Journal of Advanced Transportation 20:l, pp 73-88. Copyright 0 1986 by the Institute for Transportation Optimal Control of Headway Variation on Transit Routes Mark Abkowitz Amir Eiger Israel Engelstein Headway control strategies have been proposed as methods for correcting transit service irregularities and thereby reducing passenger wait times at stops. This paper addresses a particular strategy which can be implemented on high frequency routes (headways under Is12 minutes), in which buses are held at a control stop to a threshold headway. An algorithm is developed which yields the optimal control stop location and optimal threshold headway with respect to a system wait function. The specification of the wait function is based on the development of several empirical models, including a headway variation model and an average delay time model at control stops. A conclusion is reached that the headway variation does not increase linearly along a route, a common assumption made in many previous studies. Furthermore, the location of the optimal control stop and threshold value are sensitive to the passenger boarding profile, as expected. The algorithm itself appears to have practical application to conventional transit operations. Introduction The ability of transit buses to adhere to schedule and maintain regular headways on high frequency routes (headways under 10-12 minutes) is particularly important to transit users as service irregularities result in increased wait times and make transferring more difficult. Several control strategies for correcting service irregularities have been proposed. DR. MARK ABKOWITZ is an Associate Professor in the Department of Civil Engineering, Rensselaer Polytechnic Institute, Troy, N.Y. DR. AMIR EIGER is a principal scientist in the management sciences group of General Research Corporation in McLean, VA. DR. ISRAEL ENGELSTEIN is an operations analyst for Arthur D. Little, Inc., Washington, D.C.
74 Mark Abkowitz, Amir Eiger and Israel Engelstein Of interest in this paper are holding control strategies and, in particular, threshold-based holding control. In a threshold holding strategy, if an incoming headway at the control point is less than the threshold value, x,, the bus is held to x,; otherwise, the bus departure is not delayed. This type of strategy is appropriate when scheduled service is frequent and headways are uniform. Under these conditions, passengers are considered to arrive at stops randomly (i.e. without regard to schedule) and it can be shown (Welding, 1957) that the average wait time is: where E(W) = average wait time V(H) = headway variation H = average headway. Thus, it is apparent that control strategies aimed at reducing headway variation can affectuate reductions in passenger wait times at stops. Numerous previous studies related to the modeling of headway variation and the investigation of headway control strategies have been reported. Studies conducted by Barnett (1974) and Turnquist (1978, 1980) provide some of the conceptual groundwork for the work reported in this paper. Additional theoretical and simulation-based studies related to this subject include Osuna and Newel1 (1972), Bursaux (1979), Bly and Jackson (1974) and Kaufman (1978). A review of the above-referenced work indicates that analytical solutions to the threshold control problem are very cumbersome and require several restrictive assumptions, particularly with regard to the number of buses (headways) treated and the headway distribution generated. Furthermore, those studies which have considered impacts of control beyond the control point have assumed that these effects are distributed uniformly to all downstream stops. Consequently, the results of these studies are, in most cases, very limited. There is an explicit need to understand the factors affecting headway variation and the resulting impact of control on headway variation beyond. the control point, while maintaining an analysis framework which is representative of transit operations. In this paper, a headway variation model is developed, which is subsequently used to derive a threshold control deci-
Headway Variation 75 sion methodology. This methodology is later evaluated in terms of its effectiveness in reducing the headway variation and passenger wait times. Headway Variation Model An empirical headway variation model was developed based on Monte Carlo simulations. Simulation time periods of up to four hours and scheduled headways of three, six and nine minutes were used in order to account for the full range of headways and operational periods where thresholdbased holding strategies may be applicable. Vehicle run times to any given stop were assumed to be independently distributed random variables (Abkowitz and Engelstein, 1983a) with symmetric beta distributions (Polus, 1975) having a range from 0.7 to 1.3 of the mean run time. This range of run times permitted run time coefficients of variation from 0.05 to 0.17, which are representative of actual bus performance. The simulation output consisted of the computed headway variation for each combination of scheduled headway, run time and run time variation. (For a more detailed description of the simulation procedures, the reader is referred to the work of Abkowitz and Engelstein (1984) and Engelstein (1984).) A consistent result in all the simulation runs was that the headway variation increases rather sharply with the run time variation at low values and then tapers off. A non-linear regression analysis using the data generated by the simulation yielded the model:* where V(H) = (-12.2 + 6.95H) [l - exp[-(o.o45v(r))]] (2) n = 554 residual mean square = 4.08 V(H) = headway variation (squared minutes) V(R) = run time variation (squared minutes) fi = mean headway (minutes). *Theoretically, H2 should be used instead of I? in this expression, so that V(H)-HZ as V(R) -0. However, model estimation results using Hz yielded a residual mean square = 4.95, which was discarded in favor of the model appearing in Equation 2, on the basis of overall fit to the observed data.
76 Mark Abkowitz, Amir Eiger and Israel Engelstein V(H) 63 5P a5 36 27 18 9. 3. 00 o o n P.* D? 00 D O -00 Po0 PO 90 00 *O 0 *O 0 PPP P?* w P P 000 00 * PP o o m o 0 0 PO00 0 800 o w.* *O -Q n *Q.??PP? am 00***- 000 0 00 B 4 0 +? '.OF '?P, 0 0 0 P P *. 0 0 a * 0 0 C 0 P 0 0 i 1 9 0 P 0 i i - 3 2 5. 75-12s 17 c 22s 0-0 50. I 00 1so 200 250 V( R? Figure 1. Headway variation vs. run time variation (P-predicted headway variation, 0-simulated headway variation, *-simulated & predicted have same value). Figure 1 illustrates the observed and predicted values of the headway variation for scheduled headways of three, six and nine minutes. The model was successfully tested for other values of mean headway using additional simulation runs. Note that the application of the above model requires knowledge of the run time variation. In many practical situations, this may not be directly available and empirical models must be used. Abkowitz and Engelstein '(1983b), for example, have reported empirical results which relate the run time variation to the mean run time.
Headway Variation 77 Predicted He adway Variation 0 FI = 11 minut es + H = 8 ' H = 6 Observed Headway Variation Figure 2. Model Validation: predicted vs. observed headway variations. Model Validation Data from Route 44 in Los Angeles, California was used to validate the headway variation model of Equation (2). Route 44 was divided into links demarcated by the location of the automatic vehicle monitoring (AVM) stations. At each AVM location, data for computing the run time and headway variation was available. Sequences of successive headways corresponding to different scheduled headways were available for calculating the headway variation. Route 44 operates on six, eight and eleven minute headways during different time periods of the day and each sequence comprised a sample of nine headways from which the headway variation was computed. Larger sample sizes were not available since not all vehicles were equipped with an AVM monitor. A plot of the predicted and observed headway variation is shown in Figure 2. The plot shows a reasonably good fit of predicted to observed headway variation.
78 Mark Abkowitz, Amir Eiger and Israel Engelstein Headway Variation Downstream of the Control Point The effectiveness of a holding control strategy is dependent on its ability to reduce the headway variation at the control point and at stops downstream of the control point. It is postulated that, for a threshold-based headway control strategy, the headway variation downstream of the control point depends on the incoming headway variation at the control point, the threshold headway and the mean headway. As the threshold value increases, the departing headway variation is smaller since more buses are held at the control point, resulting in more regulated service. Monte Carlo simulations as described previously were used to generate the necessary data for estimating the headway variation model for departure headways at the control stop. Three, six and nine minute headway routes were simulated with control at points located 10, 20, 30 and 40 minutes from the route origin on successive runs. In addition, the threshold headway was varied in 0.5 minute increments from 0.0 to the scheduled headway. The simulation data was used to estimate the departure headway variation model given by: yd(h) = 0.55 ~(Zl)0.71 (H- x,)'.'~ (3) n = 357 corrected r2 = 0.94 where yd(h) = departure headway variation at control pointj x, = threshold headway (H) = incoming headway variation at control point j. Using Equations (2) and (3), one can express the headway variation at any stop i downstream of control point j as: (H) = (-12.2 4-6.95H) -12.2 4-6.95H I X exp [-0.045 (V,(R)- VJ (R))] where K(R) and c(r) are the run time variation at stops i and j, respectively. One of the immediate implications (with respect to control) of Equations (2) and (3), is that the reduction in headway variation at points 1 (4)
Headway Variation 79 Xeaawav 'Tarla c ion Figure 3. Effect of Control on Headway Variation downstream of the control point is not uniform. The maximum benefits of the control strategy are accrued by passengers at stops immediately downstream of the control point. Stops that are far from the control point may not be impacted significantly. Note in Equation (3) that the headway variation reduces to zero* when x, = H, independent of the level of headway variation before control. This is due to the fact that the distribution of headways is skewed to the right and corresponding to each increase in a headway shorter than the threshold value (xo = H), there is a simultaneous decrease in one of the following headways which is larger than H. This does not suggest that it is always better to hold to the scheduled headway since the optimal strategy depends upon the number of passengers onboard at the control stop as well as the number waiting downstream. Consider the following example of a 10-mile route with six-minute scheduled headways and hypothetical stops every 0.5 mile. Assume further that the coefficient of variation of the run time between stops is 0.17 and that the bus speeds are 10 miles per hour. Using Equations (2) and (4), the headway variation along the route is calculated without control and with control at stops four and 10, as illustrated in Figure 3. The threshold *The actual reduction depends on the number of headways used to compute the variation. In this study, no less than 15 headways were simulated in any given run.
80 Mark Abkowitz, Amir Eiger and Israel Engelstein?zssengers 3oarfing i Control at Stop 4: Total wait = 656 min. Control at Stop 10: Total wait = 538 min. Figure 4. First Passenger Boarding Profile headway of six minutes was chosen so t..at the departure headway variation is reduced to zero. Note in Figure 3 that the effect of the control in reducing the headway variation is minimal at stops that are far from the control point. This further suggests that the selection of a control point must take into consideration the stop location of those passengers waiting downstream. Figures 4 and 5 illustrate two hypothetically different passenger boarding profiles along the route. The total passenger wait time at stops was computed for each profile and each control point using Equation (1). The computed values are shown on the respective figures, from which it is evident that, for the profile of Figure 4, control at stop 10 would be more effective than control at stop four, while in Figure 5 the opposite holds true. Determination of Optimal Control Parameters - As described previously, the objective of the procedure used to determine the optimal value of the threshold headway and the optimal control stop location is to reduce the overall passenger wait and delay time along the route and at the control point, respectively.
Headway Variation 81?assenqers 3oardinp Control at Stop 4: Total wait = 500 min. Control at Stop 10: Total wait = 548 min. Figure 5. Second Passenger Boarding Profile Mathematically, the objective function can be expressed as: where (x,) = total passenger wait and delay time along the route with control at pointj T = time period under consideration (min.) < = number of passengers boarding at stop i during T nj = average number of passengers onboard at stopj N = number of stops on the route y = weighing constant for passenger delay at control stop.
(I-?) 82 Mark Abkowitz, Amir Eiger and Israel Engelstein Figure 6. Illustration off(x,; I?) and where dj(x,) denotes the average on-board passenger delay (minutes) at the control pointj derived from simulated data and given by (Abkowitz and Engelstein, 1984): dj(x,) = [3.92 4-0.0765 (H)] (534 (6) H The minimization of the system wait time, w(x,) with respect to the threshold headway x, and the control stop locationj, is formulated as the mini-rnin problem: mi,n [min v(xo)]. In this formulation, the optimal value of x, is determided atxiach stop and the stop at which control produces the minimum passenger wait time is chosen as the control point. To determine x:j = min Y(X,), one must solve a/ax, Y(x,) = 0 for all j. Differentiating Equatizn (5) and simplifying yields: where f (x,;h) = xi(h - x0)0.27 = ytn sj = 4 [15.7 + 0.3 (H)] HS q/ = 0.2 ~ - 6 (H)0.71exp(0.045 5 (R)) H & X 4 exp (-0.45 (R)) I=, (7) (9)
Headway Variation 83??. ( XJ J i c Figure 7. Total system wait, Case 1 Figure 8. Total system wait, Case 2 * X F I X oj 0 Note in the illustration off (xo; fi) in Figure 6 that Equation (7) may have zero, one or two solutions depending on the values of qj/sj and fmax = 0.4 H3.'. Four cases are considered: Case 1: qj/sj > fmax A solution to Equation (7) does not exist. Hence, H$(xo) is monotonically decreasing and the optimal threshold is x:j = I? (See Figure 7). Case 2: qjlsj = fmx Equation (7) has only one solution, namely xzj (See Figure 8).
84 Mark Abkowitz, Amir Eiger and Israel Engelstein Figure 9. Total system wait, Case 3 Figure 10. Total system wait, Case 4 X' oj fi Case 3: qjisj < f m X Equation (7) has two solutions. Let denote the smaller of the two This situation is illustrated in Figure 9 and suppose that y( H) < y(.a&). with the optimal threshold given by xzj = i.. Case 4: qj/sj < fmaxand &(I?) > Y(.', This case is illustrated in Figure 10 and x:~ = x;,. Thus, the optimal threshold headway, xzj, is given by:
Headway Variation 85 Pas s e nge rs A!&;+L& A L \ 1 r\, l\ I Passengers boarding H = 3 ain stop No. Passengers on-board control stop = 19 x* = 1.8 min O* tk = 10.6% Figure 11. Evaluation scenario 1 Passengers Passengers boarding = 9 min stop Yo. A Passengers on-board control stop = 17 xt = 9 min Figure 12. Evaluation scenario 2 I x; = xi, H A::%*= 9.258 if q,/s, >fmx or U: (HI < xi,) otherwise where all the terms have been previously defined. Note that x:, the minimal solution to Equation (7), must be determined numerically. Using the above procedure, the optimal value of the threshold headway, xz,, and the corresponding system wait time, y(x%,), are computed at each stop along the route. The optimal control stop location, j*, is easily found by selecting the stop at which y(x;j is a minimum. This completes the algorithm. Preliminary evaluation of the algorithm described above was conducted using a hypothetical 30-stop bus route. Five scenarios representing differ-
86 Mark Abkowitz, Amir Eiger and Israel Engelstein Passengers A F Figure 13. Evaluation scenario 3 Passengers - \ t 14-2 Passengers boarding fi = 6 min Szop No. A Passengers on-board control stop = 12 xg = 6 min Aw:'~ j.19%. Passengers boarding = 7 min Stop No Passengers on-board c:ntrol stop = 17 xg = 4.5 min CW" = 4.03% Figure 14. Evaluation scenario 4 ent passenger boarding and alighting profiles, and scheduled headways were examined. Bus run time characteristics were assumed to be the same for all scenarios. In each case, the optimal threshold headway and control point were found, and the percent reduction in total wait time computed directly from simulation results. Figures 1 1 to I5 illustrate the evaluation scenarios and results from which two observations emerge. First, the location of the optimal control stop is sensitive to the passenger boarding profile along the route. In general, the optimal control point is located just prior to a group of stops where many passengers are boarding. Second, the threshold headway is sensitive to the number of passengers on board the bus at the control point. In view of what has been stated previously, both of these results are to be expected.
Headway Variation 87 F as sengers Passengers boarding = stop No. A Passezgers on-board control stop = 16 x,* = 2.1 min AN" = 2.96% Figure 15. Evaluation scenario 5 Summary The effect of a threshold-based holding control strategy on reducing the headway variation at stops downstream of the control point were examined in this paper. Simulation techniques were used to develop the headway variation model which was validated using empirical data from Los Angeles, California. The model indicates that the headway variation does not increase linearly along a route, a common assumption made in many previous studies. Rather, headway variation increases sharply with the run time variation at low values and then tapers off. This result suggests that control is more effective at stops that are closer to the control point and this effect is reduced at stops further downstream. A simple algorithm was developed to determine the control stop and threshold headway which would yield the greatest reduction in total system wait time. Total wait time includes both wait time at stops and on-vehicle delay at the control stop. The on-vehicle average delay was modeled separately using simulation and, together with the headway variation model, comprised the necessary elements needed to define the system performance function to be optimized. The threshold-based headway control strategy was evaluated by simulating several scenarios of passenger boarding and alighting profiles, mean headways, etc. The optimal control resulted in a 3-10 percent reduction in total passenger wait time.
88 Mark Abkowitz, Amir Eiger and Israel Engelstein Acknowledgement This paper is based on research sponsored by the Office of Service and Management Demonstrations of the Urban Mass Transportation Administration (UMTA). The support provided by Mr. Joseph Goodman of UMTA is particularly appreciated. The assistance in model formulation and estimation provided by Mr. Jorge Galarraga was a valuable contribution to this effort. References Abkowitz, M. and Engelstein, I. (1984). Methods for Maintaining Transit Service Regularity, UMTA, Report No. NY-06-0097. Abkowitz, M. and Engelstein, 1. (1983a). Factors Affecting Running Time on Transit Routes. Transportation Research, 17A, 2, 107-113. Abkowjtz, M. and Engelstein, I. (1983b). Empirical Methods for Improving Transit Scheduling. Proceedings of the World Conference on Transport Research, Hamburg, West Germany, 844-856. Barnett, A. (1974). On Controlling Randomness in Transit Operations. Transportation Science. 8, 2, 102-116. Bly, P. H. and Jackson, R. L. (1974). Evaluation of Bus Control Strategies by Simulation. TRRL Laboratory Report No. 637, Transport and Road Research Laboratory, Department of the Environment, Crowthome, Great Britain. Bursaux, D. 0. (1979). Some Issues in Transit Reliability. Unpublished M.S. Thesis, Department of Civil Engineering, Massachusetts Institute of Technology, Boston, Mass. Engelstein, I. (I 984). Methods for Maintaining Transit Service Regularity, Doctoral Dissertation, Department of Civil Engineering, Rensselaer Polytechnic Institute. Koffman, D. (1978). A Simulation Study of Alternative Real-Time Bus Headway Control Strategies. Transportation Research Record. 663,41-46. Newell, G. F. (1974). Control of Pairing of Vehicles on a Public Transportation Route, Two Vehicles, One Control Point. Transportation Science. 8. 3, 248-264. Osuna, E. E. and Newell, G. F. (1972). Control Strategies for an Idealized Public Transportation System. Transportation Science, 6, 52-72. Polus, Avishai (1975). Measurement of Transportation System Reliability: Concepts and Applications. Unpublished Ph.D. Thesis, Northwestern University. Turnquist, M. A. (1978). A Model for Investigating the Effects of Service Frequency and Reliability on Bus Passenger Waiting Time. Transportation Research Record. 663, 70-73. Turnquist, M. A. and Blume, S. W. (1980). Evaluating the Potential Effectiveness of Headway Control Strategies for Transit Systems. Transportation Research Record. 746,25-29. Welding, P. 1. (1957). The Instability of Close Interval Service. Operational Research Quarterly. 8, 3, 133-148.