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A Reliability-based Approach for Traffic Signal Control Hong K. Lo Civil Engineering, Hong Kong University of Science and Technology Clear Water Bay, Hong Kong, P. R. China. Email: cehklo@ust.hk Submitted for PRESENTATION ONLY at TRB ABSTRACT An important consideration in traffic signal control is that traffic arrivals are not deterministic. The effect of stochastic arrivals in the past has been mainly dealt with by introducing stochastic terms in delay formulae. While convenient, this approach is somewhat indirect. Moreover, when the degree of saturation is high, the system becomes highly transient; it is debatable whether a static or time-invariant result in the form of a delay formula is applicable. In this study, instead of relying on steady state or equilibrium probability measures, we capture the transient effect by analyzing the state of the system from cycle to cycle based on a probabilistic treatment of overflow in an event tree. One can either use the approach to study an existing timing plan or to design a timing plan that satisfies a certain overflow reliability requirement. The advantage is that one does not need to resort to delay formulae. Some numerical results are included to demonstrate this approach. Word count: 5158 Tables + Figures: 9 Total equivalent word count: 5158 + 9*250 = 7408 1 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. 1. INTRODUCTION Traffic signal control has been a topic of study for many years. An important consideration is that traffic arrivals are not deterministic. The effect of stochastic arrivals has been mainly dealt with by adding stochastic terms in delay formulae. A summary of this development is surveyed in Heydecker (1995). Indeed, improving delay formulae remains a current research focus. Using the delay formulae to cater for stochastic arrivals, although convenient, is somewhat indirect. Moreover, when the degree of saturation becomes high, the system becomes highly transient; it is debatable whether a static or time-invariant result in the form of a delay formula is applicable. In this study, instead of relying on steady state or equilibrium probability measures, we develop an approach by analyzing the state of the system from cycle to cycle based on a probabilistic treatment of overflow. In fact, this approach stems from a long history. Haight (1959) was among the earliest to study overflow at a traffic signal. Based on Poisson arrivals, the study derived the probability of a specific queue size at the start of a red phase given a specific queue size at the start of the preceding green. In this present study, we are interested in developing a similar measure to describe the performance of traffic signal, namely the probability that the available green time is able to clear the approach traffic. We refer to it as the overflow reliability measure (ORM). We develop the approach based on a general arrival probability distribution and extend the consideration to include effects from previous cycles through an event tree. As is commonly known, a major concern of implementing the probabilistic approach is the required computational load. In this study, we derive analytical expressions to reduce the event tree to two distinctive recursive components, which only need to be calculated once. This result reduces the computation substantially. We include some numerical results to illustrate the applicability of this reliability-based approach. The outline of this paper is as follows. Section 2 provides the background of this problem. Section 3 depicts the framework for analyzing the overflow reliability of a traffic signal. Finally, Section 4 provides some concluding remarks and future research direction. 2. BACKGROUND In the idealized situation where traffic arrives uniformly and deterministically to a signalized intersection, one can model the traffic control system as a D/D/1 queuing regime. The timing plan to minimize the total delay per cycle can be determined by solving the following mathematical program: v ( C − gi ) 2 min TD = ∑ i i 2 (1 − vi / si ) C , gi ∑g i i +L=C (1) gi si − vi C ≥ 0, ∀i gi ≥ 0, ∀i 2 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. where TD is the total delay per cycle obtained from a deterministic analysis of the arrival and departure curves; vi , si , gi , are, respectively, the volume (in vehicles per hour), saturation flow (in vehicles per hour) , and green time (in seconds) for approach i , and C is the cycle time (in seconds). In situations where several approaches share the same phase, approach i here refers to the most critical one, in terms of having the largest vi / si ratio. The constraint si gi − vi C ≥ 0 is added to prevent the occurrence of overflow; otherwise the delay will grow from cycle to cycle without a constant value. One can prove that the optimal strategy is to provide each phase with a green time that it is just long enough to clear the traffic that arrives within the cycle (i.e., gi = vi ⋅ C / si ). The cycle time such determined is typically short. For a n -phase operation, the optimal solution of (1) is: L C= n gi = vi C / si 1 − ∑ vi / si i n (2) ∑g ⇒ +L=C Lvi / si i i gi = n 1 − ∑ vi / si i Plans with any excess green time deviate from this optimal solution, resulting in an increase in the total delay. In reality, vehicle arrivals have variability. In cycles when the arrival rates are higher than the approach’s capacity ( gi si / C ), there is an overflow. The overflow must wait for the next cycle for discharge, thus incurring additional delays. Moreover, this overflow will increase the likelihood of creating another overflow in the next cycle, which may initiate a vicious cycle. Therefore, introducing some excess green time to the optimal plan as in (2) will provide a buffer against variability. However, the buffer itself will introduce additional delays, as some of the excess green time is sometimes unutilized. The question is how big this buffer ought to be. Note that this question is also relevant to advanced traffic control systems that collect real-time traffic information, as traffic signal is about controlling the future traffic, which is not known perfectly even with the best of technology. Essentially, one still must deal with the underlying uncertainty, even though the uncertainty would be smaller with more real-time information. Let us review how this problem was addressed in the past. Webster (1958) produced one of the most influential and useful piece of work in this area, now commonly referred to as the Webster delay formula: C (1 − θ ) 2 1/ 3 X2 C d= + − 0.65 2 X 2+5θ (3) 2 (1 − θ ⋅ X ) 2v (1 − X ) v where d is the average delay per vehicle for a particular approach; the proportion of green time θ = g / C , and degree of saturation X = C ⋅ v / ( g ⋅ s ) . One may recall that the first term on the right hand side of (3) is the same as the objective function in (1) – a result of D/D/1 3 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. analysis1. The second term is the average delay obtained from assuming Poisson arrivals and deterministic departures – a result of M/D/1 analysis. The third is an adjustment factor, around 10% of the sum of the first two terms. Therefore, sometimes the Webster delay formula is simplified as: C (1 − θ ) 2 X2 d = 0.9 + (4) 2 (1 − θ ⋅ X ) 2v (1 − X ) The second term of the Webster delay formula is highly nonlinear especially when the degree of saturation X approaches one. Due to this reason, if one solves the mathematical program (1) with the Webster delay formula instead of the D/D/1 delay result, the optimal timing plan is always that X i << 1 or gi ⋅ si >> vi ⋅ C . That is, a certain buffer is added by setting the capacity gi ⋅ si higher than the mean arrivals vi ⋅ C . However, the Webster delay formula provides no information on the performance of the buffer, in terms of either overflow probability or improvement in expected delay. In fact, as we will show in Section 3.1, the performance of the buffer created with the Webster method varies with the traffic volume without a consistent measure. As the Webster delay formula approaches infinity when the degree of saturation X approaches one, delay is overestimated. This overestimation is due to the assumption that the queuing system reaches a steady state. However, as contended by Hurdle (1984), in reality, the peak period ends, and the approach volume drops long before the system reaches steady state. Over the years, there were many other delay formulae constructed. Some considered that the system could be temporarily over-saturated, hence allowing the degree of saturation to be greater than one. Examples include the TRANSYT (Kimber and Hollis, 1979) and Akcelic (Akcelik, 1981) delay formulae. Improving delay formula with field and simulation data remains as one important area of study for signalized junctions (example, Fambro and Rouphail, 1997). While these efforts are worthwhile to search for a result for practical applications, the status of this type of research can be adeptly summarized by Hurdle (1984): “No claim is made that the formulae are correct; rather they yield answers that do not violate elementary logic in the troublesome region of v/c (referred to as degree of saturation or X here in this paper) near unity where neither the steady-state nor the over- saturation models can be expected to yield reasonable results”. The above indicates that using the existing delay formulae to model over-saturated traffic is at best crude, as they aimed at finding a static or time-invariant result to capture an essentially transient problem. Therefore, using these delay formulae as an objective function in (1), in the hope that an appropriate green buffer can be constructed is an unreasonable expectation. These delay formulae will of course create a green buffer, as compared with the 1 By substituting θ = g / C ; X = vC / ( gs ) and multiplying the total approach traffic volume Cv , the first v (C − g ) 2 term on the RHS of (3) becomes , which is identical to the objective function in (1). 2 (1 − v / s ) 4 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. D/D/1 delay formula; but similar to the Webster delay formula, one has no measure on their performance. In this study, we develop an approach to directly consider the variability of traffic arrivals in the design of signal timing plans, rather than using delay formula that were independently developed with various assumptions. 3. RELIABILITY FRAMEWORK 3.1 First-order Approximation of Overflow Reliability Consider the critical approach volume (in vehicles per second) of phase i to be a random variable, Vi . Hence, the amount of traffic arriving within a given cycle is itself random, expressed as: Ψ i = Vi C , where C is the cycle time. We further assume that the amount of arrivals Ψ i follows a lognormal distribution2. For a given timing plan and hence a given set of green allocations, the maximum amount of traffic a phase can discharge is expressed as si gi . With this background, we define the overflow reliability measure (ORM) αi for approach i as the probability that traffic can be entirely discharged with the available green time; hence the approach is cleared without any overflow. As a first order approximation, assuming that there is no overflow from a previous cycle, ORM is expressed as the probability that the amount of arrivals is less than the discharge capability, expressed as: P {Ψ i ≤ gi si } = αi , (5) For the lognormal distribution considered in this study, one can convert it to a standard normal distribution via transforming the mean and variance of Ψ i (Ang and Tang, 1975). Specifically, the lognormal distribution for (5) can be written as: ln ( gi si ) − λi Φ = αi , (6) ξi where Φ ( ⋅) is the cumulative distribution function of standard normal distribution, and σ i2 ξi2 = ln 1 + µi2 , (7) 1 λi = ln ( µi ) − ξ i2 2 2 This is reasonable for heavy traffic, but the framework here is not limited to specific distributions. 5 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. where µi ,σ i2 , respectively, are the mean and variance of Ψ i , taken as µi = σ i2 = Vi C in this study3, where Vi is the mean arrival rate for the critical approach. Graphically, ORM is as illustrated in Figure 1. PDF Shaded region = P {Ψ i ≤ gi si } = α i Total arrivals Ψi λi ln ( gi si ) Figure 1 Overflow reliability measure (ORM) Note that for situations wherein the actual traffic approaches the case of deterministic arrivals, i.e., σ i → 0 and µi = Vi C , putting these conditions into (6) and (7), the fraction on the LHS of (6) becomes: 1 σ i2 ln ( gi si ) − ln ( µi ) + ln 1 + 2 ln ( gi si ) − λi 2 µi lim = lim σ i →0 ξi σ i →0 σ i2 ln 1 + µi2 ln ( gi si ) − ln ( µi ) 1 σ2 = lim + lim ln 1 + i2 (8) σ i →0 σ i →0 2 µi σ2 ln 1 + i2 µi → ∞ if ln ( gi si ) − ln ( µi ) > 0 = → −∞ if ln ( gi si ) − ln ( µi ) < 0 This is for convenience only, not a restriction. Generally, the variance of Ψ i can be a function of the cycle 3 time and mean arrival rate, or can assume any value. 6 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Putting (8) into the ORM definition (6), we have this result: ln ( gi si ) − λi → 1 if ln ( gi si ) − ln ( µi ) > 0 lim Φ = (9) ξi → 0 if ln ( gi si ) − ln ( µi ) < 0 σ i →0 gi si That is, when ln ( gi si ) − ln ( µi ) > 0 or ln >0 gi si > ViC , the ORM becomes one. On ⇒ Vi C the other hand, when gi si < Vi C , the ORM becomes zero. Hence, this probabilistic approach includes the deterministic traffic as a special case. Based on the above discussion, one can design a timing plan purely based on the ORM condition. It is expressed as: Given a set of desirable αi , determine gi and C so as to satisfy the following nonlinear simultaneous system: ln ( gi si ) − λi Φ = αi , ∀i ξi ∑g i i +L=C , (10) gi ≥ 0, ∀i where ξi , λi follow from (7). As an example, consider a junction with two competing approaches. Case 1 (2) represents the situation when both approaches have mean volumes of 600 vph (700 vph). Solving (10) with a lost time of 6 seconds and a range of overflow reliability measures, we obtain the results as shown in Figure 2. Generally, increasing the ORM requirement increases the cycle time nonlinearly. For the mean approach volume of 600 vph, at ORM = 0.5, the cycle time is 14 s; whereas at ORM = 0.99, the cycle time becomes 207 s. For the higher mean approach volume of 700 vph, the change is even more dramatic. At ORM = 0.5, the cycle time is 20 s; at ORM = 0.99, the cycle time is 473 s. One may also notice that for the same ORM, the cycle time needed for the higher approach volume of 700 vph is increased nonlinearly from the approach volume of 600 vph, especially at higher ORM values. Part of this nonlinearity is attributed to the high variance associated with the higher approach volume. The dotted line with triangle markers shows the results for the mean approach volume of 700 vph but whose variance is reduced to 75% of its mean. The nonlinear effect on cycle time is most noticeable at high ORM values. For low ORMs below 0.8, the effect of variance on cycle time is insignificant. 7 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. 500 450 700 vph, var = mean 400 600 vph, var = mean 350 Cycle time (s) 700 vph, var = 75% mean 300 250 200 150 100 50 0 0.5 0.6 0.7 0.8 0.9 1 Overflow reliability measure Figure 2 Cycle time versus overflow reliability measure One can consider the ORM condition (10) from the reverse direction. Given a certain timing plan derived from any method, (10) provides the ORM for each approach by substituting the timing plan elements gi and C . Let us examine the deterministic timing plans as determined in (2), which ignore arrival variability, wherein for every phase, the mean arrivals is set equal to the discharge capability, Vi C = si gi . Putting this condition into (10) and simplifying, we have: 1 1 Φ ln 1 + = αi . (11) 2 Vi C As the cumulative distribution function of standard normal distribution is a monotone function, (11) shows that ORM αi is inversely related to the arrivals Vi C . In other words, as the traffic volume increases, the ORM associated with the deterministic timing plan drops, or becomes less reliable. Specifically, Figure 3 shows the relationship between the ORM of the deterministic timing plans versus the mean arrivals ( Vi C ). At low volumes of arrivals, the deterministic timing plans achieve a reasonable level of ORM, which deteriorate with higher arrival volumes4. In the presence of arrival variability, this explains why deterministic timing plans are not appropriate for junctions with high traffic volumes. When the higher order effect is considered as in Section 3.2, the ORM of deterministic timing plan will be even lower as one overflow may trigger the next. 4 This deterioration of ORM is specific to the arrival distribution considered. 8 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. 0.7 0.65 0.6 ORM 0.55 0.5 0.45 0.4 0 10 20 30 40 50 Mean arrivals (veh per cycle) Figure 3 The ORM of deterministic timing plans as a function of arrivals Let us also consider the performance of the Webster timing plans in terms of ORM. One may recall that Webster’s optimal cycle time is determined as: 1.5L + 5 Co = , 1 − vi / si ∑ and the green split for each phase is apportioned according to the flow ratios: vi / si gi = ( Co − L ) . ∑ vi / si Consider the scenarios that Approach 1’s mean volume is fixed at 800 vph whereas Approach 2’s mean volumes vary from 200 to 800 vph. The saturation flow for each approach is taken as 1800 vph and the lost time 6 s. The green allocations and cycle time according to Webster’s method are provided in Table 1. Table 1 Webster timing plans with Approach 1’s volume fixed at 800 vph Approach 2 vol. (vph) Approach 1 Green (s) Approach 2 Green (s) Cycle Time (s) 200 20.4 5.1 31.50 400 24.0 12.0 42.00 600 32.6 24.4 63.00 800 60.0 60.0 126.00 Substituting the Webster timing plans in Table 1 to equation (6), we obtain the results as shown in Figure 4. Unintentionally, the Webster method allocates a higher ORM (close to 0.9) or a longer green buffer to the approach with a higher volume. As the volume from Approach 2 gradually increases, the ORM’s of both approaches drop; with the higher volume approach (Approach 1) taking a more significant drop. Eventually, when the two approaches have the same volume of 800 vph, they share the same ORM of 0.68. This result 9 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. shows that the ORM values of Webster timing plans (and hence the sizes of the green buffer) vary with the approach volumes and deteriorate with higher competing volumes. Whether this allocation is sensible or not is subject to debates. But the point is that one really has no way of altering this result within the Webster method. The same can be said about the other methods, as overflow reliability is not considered; they only considered this problem of stochastic arrivals indirectly through the delay formulae. ORM of Webster Plans: Approach 1 Volume fixed at 800 vph 1 0.9 0.8 0.7 0.6 ORM ORM (Approach 1) 0.5 ORM (Approach 2) 0.4 0.3 0.2 0.1 0 0 200 400 600 800 1000 Approach 2 Volume (vph) Figure 4 The ORM of the two approaches under varying loadings To attain a high ORM, as seen in the example of Figure 2, very often a long cycle time is required. This may not be practical. To rectify this implementation problem, one may insert constraints on either the green times or the cycle time. But these constraints will restrain the ORM attainment. Returning to the example in Figure 2, if one were to limit the cycle time to 180 s, for example, then for an approach volume of 700 vph, it would not be possible to achieve an ORM of 0.99. One may, however, modify the simultaneous system (10) to the following mathematical program to maximize the combined ORM subject to the timing plan constraints, expressed as: max ∑ wiαi gi i P {Ψ i ≤ gi si } = αi ,∀i ∑g i i +L=C , (12) 0 < g min ≤ gi , ∀i 0 < C ≤ Cmax wi > 0, ∀i 10 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. where wi is a weight allocated to each approach to allow for differential treatment of the approaches; g min , Cmax are allowable extreme values for green time and cycle time, respectively. In this mathematical program, the ORM αi , instead of being set a desirable value as in (10), is optimized. One may use the expressions (6)-(7) developed in this study for lognormal distribution in the place of the ORM condition. Or one may choose other types of distributions for this purpose. 3.2 Higher-Order Overflow Reliability The previous discussion simplifies the analysis of overflow by considering only the arrivals and the discharge capability of green time within one cycle. In general, any existing overflow or residual queue will trigger a higher probability of overflow in the following cycle, as part of the discharge capacity is used for clearing the previous overflow. To extend the consideration of this overflow effect between subsequent cycles, we classify the status of an approach by either overflow ( O ) or underflow ( U ) and define the overflow as H , which is a random variable. In the following, for ease of notation, we drop the subscript i associated with each approach but add the superscript j to represent that the variable being considered is in the j th cycle. Thus, H j represents the overflow at the end of the j th cycle; U j = P {H j = 0}, O j = P {H j > 0} , respectively, denote the cases of underflow and overflow at the end of j th cycle. Generally, one can write: H j = max ( 0, VC j + H j −1 − gs ) (13) where VC j is the arrivals within cycle j ; H j −1 is the overflow from the previous cycle; and gs is the discharge capacity available for the phase. We begin the analysis with an underflow for j = 0 . The event tree of the overflow/underflow occurrences is shown in Figure 5, in which each darkened (hollow) dot represents an overflow (underflow). In addition, we notate the state of each event according to the cycle number and its order counting from the top of the tree, shown as S ( j, k ) in Figure 5. According to this arrangement of the event tree, when k is an odd (even) number, the state of the system is in U j ( O j ). Each cycle (except Cycle 1) is associated with multiple occurrences of the overflow and underflow states. However, the occurrence of each of these states, even within the same cycle, has a different probability. One needs to trace down the specific branch of this event tree to determine the probability of a particular state by means of conditional probabilities. Letting f ( x ) be the probability density function5 of the arrivals within one cycle, the results for the 1st cycle are: 5 The following derivation is general to all probability distributions. 11 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. gs U = P {H = 0} = P {VC ≤ gs} = 1 1 1 ∫ f ( x )dx 0 ∞ O1 = P {H 1 > 0} = P {VC 1 > gs} = ∫ f ( x )dx gs S(3,1) S(2,1) S(3,2) Legend: Overflow S(1,1) S(3,3) Underflow S(2,2) S(3,4) S(3,5) S(0,1) S(2,3) S(3,6) S(1,2) S(3,7) S(2,4) S(3,8) Cycle 1 Cycle 2 … Figure 5 The event tree of the overflow-underflow occurrences For the second cycle, considering that the arrivals are independent between cycles, we have these results: gs 2 U U = P {U ,U } = P {CV ≤ gs} P {CV ≤ gs} = ∫ f ( x )dx 1 2 2 1 2 1 0 ∞ gs U O = P {O ,U } = P {CV > gs} P {CV ≤ gs} = ∫ f ( y )dy ∫ f ( x )dx 1 2 2 1 2 1 gs 0 { O1U 2 = P {U 2 , O1} = P VC 2 + (VC 1 − gs ) ≤ gs, VC 1 > gs } = P {VC 2 + VC 1 ≤ 2 gs,VC 1 > gs} 2 gs 2 gs − x = ∫ f ( x ) dx ∫ gs 0 f ( y ) dy 12 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. { O1O 2 = P {O 2 , O1 } = P VC 2 + (VC 1 − gs ) > gs,VC 1 > gs } = P {VC 2 + VC 1 > 2 gs,VC 1 > gs} ∞ ∞ = ∫ f ( x )dx ∫ f ( y ) dy gs 2 gs − x One can continue this procedure for the third cycle and so on. It will get more tedious as one considers more cycles. Because even the arrivals are independent from cycle to cycle, the overflow relationship, expressed in (13), depends on the overflow from the previous cycle, which is in turn a function of those in earlier cycles. In general, it is not possible to reduce the procedure via a Markovian type of transition probabilities between only two subsequent cycles. Nevertheless, it is possible to reduce the event tree by breaking it into components and deriving appropriate conditional probabilities for each component states. Each of the states in the event tree belongs to either (A) a state subsequent to an underflow cycle or (B) part of an overflow chain as shown in Figure 6. In fact, the first overflow cycle in an overflow chain belongs to both (A) and (B). The rest of the states belong to either (A) or (B) exclusively. Underflow Prior to the chain 1st branch-off underflow 1st overflow in the chain 2nd overflow 3rd overflow … States subsequent States in an overflow to an underflow chain Figure 6 States in the event tree Basically, for any cycle j that has attained an underflow status (i.e., there is no overflow), then the next cycle j + 1 has a fresh start; overflow (underflow) becomes entirely a matter of arrivals versus discharge capability of the available green. Mathematically, this can be expressed as: 13 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. P {VC j +1 ≤ gs,U j ,...} P {VC j +1 ≤ gs} P {U j ,...} P {U j +1 | U ,...} = j = P {U j ,...} P {U j ,...} . (14) gs = P {VC j +1 ≤ gs} = ∫ f ( x ) dx 0 In a similar manner, we obtain: ∞ P {O j +1 | U ,...} = P {VC j j +1 > gs} = ∫ f ( x ) dx . (15) gs Referring to the event tree in Figure 5, the conditional probability can be rewritten as: gs P {S ( j + 1, 2k − 1) | S ( j , k ) ,...} = ∫ f ( x ) dx, ∀k ∈ odd, j, = 0,1,2,... 0 ∞ (16) P {S ( j + 1, 2k ) | S ( j, k ) ,...} = ∫ f ( x ) dx, ∀k ∈ odd, j, = 0,1,2,... gs For any cycle j that starts an overflow chain, the overflow influence will carry forward to this next cycle and so on until a future cycle that returns the system to an underflow status. At that point, the system renews itself, where the overflow influences from previous cycles completely disappear. Let m represent the first cycle that starts the overflow chain that contains cycles m + 1, m + 2,..., m + l ; whereas m − 1 is the underflow cycle prior to the overflow chain. In Figure 5, such examples of overflow chains include S (1,2) - S (2,4) - S (3,8) - S (4,16)-,... ; S (2,2) - S (3,4) - S (4,8) - ... , etc. The conditional probability of the first overflow cycle in an overflow chain is determined by (15). The conditional probability of the second overflow cycle in the chain given the first overflow cycle is: P {O m +1 , O m ,U m −1 ,...} P {O m +1 m | O ,U m −1 ,...} = P {O m ,U m −1 ,...} = { P VC m +1 + (VC m − gs ) > gs,VC m > gs P {U m −1 ,...} } . (17) P {VC m > gs} P {U m −1 ,...} ∞ ∞ ∫ f ( x1 ) dx1 ∫ f ( x2 ) dx2 2 gs − x1 = gs ∞ ∫ f ( x ) dx gs 1 1 And the conditional probability of the first branch-off underflow cycle (see Figure 6) is expressed as: 14 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. P {U m +1 | O m ,U m −1 ,...} = 1 − P {O m +1 | O m ,U m −1 ,...} , m +1 m { m −1 where the expression P O | O ,U ,... } is as determined in (17). For the third overflow cycle in the chain, the conditional probabilities of a continued overflow and a branch-off underflow are, respectively: m+2 m +1 ∑VC i > 3gs, ∑VC i > 2 gs, VC m > gs P P {O m + 2 | O m +1 , O m ,U m −1 ,...} = i=m m +1 i =m ∑VC > 2 gs, VC m > gs i P i =m ∞ ∞ ∞ ∫ f ( x1 ) dx1 ∫ f ( x2 ) dx2 ∫ f ( x3 ) dx3 2 gs − x1 3 gs − x1 − x2 = gs ∞ ∞ ∫ f ( x1 ) dx1 ∫ f ( x2 ) dx2 gs 2 gs − x1 P {U m + 2 | O m +1 , O m ,U m −1 ,...} = 1 − P {O m + 2 | O m +1 , O m ,U m −1 ,...} In general for the l overflow cycle in the chain, the conditional probabilities for a continued overflow and a branch-off underflow are, respectively: ∞ ∞ ∞ ∫ f ( x ) dx ∫ gs 1 1 2 gs − x1 f ( x2 ) dx2 ... ∫ l f ( xl +1 ) dxl +1 ( l +1)⋅ gs − xi P {O m +l | O m +l −1 ,..., O m ,U m −1 ,...} = ∑ 1 ∞ ∞ ∞ (18) ∫ f ( x ) dx ∫ gs 1 1 2 gs − x1 f ( x2 ) dx2 ... ∫ l −1 f ( xl ) dxl l ⋅ gs − ∑ xi 1 P {U m + l | O m + l −1 ,..., O m ,U m −1 ,...} = 1 − P {O m + l | O m + l −1 ,..., O m ,U m −1} . (19) Due to their recursive nature, these conditional probabilities can be further simplified to this useful expression: 15 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. P {O m +l ,..., O m | U m −1 ,...} = P {O m | U m −1 ,...} P {O m +1 | O m ,U m −1 ,...}... P {O m +l | O m +l −1 ,...,U m −1 ,...} ∞ ∞ ∞ ∞ ∞ ∞ ∫ f ( x1 ) dx1 ∫ f ( x2 ) dx2 ∫ f ( x1 ) dx1 ∫ f ( x2 ) dx2 ∫ f ( x3 ) dx3 = ∫ f ( x1 ) dx1 2 gs − x1 2 gs − x1 3 gs − x1 − x2 ⋅ ⋅ ... gs gs ∞ ∞ ∞ ∫ f ( x ) dx ∫ f ( x ) dx ∫ f ( x2 ) dx2 gs 1 1 1 1 gs gs 2 gs − x1 ∞ ∞ ∞ ∫ f ( x ) dx ∫ gs 1 1 2 gs − x1 f ( x2 ) dx2 ... ∫ l f ( xl +1 ) dxl +1 ( l +1)⋅ gs − ∑ xi ... ∞ ∞ ∞ 1 ∫ f ( x ) dx ∫ gs 1 1 2 gs − x1 f ( x2 ) dx2 ... ∫ l −1 f ( xl ) dxl l ⋅ gs − ∑ xi 1 Finally, by canceling terms, we have: ∞ ∞ ∞ P {O m + l ,..., O m | U m −1 ,...} = ∫ f ( x1 ) dx1 ∫ f ( x2 ) dx2 ... ∫ f ( xl +1 ) dxl +1 (20) gs 2 gs − x1 l (l +1)⋅ gs − ∑ xi 1 This expression finds the conditional probability from an initial underflow cycle to any overflow cycle of an overflow chain. As all the overflow chains are identical in terms of this conditional probability, one only needs to determine (20) once for the entire event tree. We now have derived the closed-form conditional probabilities of all the states in the event tree. One can then map these conditional probabilities expressed in terms of U m , O m , etc. back to the state notation S ( j, k ) . For example, for the chain S (1,2) - S (2,4) - S (3,8) - ... , set m = 1 ; S (1,2 ) = O1 , S ( 2,4 ) = O 2 , etc. In general, the arrangement of the event tree in Figure 5 is such that for any cycle m − 1, m = 1,2,... , an odd k represents an underflow state (i.e., S ( m − 1, k ) = U m −1 ). As the first cycle of an overflow chain always follows from an underflow state, its corresponding position in the event tree is therefore at S ( m,2k ) , m = 1,...;k = 1,3,... . Or, the first overflow cycle in an overflow chain can be expressed as: S ( m,2k ) = O m , m = 1,2,...;k = 1,3,... where 2k is subject to the maximum number of states possible for that cycle (i.e., 2k ≤ 2m ). For a particular overflow chain starts at cycle m ' and state 2k ' , each of the subsequent overflow cycles within this chain is then: S ( m '+ p,2 p +1 k ' ) = O m ' + p , p = 1,2,... . And each of the branch-off underflow cycle is S ( m '+ p,2 p +1 k '− 1) = U m ' + p , p = 1,2,... . Both of these expressions are subject to the maximum number of states for the particular cycle of concern. In effect, although there are many overflow chains in this event tree, they are all identical in terms of the conditional probabilities as expressed as (18)-(19). Thus, one only needs to calculate these expressions once and substitute their values back to the corresponding positions in the event tree. 16 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. As explained earlier, there are multiple occurrences of the overflow/underflow states at the end of each cycle. Therefore, to find the probability of a specific state, say S ( j, k ) , at the end of cycle j , one must trace the corresponding branch of the event tree that leads to S ( j, k ) and multiply the conditional probabilities along the branch. Mathematically, this is expressed as: { } { } P S ( j, k j ) = P {S (1, k1 )} P {S ( 2, k2 ) | S (1, k1 )}...P S ( j, k j ) | S (1, k1 ) ,...S ( j − 1, k j −1 ) , where a subscript is added under k to specify the particular state at each cycle along the branch of the event tree that leads to S j, k j . ( ) Finally, the total probability of overflow (underflow) at the end of the j cycle can be determined by summing the probabilities of all the overflow (underflow) states at cycle j , expressed as: 2 j −1 P {O j } = ∑ P {S ( j,2k )}, (21) k =1 2 j −1 P {U j } = ∑ P {S ( j,2k − 1)} (22) k =1 Of course, one can choose to determine the value of either P O j { } or P {U } and use the j { } { } relationship P O j + P U j = 1 to find the value of its counterpart. According to its definition in Section 3.1, the ORM at the end of the j th cycle is equivalent to P U j { } as determined in (22). In summary, if one is interested in estimating a first order approximation of ORM, one only needs to apply (5) to a specific arrival distribution. An example of the lognormal distribution is illustrated in (6)-(7). On the other hand, if one intends to include higher order effects for a more accurate answer, one needs to use (22) and the related development discussed above. As should be obvious, the complexity of including higher order effect increases significantly. 3.3 Numerical Comparison To study the effect of higher order considerations, we examine the ORMs of a series of consecutive cycles subject to the same mean traffic volumes. Specifically, we consider a junction of two competing approaches; each has a mean volume of V = 600 vph (or 1/6 vps). We approximate the arrivals with a lognormal distribution with its mean and variance both equal VC , where C is the cycle time. The lost time is taken as 6 seconds. The effective green time is split between the two approaches. We study a range of cycle time as shown in Table 2. In particular, two special cases are included: the timing plans calculated according to Webster’s method (Scenario 6) and a purely deterministic method (Scenario 7) as discussed in Section 2. 17 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Table 2 Study Scenarios Scenario Cycle Time (s) 1 180 2 150 3 120 4 90 5 60 6 (Webster) 42 7 (Deterministic) 18 The results are plotted in Figure 7. The x-axis plots the cycle times of the seven scenarios whereas the y-axis plots their corresponding ORM values. The series of lines represent the results for different numbers of cycles under consideration. For example, the top line shows the ORM values if only one cycle is considered. This is referred to as the first order approximation in this study (see Section 3.1). It tends to overestimate the ORM as it ignores the fact that any existing overflow will lead to a higher probability of subsequent overflows. In general, the ORM values after more cycles of considerations (referred in this study as the higher order effect) are lower. As expected, this lowering effect is more severe for short cycles than for long cycles. In the scenario of a long cycle time (C=180), there is no significant difference between the ORM of the 1st cycle and that of the 5th cycle. For the given loadings, the same can be said down to a cycle time of 90 seconds. For these cases, one does not lose much accuracy in using the 1st order approximation. For the Webster timing plan, the ORM drops from 0.79 after the 1st cycle to 0.60 after the 5th cycle, a 24% reduction. For Scenario 7 (the deterministic method), the drop is even more significant, from 0.61 (1st cycle) to 0.24 (5th cycle) or a 61% reduction. Another observation from these results is that the drop in ORM between subsequent cycles gradually decreases. For example, for the case with the most significant drop (Scenario 7 when the cycle time is 18 s), its drop in ORM from the 1st cycle to the 2nd cycle, the 2nd cycle to the 3rd cycle, and so on, are respectively, 0.17 (=0.61-0.44), 0.1, 0.06, 0.04. This shows that the ORM differences between consecutive cycles gradually diminish after a relatively small number of cycles. It appears that one does not need to include a very long series of cycles to obtain reasonable convergent results. Nevertheless, one should be cautious in generalizing this result, as there is no guarantee that the final ORM will converge to an asymptotic value especially for short cycles. 18 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. ORM Value Versus Cycle time: Higher Order Effects 1.000 0.900 0.800 0.700 1st Cycle 2nd Cycle ORM Value 0.600 3rd Cycle 0.500 4th Cycle 0.400 5th Cycle 0.300 0.200 0.100 0.000 0 50 100 150 200 Cycle Time Figure 7 ORM values versus cycle time 4. CONCLUDING REMARKS This study developed a methodology to analyze the overflow reliability of a signalized intersection. One can either use the method to study an existing timing plan or use it to design a timing plan to satisfy a certain overflow reliability requirement. We extended the approach from a first order approximation to higher order considerations. The results showed that the effects of higher order considerations are important for short cycles. The results also showed that whereas one can always improve the overflow reliability with a longer cycle time, its effect diminishes gradually. That is, when the overflow reliability is already at a high value (say, 0.9), it takes an extended increase in cycle time for further improvement. On the other hand, at low overflow reliability (say, 0.5), one can gain substantially in overflow reliability with a modest increase in cycle time. Finally, while this approach provides an additional dimension in analyzing and designing signal timing plans, overflow reliability by itself may not form a complete set of 19 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. performance measure. However, as this approach is developed formally from a probabilistic framework, it can be extended to include considerations of expected delay and expected queue length within a formal structure. The advantage of this approach is that one does not need to resort to delay equations developed in this past, which are approximate representations to reality. This is the focus of our current study. ACKNOWLEDGEMENT The assistance of K. W. Yuen for part of the computational work is acknowledged. The author is grateful for helpful suggestions and discussions from Chan Wirasinghe and Ben Heydecker. This research is sponsored by the Hong Kong Research Grant Council’s direct allocation grant RGC-DAG97/98.EG03 and Competitive Earmarked Research Grant HKUST6105/99E and Sino Software Research Institute Awards, SSRI98/99.EG02 and SSRI99/00.EG02. REFERENCES Akcelik, R. 1981. Traffic Signals: Capacity and Timing Analysis. Research Report 123. Australian Road Research Board, Victoria. Ang, H. and W. Tang. 1975. Probability Concepts in Engineering Planning and Design. John Wiley. New York. Fambro, D. and N.N. Rouphail. 1997. Generalized Delay Model for Signalized Intersections and Arterial Streets. Transportation Research Board Annual Conference Paper 970823. Washington, DC. Haight, F. 1959. Overflow at a Traffic Light. Biometrika, 46, 420-24. Heydecker, B.G. 1995. Treatment of Random Variability in Traffic Modelling. Paper presented at the Workshop on Traffic and Granular Flow, Julich, 9-11 October, 1995. Kimber, R.M. and E.M. Hollis. 1979. Traffic Queues and Delays at Road Junctions. Laboratory Report 909. Transport and Road Research Laboratory, Crowthorne, Berkshire, England. Webster, F. 1958. Traffic Signal Settings. Road Research Technical Paper No. 39, Road Research Laboratory, Her Majesty’s Stationery Office, London, London. 20 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal.

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