Tronrpn Rex-B Vol ISB. pp I::-13b. 19x1 0191.?615’0401?7-1250?.00 0 Prmted I” Great Br~tam Perpamon Press Lid ANALYTICAL MODEL OF TRAFFIC DELAYS UNDER BUS SIGNAL PREEMPTION: THEORY AND APPLICATION JESSE JACOBSON U.S. Department of Transportation. Transportation Systems Center. Kendall Square, Cambridge, MA 02142, U.S.A. and YOSEF SHEFFI Department of Civil Engineering, Massachusetts Institute of Technology. Cambridge, MA 02139. U.S.A. (Receioed 1 June 1979: in revisedform 7 February 1980) Abstract-Major emphasis has been placed in recent years on the improvement of the operations of existing transportation facilities. using Transportation Systems Management strategies. Accordingly, preferential treatment of high occupancy vehicles is playing an increasing role in transportation projects. This paper deals with one of these strategies. the prioritv treatment of buses at signalized intersections. Such treatment is aimed at improving the capacity of intersec- tions. The paper develops an analytical model of delays at signalized intersections under a bus preemption scheme. The analysis is presented for the simplest case, i.e., two intersecting one-way streets. The results suggests that the benefits of bus preemption can be increased by properly adjusting several design parameters such as cycle and phase duration of the preempted phases as well as the non-preeempted parameters. The model outlined in this paper is applicable to any situation in which stochastic variation is introduced into the signal cycle as well as to bus preemption. Consequently, other potential applications of the model include the design/analysis of traffic actuated signals, and pedestrian actuated signals. 1. INTRODUCTION This paper presents a methodology for the analysis and the design of bus preemption at an isolated signalized intersection. Bus preemption falls in the category of projects aimed at priority treatment of high occupancy vehicles (HOVs) such as special HOV lanes, reduced tolls for HOVs and other similar concepts. The objective of such strategies is to increase the perceived advantage of HOVs relative to single occupancy vehicles and therefore divert users from the private automobile to HOV. Such objectives are particu- larly important in light of the greater energy and environmental efficiency of HOVs. This paper develops an analytical model of the delays to bus passengers and auto- mobile occupants at a signalized intersection under bus preemption. While most previous modelling efforts in this area have utilized discrete event simulation, this paper develops an approximate analytical approach which enables the analyst to investigate the effects of several design parameters on the total intersection delay. Thus cycle and phase durations, as well as the preemption strategy parameters themselves, can be set to minimize the intersection delays under preemption. The analysis costs are small compared to simula- tion, and thus many design concepts can be tested and sorted very efficiently, and detailed design trade-offs can be fully explored. As expected, the results of the experiments showed that bus preemption reduces the total delay, expressed in person-seconds per hour of operation, when both bus occupancy and the flow of buses are high. However, it is shown that, by adjusting several design parameters, the benefits can be increased substantially. The reason for this finding is that the optimal signal setting without preemption differs significantly from the optimal setting with preemption. It is also shown that, contrary to common engineering wisdom, bus preemption is beneficial even (and more so) when the cross traffic is high. The literature on bus preemption can be divided into two categories: experimental and modeling. These studies are discussed in the remainder of this section. 127
ANALYTICAL MODEL OF TRAFFIC DELAYS UNDER BUS SIGNAL PREEMPTION:
THEORY AND APPLICATION
JESSE JACOBSON U.S. Department of Transportation. Transportation Systems Center. Kendall Square,
Cambridge, MA 02142, U.S.A.
and
YOSEF SHEFFI
Department of Civil Engineering, Massachusetts Institute of Technology. Cambridge, MA 02139. U.S.A.
(Receioed 1 June 1979: in revisedform 7 February 1980)
Abstract-Major emphasis has been placed in recent years on the improvement of the operations of existing transportation facilities. using Transportation Systems Management strategies. Accordingly, preferential treatment of high occupancy vehicles is playing an increasing role in transportation projects. This paper deals with one of these strategies. the prioritv treatment of buses at signalized intersections. Such treatment is aimed at improving the capacity of intersec- tions. The paper develops an analytical model of delays at signalized intersections under a bus preemption scheme. The analysis is presented for the simplest case, i.e., two intersecting one-way streets. The results suggests that the benefits of bus preemption can be increased by properly adjusting several design parameters such as cycle and phase duration of the preempted phases as well as the non-preeempted parameters. The model outlined in this paper is applicable to any situation in which stochastic variation is introduced into the signal cycle as well as to bus preemption. Consequently, other potential applications of the model include the design/analysis of traffic actuated signals, and pedestrian actuated signals.
1. INTRODUCTION
This paper presents a methodology for the analysis and the design of bus preemption at an isolated signalized intersection. Bus preemption falls in the category of projects aimed at priority treatment of high occupancy vehicles (HOVs) such as special HOV lanes, reduced tolls for HOVs and other similar concepts. The objective of such strategies is to increase the perceived advantage of HOVs relative to single occupancy vehicles and therefore divert users from the private automobile to HOV. Such objectives are particu- larly important in light of the greater energy and environmental efficiency of HOVs.
This paper develops an analytical model of the delays to bus passengers and auto- mobile occupants at a signalized intersection under bus preemption. While most previous modelling efforts in this area have utilized discrete event simulation, this paper develops an approximate analytical approach which enables the analyst to investigate the effects of several design parameters on the total intersection delay. Thus cycle and phase durations, as well as the preemption strategy parameters themselves, can be set to minimize the intersection delays under preemption. The analysis costs are small compared to simula- tion, and thus many design concepts can be tested and sorted very efficiently, and detailed design trade-offs can be fully explored.
As expected, the results of the experiments showed that bus preemption reduces the total delay, expressed in person-seconds per hour of operation, when both bus occupancy and the flow of buses are high. However, it is shown that, by adjusting several design parameters, the benefits can be increased substantially. The reason for this finding is that the optimal signal setting without preemption differs significantly from the optimal setting with preemption. It is also shown that, contrary to common engineering wisdom, bus preemption is beneficial even (and more so) when the cross traffic is high.
The literature on bus preemption can be divided into two categories: experimental and modeling. These studies are discussed in the remainder of this section.
127
128 JESSE JACOBSON and YOSEF SHEFFI
Elias (1976), in a bus preemption study, in Sacramento, California, reports results which are similar to those reported by TJKM (1978) which evaluated a similar project in Concord, California. Both studies demonstrated benefits derived from preemption and show that, with low bus frequencies, the added delays to automobile occupants are negligible. The U.K. Department of the Environment (1975) reports the delay reduction due to preemption in Derby, England, and argues that, even with relatively high bus frequencies, the reduction in the delay for bus riders is significantly larger than the additional delay to automobile occupants.
UMTA’s Service and Methods Demonstration Program supported a bus preemption demonstration project in Miami (Courage er al., 1977) which brought about a 20”/ reduction in total system’s delays.
Cottinet (1977) investigated and compared three preemption strategies in an experi- ment in Nice, France. The first strategy allows an incoming bus to change the signal to green whenever it hits the detector (allowing, of course, for some mandatory phases). The second strategy shortened the red period only, while the third allowed only for an extension of the green period. Since the first strategy was reported superior to the last two, it was chosen to concentrate on it in the present research. However, the method- ology developed in this paper is directly applicable to the remaining two strategies.
The modelling of bus preemption has predominantly utilized discrete event simulation techniques. El-Reedy and Ashworth (1978) investigated possible improvement in isolated intersection performance when the bus signal control system is superimposed to an optimal fixed setting. The optimal fixed setting was achieved with the use of the TRAN- SYT program (Robertson, 1969), which was also used in some of the above mentioned experimental studies (e.g., Cottinet in Nice). El-Reedy and Ashworth investigated the delay reduction both when the underlying fixed setting was designed to minimize vehicle delay and when it was designed to minimize person delay, a concept which we adopted in this paper. Their findings agree with the above mentioned experimental studies for low bus flows. However, they argue for a fixed control with higher bus flows. The possibility of altering the underlying fixed setting once the bus preemption control is installed was not tested. As shown in this paper, the preemption benefits can be substantially increased by changing the underlying signal setting once preemption is installed. In fact, a reference where such a concept was used or even considered could not be found.
Other researchers have used discrete event simulation to study bus preemption (for example see Ludwick, 1976). This reference suggests that bus preemption should be considered for high frequency service only, a recommendation which is in line with observations in a later section of this paper, which suggest that the benefits from bus preemption grow as the proportion of buses in the preemption direction grows. More general conclusions with regard to the benefits of bus preemption were reached by Lieberman et al. (1977) who did not disaggregate their recommendations by automobile and bus flows.
Simulation models were also used to study related problems such as traffic and pedes- trian actuated signals. These problems are related to bus preemption since one of the main causes of delay under such dynamic controls is the stochasticity introduced into the signal setting by the random actuation. Thus, in all these cases. the phase durations faced by the vehicle approaching the intersections are random variables. This is the basis for the approximate analytical model developed in the paper.
The inherent level of complexity and the large cost associated with simulation metho- dologies restricts their use to either researvch applications or the detailed design stage. An analytical model of bus preemption was developed by vanBilderbeek (1969), assuming deterministic vehicle arrival process. Even though this approach gives the reader more insight into the problem, as compared to simulation, the use of deterministic arrival process would invariably cause an underestimate of the delay. A more recent paper by Allsop (1977) analyzes the optimal phase setting under different bus preemption strate- gies. While Allsop’s paper discusses a large set of preemption strategies. and provides an algebraic solution for the optimal phase duration. the approach assumes that the best
Analytical model of traffic delays 129
preemption strategy is one that minimizes dela? to HOVs. without taking into consider- ation delays to private vehicles.
This paper develops a simple stochastic model of total intersection delay (i.e. person and vehicle delay) under bus preemption. The model can be used either as a screening tool. to test possible candidate intersections. or as a design tool. to set the preemption and signal parameters such that total delay is minimized.
The model is developed in the next two sections of the paper. Section 2 lays the foundation by developing the probability density function of the non-green period under preemption, while Section 3 deals with the delay components and the total delay at a preempted intersection. The use of the model to minimize delays is demonstrated in Section 4, which also includes sensitivity analysis of the model’s inputs. Section 5 con- cludes the paper with a summary and suggestions for further research.
2. THE NON-GREEN PERIOD PROBABILITY DENSITY FUNCTION
As mentioned in the previous section, under preemption, the non-green period experi- enced by a motorist arriving at an intersection is a random variable. This section de- velops the probability density function of this random variable. The parameters of the density functions depend. obviously, on the specific preemption strategy utilized. This section develops the probability density function of the non-green period on the main direction of flow, i.e., the direction of the buses.
The scenario of the analysis is an isolated intersection with a one-way main street (on which the buses operate) and a one-way cross street (with no buses). The preemption strategy under study permits both an extension of the duration of the green, and a shortening of the duration of the red. The “location” of each cycle on the time axis remains intact, whether the cycle is preempted or not (i.e. no clock acceleration and deceleration are allowed). If a bus is detected during a green period, the green can be extended (up to a maximum duration) to allow the bus to clear the intersection. If a bus is detected during the red period, the red period can be shortened (providing a minimum green period for the cross traffic).
The various possibilities are illustrated in Fig. 1. The following notations are used in the figure (the dimensions are seconds).
TC = Total cycle length, NG* = The non-green period on the main street without preemption (termed the
underlying non-green period) TE = The maximum allowable green extension TS = The minimum green period for the cross traffic TA = Amber and all red periods.?
As noted in the figure, it is assumed, in line with the assumptions on the “location” of the cycle, that TE + TS < NG* - 2TA. In other words, the cycle can accommodate a bus preemption occurring at the end of the green period. It is also assumed that the time from detection to bus passage is no longer than TE; in other words, TE is at least as long as the time it takes for the bus to traverse the intersection when the road segment between the detector and the intersection is congested. The time it takes the bus to traverse the intersection (since detection) under free flow conditions is designated TF and it is used later to compute HOV delays.
The arrows in Fig. 1 indicate possible detection events within the cycle. In case “a”, the cycle would not be preempted, even though a bus is detected, since the remaining green is sufficient for clearing the intersection. A second possibility where a bus detection would not cause preemption (this possibility is not shown in Fig. 1 but is included in case “a”) is
*It is assumed, for reasons of clarity. that the amber period appears in each flow direction after both the green and red periods. The results of the analysis are, however, not affected by assuming that in amber period appears at the end of the red period as well.
TB-B Vol ISB. No. 2-E
130 JESSE JACOBSON and YOSEF SHEFFI
Fig. I. Preemption strategy
when a bus is detected during the last TA period of a cycle, i.e. when the cycle is already in the process of turning green. In case “b”. the remaining underlying green duration is less than TE and the green period is extended by an amount equal to the difference between TE and the remaining underlying green. In case “c”. a bus is detected during the red period (or TA between the green and the red periods) but before the minimum green for the cross traffic (TS) has been completed. In this case, the minimum green for the cross traffic is completed and the signal turns green for the remainder of the system. In case “d”, a bus has been detected during the red period, after the minimum (TS) has been completed. and the signal would turn green immediately (obviously following an amberi all-red period). The signal turns green immediately. in this case, to allow any queue in front of the bus to dissipate.
The probability density function of the length of the non-green period, NG, given that a bus is detected during the cycle under consideration, can now be developed. Let the random non-green period be denoted by NG and the event “a bus is detected during TC” be denoted by B.
Figure 1 illustrates the conditional probability density function P(NG i B), given the aforementioned cycle parameters and preemption strategies.
The probability of NG = NG* (given B) is the probability of case “u” of Figure 1. Assuming that bLs detection times. given B. are random. P(NG = NG* I B) = (TC - NG* - TE + TA).‘TC. The length of the non-green period is between (KG* - TE) and NG* in case -‘h”, equal to (NG* - TS) in case “c”. and
Analytical model of traffic delays 131
ITC-SC*-TE+TA),'TC -
iTA*TSl/lC-
IITC-
i,'TC- I
TS*ZT.A SC"-TE FX* c KG _-
Fig. 2. Conditional probability density function for NG.
between (TS + 3TA) and NG* in case “d”. In cases “b” and non-green period is determined by the time of detection.
Using the same consideration as in case “a” above,
“d” the length of the
and assuming that (NG* - TS - 2TA) > TE the probability density function P(NG I B) is given by:
P(NGIB) =
1
(TC - NG* - TE + TA)/TC for NG = NG* ‘/TC for (NG* - TE) 5 NG I NG*
l/TC for (TS + 2TA) I NG I (NG* - TE) (TS + TS)/‘TC for NG = (T’S + 2TA)
(1)
In order to compute the unconditional probability density function of the non-green period, P(NG). the bus arrival rate has to be introduced. To simplify the computation of the model’s results, it is assumed that the preemption response is determined only by the first bus to be detected in a given cycle. In other words, a cycle is not preempted “twice” during one cycle. Thus, for example, if a bus is detected during the beginning of the green period the preemption would not be activated, even if a second bus appears during the same cycle. Note that this assumption can be relaxed, and distributions similar to the one in Fig. 2 can be developed for the case of two and three buses in the cycle, using a similar logic. However, the impact of the relaxation of this assumption depends on the preemp- tion strategy and on the relative magnitudes of the bus arrival rate and the cycle dur- ation. If the mechanism allows for multiple preemption or if both the bus arrival rate and the cycle duration are relatively high, this assumption can be relaxed.
Utilizing the single preemption assumption, event B is equivalent to “one or more buses arriving at the intersection during TC”. Under these conditions, the probability density function of NG is:
P[NG] = P[NGI B]Pr[B] + (1 - Pr[B])NG* (2)
where Pr[B] is the probability of event B. Let &, denote the bus arrival rate in buses per hour; the arrival rate during one cycle is i,, . TC/3600. Assuming a Poisson (;lJ arrival process of buses, the probability of one or more buses detected during a given cycle is:
Pr[B] = 1 - exp( - A,, . TC/3600). (3)
Using (3), (2) and (l), the probability density function of NG can be computed. Note that the assumption of a Poisson arrival process is more realistic in an intersec-
tion where several bus lines are using the main street, since in the case of one single bus line, the vehicle bunching could invalidate this distributional assumption. However, note
132 JESSE JACOBSEN and YOSEF SHEFFI
that any other distribution of Pr[B] can be used, since the model is solved numerically any how, and the special properties of this Poisson process are not used in the derivation of the results.
3. THE DELAY AT AN INTERSECTION WITH RANDOM PHASES
The problem of intersection delay has been studied by many researchers, in particular Newell (1965) and Miller (1963) without developing an exact solution. Assuming a Poisson arrival process for the vehicles approaching an intersection Webster (1958) obtained a solution by simulation and curve-fitting.
Our approach is based on an expression for the mean vehicle delay obtained by McNeil (1968). His expression for the mean wait time per vehicle (given fixed phase pattern), Ew] is:
r - Fraction of non-green period. i.e. r = NG*/TC. k - Vehicle arrival rate.
lu -1 - Service rate per vehicle (assumed fixed).
p - The arrival rate-service rate ratio, i.e. p = i/p. Q(0) - Number of vehicles at the end of the green period.
I - Dispersion index for the arrival process, i.e. I = VAR [N(t)]/E[N(t)], where N(t) is the number of vehicles in time t.
This formula applies to large number of cycles with fixed periods. In order to treat this formula as a conditional expectation (i.e. Ew ING]), the definition of r should be changed to r = NG/TC. However, as mentioned before, (4) still applies to a large number of cycles with fixed phases. The parameter that captures this effect is E[Q(O)], which is the expected number of vehicles at the beginning of the non-green phase. Using an assumption of simple Poisson (E.) arrivals. McNeil (1968) shows that:
where trt is the number of vehicles that may clear the intersection in a green period, i.e. m = (( I - r)TC . LO. where ( ) means ‘rounded to the closest integer.” This formula is quite robust to different values of rn, i.e., variable non-green periods. Thus, it is assumed that (5) can be interpreted as the total expected wait time given NG. Note that in order to simplify the analysis. the service rate of vehicles queuing at the intersection (l/p) has been assumed independent of the queue length. While the non-stochasticity condition can be relaxed. the probability generating function of the inter-departure times. has to be assumed identical for all the vehicles in the queue. Thus, service rates which depend on the queue lengths cannot be accommodated by the model.
In order to find the delay under preemption, we have to solve for the expectation of the wait time given NG (eqn 4) over the density of NG (2) with (1) and (3) substituted properly. This is done through a simple numerical integration of the density function in (1).
An expression for the delay along the cross direction can be developed in a fashion similar to the logic used for the main direction. The probability density function of the non-green period for the cross direction. NG’. is identical to that of NG with
NG = TC + 2TA - NG’
Following (2). the unconditional density function of NG’ becomes
where Pr[B] is given by (3). The average delay is computed in a manner analogous to the computation ofthe delay on the main direction. integrating (4) over the density given by (7).
The delay to HOVs when preemption is not effective is assumed to be equivalent to the private vehicles’ delay. Under preemption. HOVs are delayed due to automobiles queued at the intersection in front of the HOV. The following analysis of these delays is performed for each of the preemption cases described in Fig. 1.
Consider case “a”, as depicted in Fig. 1 (i.e. excluding the special case of a bus arrival during the last TA period of a cycle-denote this case “ii”) and assume that a bus is detected t seconds after the beginning of the green period. On the average. the number of vehicles waiting at the beginning of the green is E [Q(O)] -t i:NG* (ignoring the possibi- lity that the preceding cycle has been preempted). Thus the average number of vehicles in front of the bus at the detection instant is: E(Q(O)] + i:NG* + (3. - p)E[rJ, and the time it would take this queue to dissipate is (E[Q(O)J + (i - p)E[t]I. If the time from detection fo crossing the intersection is TF under free flow conditions. the delay under this case, D ) ii. is:
Note that this analysis applies to case “b” as well. The mean arrival time for this event (6 u b) is E[tJ = 1,/2 (TC - NG*). Thus
Eli u b = max:[E[Q(O)] + i.NG* + (j. - p)(TC - NG*)/2& - TF. 0:. (9)
The likelihood of this event (given a detection) is (TC - NG*)/TC. Under the special case that the bus is detected during the last TA period of a cycle (denoted “a”‘) the number of automobiles waiting in front of the bus, on the average is [E[Q(O)] + i (NG* - l/2 TA)] and the average delay per bus is:
The likelihood of such a case is, of course, TA!TC (given a detection). Under case “c”, assume that an HOV is detected r seconds after the end of the green period. The average number of automobiles queuing at the intersection at the detection instant is E[Q(O)] + E[t]. Using the same arguments as above, the mean delay in this case, Dl c. is:
El c = max {(TS + TA)/2 + [E[Q(O)] a+ R(TS + TA)/2]p - TF, 0). (11)
where E[t] = (TS + TA)/2 for case “c”. The likelihood of case “c” is (TS + TA)TC (see Fig. 1).
Using similar arguments (and measuring t from the end of the green period) the following expression for the delay under case “s’, D (d. can be developed.
Thus, the average delay for an HOV under preemption. b. is:
D = I/TC[(TC - NG*)*DlG u b + TA*DIa’ + (TS + TA).Djc
+ (NG* - 2TA - TS).Dlq (13)
where the conditional delays are given by (9~(12). Assuming HOV and automobiles occupancy factors, the total person delay can be computed for each combination of design parameters.
Note that in developing expression (13) a deterministic approximation of queuing theory has been used. This expression underestimates the average HOV delays under preemption.
A program carrying out the above calculations has been coded and is available from the authors. The program accepts as input the cycle parameters (TC, NG* and TA). the preemption strategy parameters (TE, TS), the private vehicles flow parameters (Remain, E,,,,, b I) and the bus arrival rate (&). In this program TF is calculated as a function of TE (assuming that TE exactly equal the time from detection to intersection clearance
134 JESSE JACOBSON and YOSEF SHEFFI
when the road segment in front of the HOV is congested). If ur is free flow speed for HOVs and 1 is the average spacing in congested conditions, it is easy to see that: TF = (TE . Q/p. pf), Thus, in order to compute TF, l and I.+ have to be input in addition to TE and p. The program returns the total person delay under preemption on the main and cross roads. It can be used to compare the delays for automobiles on the main road between various preemption or no preemption strategies. However, the figure of merit in this study is the total delay to HOV riders and occupants of the private vehicles on the main direction as well as occupants of the private vehicles in the cross direction.
4. APPLICATIONS AND SENSITIVITY ANALYSIS
This section analyzes the results of the minimization of total delay at an intersection under bus preemption. The effect of varying only a subset of the design parameters is discussed, as well.
The scenario for this analysis is a one-way main street on which the buses operate, and a one-way cross street on which there are no buses. The figure of merit, i.e. the one to be minimized is total person delay (or the average delay per person) measured in seconds per hour. Other indicators of the intersection performance, such as queue lengths, and delays to both private vehicles (at each branch) and bus patrons are computed as well. Note that, in order to estimate person delay rather than vehicle delay, the load factors of all vehicle types has to be input into the model.
The search for the optimal signal control under different operating strategies and vehicle flows, is carried out heuristically using a grid search. A rigorous proof of unimo- dality of the total delay function is not within the scope of this paper. However. extensive working experience, and analogy to the fixed time signal problem, suggest that the deiay function is in fact unimodal and the optimal setting corresponds to a global minimum.
The model applications discussed in this section include an unconstrained delay mini- mization and a comparison between minimization of vehicle delay and minimization of person delays.
4.1 Unconstrained minimization The optimal signal setting is the combination of cycle length (TC), phase duration
(NG*), and maximum green extension on the main street (TE) which minimizes the total person delay. For the purposes of this analysis, however, the cycle length is fixed at 60 sec. and the “amber/all-red” period at TA = 5 sec. Vehicle occupancies are assumed to be 30 passengers per bus and one occupant per automobile. and the service time is assumed to be I/,u = 2 sec. per vehicle per lane. The cross traffic arrival rate has been fixed at A,,,, = 1000 vehicles per hour while two levels of traffic arrival rate on the main street are assumed: kmai,, = 1000 and 1200 vehihr. The bus arrival rate is assumed to be 30 buses per hour.
With regards to the above mentioned parameter values, only one assumption cannot be easily changed-the assumption regarding the service time. Even though the value of p can be set at any number, the observed dependence of the discharge rate on the queue length (e.g. Greenshields et al., 1947) cannot be incorporated in the model without major modifications. However, these considerations were deemed beyond the scope of the current analysis and were not incorporated in the model. Moreover, the emphasis here is on the comparison of the signal settin g with and without preemption and. since the “before” and “after” setting is determined by the same formulae, the comparisons are still valid.
Table 1 gives the optimal signal setting for the preemption and non-preemption cases. for the above mentioned parameter values. In both cases. the control parameters (phase duration and the maximum green extension) were set as to minimize total person delay. As mentioned above, the comparisons were conducted for &,ain equal 1000 and 1700 veh/hr.
Anal! wal model of traffic delays 135
Table 1. Optima! sIgnal settmg for preemption and no preemption
= 1 000 veh./hr = 1,000 vehlhr
Expected queue Main length for Direction
10.2 14.0 14.4 19.7
private vehi- cles cross I 1
Direction 16.9 14.2 22.4 20.5
Expected Wait time for HOVs [SeCl 36.9 7.5 43.1 12.7
Total person delay jseclhr ] 130810 108238 171010 156209
Change in total person delay with respect to no preemption [Xl
.NA -17.2 NA -8.7
For both flow levels the duration of the underlying green phase (TC - NG*) decreases with preemption. Accordingly, the expected wait time for private automobiles increased on the main street while decreasing on the cross street. The common engineering wisdom is that the reverse is true, i.e. that private automobiles in the preemption direction benefit from the preemption while the cross traffic is subject to excessive delays. This, however, points out the difference between this paper’s approach and the more common one which does not call for a resetting of the signal parameters once preemption is implemented. Once the signal is reset for preemption. the delay to the automobiles on the main street increases since the delay of buses is significantly decreased and the signal is set to minimize total person delay. It is interesting to note that. once the signal is optimized for preemption, the expected queue lengths for the private automobiles are identical .for the main and cross directions. Thus. if the arrival rate of private automobiles is identical in both branches. the expected wait time would be the same.
An additional point worth noting is that, as evident from Table 1, the benefits of bus preemption decreases when the proportion of buses with respect to private automobiles (in the preempted direction) decreases (assuming constant cross-flow). Thus the share of buses in the main direction traffic stream should be one of the determinants of the decision to implement bus preemption.
To summarize this experiment, note that significant gains in total person delay can be obtained by considering all the parameters of a preemption strategy, including the under- lying fixed signal setting. The following section investigates the impact of including the underlying (non-preempted) phase periods among the design parameters, as person delay and vehicle delay minimization are compared.
4.2 A comparison OIT vehicle and person delay minimization In this experiment. the total cycle length is assumed to be set at TC = 60 sec. The
traffic flows are assumed fixed at imain = i.,,,,, = 1000 veh/‘hr and & = 30 buses/hr
136 JESSE JACOBSIN and YOSEF SHEFFI
throughout this experiment. For these conditions, the following four cases are analyzed:
Case A No preemption; the signal is set to minimize total person delay. Case B No preemption; the signal is set to minimize total vehicle delay. Case C Preemption; the phases are equal to Case A above: TE is set in order to
minimize total person delay. Case D Preemption; phases and TE are set in order to minimize total person delay.
Results of this experiment are reported in Table 2. The total person delay in Case B is higher than in Case A since Case B does not
account for the higher vehicle occupancy of HOVs relative to automobiles. This, of course, is well known and it is interesting to note that one of the latest versions of the signal optimization program TRANSYT can be specified to minimize person delay rather than vehicle delay.
The results of Cases C and D are also obvious. As more design parameters are allowed to vary, and the mathematical program is further relaxed, the benefits are increasing, i.e., the total person delay is lowered. Case C represents a situation where the underlying fixed signal setting is not considered as a design variable, and only the maximum green extension (which is a preemption parameter) is. The inclusion of the phase durations in the design variables (Case D) increases significantly the benefits associated with preemp- tion. In the latter case it is interesting to note the magnitude of the reduction once the phase duration is included in the decision variables. Obviously, even greater savings can
Table 2. Optimal signal setting for preemption and no preemption [fixed cycle length]
A - NO preemption
B - No preemption optimization of total vehicle delay
C - Preemption, assuming (Tc-NG*) is equal to case A.
D - Preemption
x maI” = 1000 vehlhr
i cross = 1000 vebfhr
Analytical model of traffic dela! F 137
be expected if the total cycle duration were included among the design parameters. Nevertheless. the large magnitude of the savings with respect to Case A (17.3%) reaffirms the earlier statement regarding the increase in benefits with the increasing proportion of buses in the main traffic direction. As this proportion grows. more benefits can be expected. Note also that when the phase duration is adjusted. the expected queue lengths from private vehicles are identical for both intersection branches, as mentioned in Section 4.1.
The equality of the queue lengths between the two intersection branches can be used to quickly assess the potential for improvement of the intersection operations. This can be done by observing the queues and recording average queue lengths at several periods of the day. Those periods where the queue lengths are uneven can be corrected by re- optimizing the underlying phase durations. This is evident from comparing the queue lengths in Cases C and D of Table 2.
5. SUMMARY AKD CONCLUSION
This paper has presented an analytical framework for the analysis of bus preemption at an isolated intersection. The model developed in Section 2 and 3 predicts the total person delay at a two-branch intersection, given the automobile and bus flows, the preemption strategy and the underlying signal setting. The model treats the green period experienced by an automobile as a random variable. the density function of which is developed in Section 2. Section 3 uses some existing results to develop expected delays conditional on the amount of green available in each cycle; the expected delay formulae are developed by taking this expectation with respect to the above-mentioned density function.
The model is used in Section 4 to minimize the total delay to automobile and bus occupants by parametrically varying the design variables. It is important to emphasize here that the figure of merit here is the total person delay rather than the vehicle delay, since the preferential treatment of high occupancy vehicles is of essence in bus preemp- tion.
The approximate-analytical solution for the intersection delays under preemption has permitted the search for the optimal signal setting and preemption strategies utilizing the full range of design parameters. As shown in Section 4, the inclusion of the underlying (non-preempted) signal setting parameters in the problem has improved the solution considerably. This, of course, is to be expected since when all design parameters are treated as variable, the minimization problem is a relaxation of the more traditional one, when only the preemption strategy is considered.
The model developed here also permits the exploration of many important tradeoffs which can provide insights as to the appropriateness of bus preemption in certain inter- sections. For example, it seems that the benefits associated with bus preemption are rela- tively small when the traffic flow in the preemption direction is much higher than the cross traffic flow. It is reasonable to assume that, at the limit, when there are buses alone in the preemption directions, the benefits would be the greatest (in this situation, bus preemption can be thought of as a vehicle actuated signal). Another example of the model’s use is the expected conclusion that preemption is more beneficial where the rate of arrival of buses is higher. This is obvious since the delays to bus occupants are sharply reduced by the preemption.
The model presented in this paper can be extended and improved in several directions. The first one is the incorporation of the model in a formal optimization procedure for the design of preemption. The second one is the correction for queue length-dependent service rates for the private vehicles, as mentioned in Section 4. A third refinement of the model can be achieved by considering multiple preemption at any given cycle. Such a model can be further extended to include preemption from several directions (e.g., on a two-way main street). Another possible improvement can be achieved by modelling the arrival process of vehicles in batches. Such a model can be easily constructed using the
138 JESSE JACOBSEN and YOSEF SHEFFI
results for intersection delay with batch arrivals given by McNeil (1968). This can account for arrivals which follow the discharge from a signalizzed intersection located upstream. Such a model can serve as a basis for modeiling a series of intersections with bus preemption.
As mentioned in the introduction, the general methodology is applicable whenever stochastic variation is introduced into a fixed signal setting e.g. pedestrian-actuated signals and traffic actuated signals.
Acknowledgemenrs-This paper was prepared as part of the Service and Methods Demonstration Program of the Urban Mass Transportation Administration. We wish to acknowledge the review and comments of Howard Slavin and David Spiller of TSC.
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