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Reliability-Based Approach for Traffic Signal Control

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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




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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




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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



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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 )


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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.



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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.



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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.




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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.



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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


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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




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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.



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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.




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