicle delay of a signal timing plan gp ðkÞ for cycle k under scenarios of inflow and outflow rates in both cycles k and k þ 1. The two-stage SP model solves for the signal timing plan of cycle k that minimizes the expected total vehicle delay in cycles k and k þ 1. The first stage of the SP model, which considers the green time allocation in cycle k, is

min En Q ðgp ðkÞ; nx Þ gp ðkÞ X gp ðkÞ 6 g s:t:

ð9aÞ ð9bÞ

p2P

gp ðkÞC P g pmin ; 8p 2 P gp ðkÞ P 0; 8p 2 P;

ð9cÞ ð9dÞ

where Q ðgp ðkÞ; nx Þ is the optimal objective function value of the second stage problem under first stage solution gp ðkÞ and scenario x. According to (1), the second stage scenario subproblem can be formulated as

Q ðgp ðkÞ; nx Þ ¼ min

kþ1 X X X

  x ð2ðk þ 1Þ  2t þ 1Þ kx m ðtÞ  lm ðtÞ

t¼k p2P m2M P p x ðkÞsp; s:t: x m ðkÞ 6 m ðkÞ;

l g 8m 2 Mp ; 8p 2 P x x lm ðkÞ 6 X m ðkÞ=C þ km ðkÞ; 8m 2 M x x Xx 8m 2 M m ðk þ 1Þ ¼ X m ðkÞ þ km ðkÞC  lm ðkÞC; X gp;x ðk þ 1Þ 6 g

ð10aÞ ð10bÞ ð10cÞ ð10dÞ ð10eÞ

p2P

gp;x ðk þ 1ÞC P g pmin ; 8p 2 P lxm ðk þ 1Þ 6 gp;x ðk þ 1Þsp;mx ðk þ 1Þ; 8m 2 M p ; 8p 2 P x x x lm ðk þ 1Þ 6 X m ðk þ 1Þ=C þ km ðk þ 1Þ; 8m 2 M gp;x ðk þ 1Þ P 0; 8p 2 P Xx m ðk þ 1Þ P 0;

8m 2 M x x lm ðkÞ; lm ðk þ 1Þ P 0; 8m 2 M;

ð10fÞ ð10gÞ ð10hÞ ð10iÞ ð10jÞ ð10kÞ

p;x x where lx ðk þ 1Þ; X x m ðkÞ; g m ðk þ 1Þ; lm ðk þ 1Þ are decision variables under scenario x. Constraints (10b) and (10c) depict the actual outflow rates in cycle k under scenario x. Constraint (10d) updates the vehicle queue length at the beginning of cycle

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k þ 1. Constraints (10e)–(10h)) are traffic constraints of cycle k þ 1 with respect to the realized inflow and outflow rates under scenario x. Constraints (10i, 10j and 10k) are nonnegativity constraints. Theoretically, solving a multi-stage SP may increase the accuracy of the evaluation of recourse function(s), if we can obtain accurate data on the future scenarios, especially the conditional probabilities among scenarios of future stages. However, in practice, it is very difficult to obtain accurate data for several signal timing periods ahead. Therefore, solving multi-stage SP does not necessarily improve the accuracy. Also considering the computational time issue, we therefore propose the use of two-stage SP instead of multi-stage SP. 2.3. Solution method of the two-stage SP model While the recourse function in the master problem (9) remain convex, it has no closed form expression. Further, the evaluation of a given first stage solution requires solving for second stage subproblems of the entire sample space, which is computationally demanding, especially when the sample space is large. Over the past decades, there have been algorithms developed, by combining statistical sampling methods with mathematical programming, to solve two-stage stochastic programs, among which are the classic Benders’ Decomposition (Benders, 1962) and algorithms developed upon it, such as L-Shaped Method (Van Slyke and Wets, 1969), Regularized Decomposition (Ruszczynski, 1986; Ruszczynski and Swietanowski, 1997), Stochastic Decomposition (Higle and Sen, 1991; Higle and Sen, 1994; Higle and Sen, 1996; Higle and Sen, 1999). Another approach is to replace the expectation in the master problem (9a) with a sample mean estimator with a (much) smaller sample size (as compared to the original sample space). This is often referred to as Sample Average Approximation (Kleywegt et al., 2001), or also known as Sample Path Optimization (Fu et al., 1996; Robinson, 1996). Higle and Zhao (2012) provides a numerical comparison among the above two classes of solution approaches. In practice, the inflow and outflow rates are continuous random variables. A computationally convenient method to solve the two-stage SP model is to use the sampling approach. That is, we generate a random sample S and use the sample mean estimator to approximate the expectation of the recourse function value in the original scenario space. Further, we observe that the number of random variables in the problem under study is relative small. Therefore, we convert the corresponding two-stage SP model (9) and (10) to its deterministic equivalent problem (DEP, Kall and Wallace, 1994; Birge and Louveaux, 1997) and then solve it using a standard commercial LP solver. Choosing an appropriate sample size jSj is a trade-off between computational time and solution quality. Generally, a large sample size may improve the solution quality but also increase computational time. We will discuss on the sample size jSj in Section 3.4. 3. Test Case I In this section, we use a test case to examine the performance of our SP model. The setting of the test case is presented in Section 3.1. We compare the performance of the SP model with that of the deterministic LP model in Section 3.2. In Section 3.3, we vary the inflow rates and reevaluate the performance of the LP and SP models, aiming to investigate when we should apply the SP model rather than the LP model. We further discuss the selection of an appropriate sample size to solve the SP model in Section 3.4. 3.1. Experiment design We use the same phase configuration as in Liu et al. (2008) and Zhao et al. (2011). As shown in Fig. 2, we study an intersection with three signal phases: phase 1 allows vehicles on the west–east (W–E) road to go straight or turn right; phase 2 allows vehicles on the W–E road to turn left; and phase 3 is for the vehicles on the north–south (N–S) road. We assume the

Phase 1

Phase 2

Phase 3

N

Fig. 2. The studied intersection and its phases.

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traffic flow on the W–E road is large, and increases significantly during peak hours. On the contrary, the traffic flow on the N– S road is small, and remains stable all the time. The basic parameters are set as: cycle length C ¼ 120 s, total allowable green time ratio g ¼ 0:9, and minimum green time g p;min ¼ 18 s, 8p 2 P. We assume all the inflow and outflow rates follow truncated normal distributions. The normality of the outflow rate distribution has been validated in Tan et al. (2013). However, the proposed model in this paper is not restricted to the normal distribution assumption and can be applied to other distributions. We use  km ðkÞ and r2m ðkÞ to denote the mean and standard deviation of the inflow rate of stream m in cycle k. The values of these parameters are shown in Table 2. The inflow rate distributions of the W–E road form a peak around cycle 8, and those of the N–S road remain the same in all the cycles. We ignore the (minor) flow fluctuations of the N–S steams for simplicity of illustration. The performance difference between the LP model and SP model can be readily observed in this simplified case. We assume the inflow rates of turning streams are 1=4 of the corresponding through streams. We set the saturation outflow rate spm  Nð2100; 1802 Þ (veh/h), for every stream m and phase p, which is stationary over time. According to Tan et al. (2013), the standard deviation of the outflow rate is about 60 (veh/h), when no pedestrian or bicycle intrusions occur. In our experiments, we triple the standard deviation as 180 (veh/h) to reflect the possible intrusions. For variance reduction in comparison, we use common random seeds when generating the inflow and outflow rates, and run the simulation for 100 replications. In this experiment, we set the sample size as 100 in the sampling heuristic. To compare the two signal timing models, we use the following performance indicators: (i) The total vehicle delay D: as defined in (1). (ii) The total queue length LðkÞ: the sum of the queue lengths in all directions at the end of cycle k, i.e., P LðkÞ ¼ m2M X m ðk þ 1Þ. (iii) The throughput TðkÞ: the cumulative number of vehicles that go through the intersection at the end of cycle k, i.e., P P TðkÞ ¼ kt¼1 m2M lm ðtÞC.    LðkÞ, and TðkÞ to denote the average values of total queue length, throughput, and total delay over the 100 repWe use D; lications. We use superscripts ‘‘LP’’ and ‘‘SP’’ to distinguish the performance indicators of the two models, for example, T SP ðkÞ and T LP ðkÞ are the (expected) throughput of the SP and LP models at the end of cycle k. Note that, while the performance indicators of the ‘‘LP’’ model can be evaluated deterministically, those of the ‘‘SP’’ model are the expected values. However, in the numerical experiments, both sets of performance indicators are dependent on the sampled scenarios in the replications. 3.2. Comparison between the LP and SP models  LP ¼ 3678:0 min, and We first compare the total vehicle delays of the two models. The experiment results show that D SP  D ¼ 3339:6 min. The SP model reduces the total vehicle delay by 9.2%. Paired t-test indicates the significance of improvement of the SP model over the LP model (p ¼ 0:03). Table 2 The means and variances of vehicle inflow rates (veh/h) in Test Case I. Stream Cycle

Mean

SD

Mean

SD

Mean

SD

Mean

SD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1020 1020 1020 1080 1140 1200 1260 1320 1320 1260 1140 1020 960 900 840 840 840 840 840 840

204 204 204 216 228 240 252 264 264 252 228 204 192 180 168 168 168 168 168 168

255 255 255 270 285 300 315 330 330 315 285 255 240 225 210 210 210 210 210 210

51 51 51 54 57 60 63 66 66 63 57 51 48 45 42 42 42 42 42 42

255 255 255 270 285 300 315 330 330 315 285 255 240 225 210 210 210 210 210 210

51 51 51 54 57 60 63 66 66 63 57 51 48 45 42 42 42 42 42 42

600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600

120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120

Please cite this article in press as: Tong, Y., et al. Stochastic programming model for oversaturated intersection signal timing. Transport. Res. Part C (2015), http://dx.doi.org/10.1016/j.trc.2015.01.019

160 140 120 100 80 60 40 20 0

LP SP 2

4

6

8

T¯ SP (k) − T¯ LP (k)

Y. Tong et al. / Transportation Research Part C xxx (2015) xxx–xxx

¯ L(k)

8

10 12 14 16 18 20

Cycle k

(a) Total queue length

16 14 12 10 8 6 4 2 0

2

4

6

8

10 12 14 16 18 20

Cycle k

(b) Throughput difference

Fig. 3. Comparison of the LP and SP models in Test Case I.

Fig. 3 compares the SP and LP models by the total queue length and throughput. Fig. 3(a) plots the average total queue length  LðkÞ of the 100 replications, which shows that the maximum queue length of the SP model can be 7:0% lower than that of the LP model. Fig. 3(b) plots the differences between the SP’s and LP’s throughput, i.e., T SP ðkÞ  T LP ðkÞ, which shows that the SP model allows more vehicles to go through the saturated intersection in cycles immediately after the peak cycle. In this test case, the saturation outflow rates in all directions follow the same distribution. Therefore, no matter how we allocate the green time, the total throughput will be the same as long as we make full use of the green time in all phases. However, we have the minimum green time constraint. When the time needed to clear the queue is less than the minimum green time, the remaining part of the green time will be wasted. To make full use of the green time, we need to avoid a too short queue in any phase. In the SP model where we solve the current cycle with the considerations of the possible scenarios in the next cycle, we can avoid the potential loss of green time as described above. On the other hand, this is not possible in the LP model where we only deterministically solve for the signal timing plan for one cycle. This explains why the SP model can result in better performance as shown above. 3.3. Comparison of the LP and SP models under various inflow rates Since solving an SP model requires more computational time than solving an LP model, it is worthy to use the SP model only if the SP model offers significant improvement than the LP model. In this section, we vary the inflow rates in Table 1, and compare the performance of the SP and LP models. Particularly, based on the inflow rates in Test Case I, we vary the means of traffic inflow rates in W–E direction (the first stream in Table 2, denoted as DkWE in the follows). All other parameters remain the same as in the previous configuration. Moreover, we define the ‘‘critical ratio’’ to measure the degree of congestion at the intersection. The critical ratio in cycle k is defined as

ck ¼

X p2P

maxp m2M

 Eðkm ðkÞÞ : Eðspm Þ

ð11Þ

Intuitively, the critical ratio ck represents the expected green time required to depart the arrived vehicles in a time unit (e.g., one signal cycle). In other words, we need ck C effective green time to depart the expected arrived vehicles in cycle k. Due to the effect of lost green time, when ck > g, only part of the arrived vehicles can depart in the cycle, and queues will accumu as the average of critical ratios in all cycles within the planning horizon, i.e., late. We also define the average critical ratio c

c ¼

Pn

c

k¼1 k

n

:

ð12Þ

Fig. 4 shows the total throughputs of the LP and SP models under various inflow rates and the corresponding average crit on the horizontal coordinates. Since the total throughput is ical ratios. We label both DkWE and average critical ratio c bounded by both the number of arriving vehicles and the intersection capacity, it increases initially as the inflow rate increases. The throughput becomes stable when inflow rate is high, bounded by the intersection capacity. We notice that  2 ½0:90; 1:15. Consider that g ¼ 0:90 in the experthe SP model can better utilize the intersection capacity, especially when c  is slightly larger than the allowable iment, we notice that the improvement is significant when the average critical ratio c green time ratio g. This phenomenon can be intuitively explained as follows. When the inflow rate is low, most arrived vehicles can be cleared in each cycle, and both the LP and SP models perform well. In the opposite, when the inflow rate is high, the throughput is mostly constrained by the capacity of the intersection and queues accumulate in all directions, such that applying different signal timing strategies does not make much difference. When the inflow rate is slightly larger than the intersection capacity, i.e., the queue lengths at the intersection are neither too short nor too long. the SP model can generate signal timing plans with better performance. This analysis provides insights on when to use the SP model. Please cite this article in press as: Tong, Y., et al. Stochastic programming model for oversaturated intersection signal timing. Transport. Res. Part C (2015), http://dx.doi.org/10.1016/j.trc.2015.01.019

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Y. Tong et al. / Transportation Research Part C xxx (2015) xxx–xxx

γ¯ 2880

0.9

1.0

1.1

1.2

1.3

Total throughput

2860 2840 2820 2800 2780 2760 2740 LP SP

2720 2700 -100

0

100

200

300

400

500

600

700

800

900 1000

ΔλW E Fig. 4. Comparison of the LP and SP models under various inflow rates.

3.4. The required sample size for solving the SP model In this section, we examine the impact of sample size. Table 3 summarizes the test results on various sample sizes. For each sample size, we record the mean and standard deviation (denoted as ‘‘SD’’ in the table) of the computational time, expected total vehicle delay, total throughput, and maximum queue length over 100 replications. The computational time is the average solution time for each cycle, tested on a computer with Intel i5 2.80 GHz processor and 8 GB RAM. We use IBM ILOG Cplex 12.4 to solve both the LP and the deterministic equivalent problem (DEP) of the SP problems. Since the SP model should be frequently solved by the signal controller in online signal timing control, the computational time has to be sufficiently short compared to the signal times. Note that the computational time increases dramatically as the sample size grows. Fig. 5 shows the throughput differences between the SP and LP models under various sample sizes. Although all performance measures get better as the sample size grows, the improvement of performance is not significant when sample size is larger than 50. We find that using 50 as sample size in our problem can lead to a good trade-off of computational efficiency and signal timing performance. As compared to using sample size of 100, it reduces 64.7% of the computational time (only 30.03 ms), but with only 0.27% difference in total vehicle delay, 0.00% difference in total throughput, and 0.19% difference in maximum queue length. In practice, when we need to design signal timing plans for isolated intersections, we can similarly test on different sample sizes with the possible inflow/outflow rates to find the appropriate sample size.

Table 3 Performance of the SP model in the base experiment under different sample sizes. Sample size

Total delay (min)

Total throughput

Maximum queue length

SD

Mean

SD

Mean

SD

Mean

SD

5.54 6.86 8.70 30.03 84.96

1.13 1.03 1.32 3.11 13.80

3890.0 3434.6 3382.0 3348.7 3339.6

1126.9 1075.3 1085.4 1086.0 1104.0

2784.6 2799.7 2801.4 2802.5 2802.2

33.1 35.8 36.0 36.3 36.0

176.3 161.4 159.7 158.6 158.3

42.5 41.5 41.6 42.0 42.6

T¯ SP (k) − T¯ LP (k)

1 5 10 50 100

Computational time (ms) Mean

15

|S|=100 |S|=50

10

|S|=10

5

|S|=5

0

|S|=1

-5 2

4

6

8

10

12

14

16

18

20

Cycle k Fig. 5. Throughput differences between the LP and SP models in Test Case I under various sample sizes.

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4. Test Case II In this section, we compare the SP model with the model proposed in Lin et al. (2011) so as to provide insights on improvements from green time utilization and queue balancing perspectives. 4.1. Experiment design In order to illustrate queue balancing more clearly, we modify the phase configuration from Test Case I, removing all turning movements, but only keeping the four straight streams. The phase configuration in Test Case II is shown in Fig. 6. As the phase configuration has been modified, we increase the mean and standard deviation of the inflow rates on the W–E road to meet the intersection oversaturation condition. The inflow rates in Test Case II are shown in Table 4. Other parameters remain the same as in Test Case I. 4.2. Comparison between SP model and queue-balancing model The queue balancing idea was discussed in Lin et al. (2011), in which the queue state is defined as the vector of queue lengths in all directions. In this test case, we let Q NS denote the queue length in the N–S direction, Q WE denote the queue length in the W–E direction, and ðQ NS ; Q WE Þ represent the queue state. We can construct a two-dimensional coordinate system to plot the queue state from cycle to cycle. In Lin et al. (2011), it was shown that, the Q NS =Q WE ratio at all time should keep nearly constant for balanced queues at oversaturated intersections. That is, the desired queue proportion line should be a line passing through the origin point ð0; 0Þ. Because of the

Phase 1

Phase 2

N

Fig. 6. The phase configuration in Test Case II.

Table 4 The means and variances of vehicle inflow rates (veh/h) used in Test Case II. Stream Cycle

Mean

SD

Mean

SD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

1224 1224 1224 1296 1368 1440 1512 1584 1584 1512 1368 1224 1152 1080 1008 1008 1008 1008 1008 1008 1008

245 245 245 259 274 288 302 317 317 302 274 245 230 216 202 202 202 202 202 202 202

600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600

120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120

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Queue length

200 150 100 50 0 1 2 3 4 5 6 7 8 9 1011121314151617181920

1 2 3 4 5 6 7 8 9 1011121314151617181920

QN S

QW E

(a) The SP model Queue length

200 150 100 50 0

1 2 3 4 5 6 7 8 9 1011121314151617181920

1 2 3 4 5 6 7 8 9 1011121314151617181920

QN S

QW E

(b) The LP model Fig. 7. Box plots of queue lengths of the SP and LP models in Test Case II.

fluctuating vehicle arrivals, the empirical queue states will not fall exactly on the desired queue proportion line from cycle to cycle. So, Lin et al. (2011) proposed a control policy that maintains queue balancing in the intersection. This control policy is queue-based and implicitly minimizes the maximum queue length at each cycle. Lin et al. (2011) further showed that the queue balancing policy indeed generates the signal timing plan with shorter total vehicle delay. Similar to Lin et al. (2011) and Park et al. (2001), Fig. 7 shows the box plot of queue lengths in the N–S and W–E directions generated by the LP model proposed in Liu et al. (2008) and the SP model proposed in this paper. Clearly, the N–S and W–E queues are more balanced under the SP model, and the LP model tends to lead to imbalanced queues, especially in the W–E directions. Fig. 8 shows the scatter plot of all queue states ðQ NS ; Q WE Þ in our experiment. We can assess the queue balance using the

200

200

150

150

QWE

QWE

linear regression R2 of these queue states. A larger R2 value indicates more balanced queues. We observe that the queues are more balanced and queue states are closer to the regression line under the SP model. In fact, the signal timing plan generated by the SP model results in less deviations from the desired queue proportion line. In Figs. 7 and 8, we can find that while the objective in our SP model is to minimize the expected total delay, the corresponding signal plan leads to more balanced queues. On the other hand, the approach in Lin et al. (2011) is queue-based but meanwhile results in shorter vehicle delay. These two approaches appear to reach the same goals yet from different approaches.

100

50

100

50

0

0 0

50

100

150

200

QNS

(a) The SP model (R2 = 0.635)

0

50

100

150

200

QNS

(b) The LP model (R2 = 0.401)

Fig. 8. Scatter plots of queue states of the SP and LP models Test Case II.

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5. Conclusions Recent studies have shown that both the inflow and outflow rates of an oversaturated intersection can be stochastic due to random traffic demand, random driver behaviors, and pedestrian and/or bicycle intrusions. Neglecting the uncertainty in inflow and outflow rates may lead to poor control performance. Several robust fixed-time signal control models and adaptive signal control models that adopt greedy algorithms have been proposed. Different from existing approaches, our two-stage stochastic programming (SP) model extends the deterministic linear programming (LP) model for the signal timing plan of oversaturated intersections in Liu et al. (2008) and Zhao et al. (2011). The proposed model aims to minimize the expected total vehicle delay under both inflow and outflow uncertainties. Numerical experiments show that the SP model can generate signal timing plans that better utilize the green time and offer significantly improved intersection performances, because the SP model incorporates the knowledge about the possible fluctuations of the inflow and outflow rates in the current and next cycles. Since SP models require more computational time, we perform numerical experiments to study on traffic conditions when the SP model may result in more significant improvements over the LP model. Results of our numerical experiments indicate that improvements of the proposed SP model are more significant when the inflow rates are slightly larger than the intersection capacity (measured by saturation outflow rates). When the inflows rates at all directions are much smaller or larger than the saturation outflow rates, the SP and LP models produce signal timing plans with similar performances. Furthermore, we provide discussions on the effect of sample size on computational time and solution quality, too. Results indicate that the good solution to the SP model can be obtained with relatively small-sized samples. As we compare the SP model with the queue balancing model proposed in Lin et al. (2011), we further observe that the SP model generates signal timing plans that result in more balanced queues at the intersection than the LP model. In this paper, we only focus on isolated oversaturated intersections. An important yet challenging task is to extend the proposed methodology to deal with oversaturated arterials and networks in the future. We will discuss this problem in our coming reports. 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