EVALUATING POTENTIAL EFFECTIVENESS OF HEADWAY
Transportation Research Record 74625Evaluating Potential Effectiveness ofHeadway Control Strategies forTransit SystemsMark A. Turnquist and Steven W. BlumeHolding strategies for control of headways between transit vehicles areoften considered as a means of improving the reliability of transitservice. This paper describes simple tests that can be used to identifysituations for which control is potentially attractive. These tests dependonly on a simple measure of headway variability and the proportion oftotal passengers who will be delayed as a result of the holding strategy.Thus, this analysis provides transit operators with a simple screeningmodel to evaluate potential effectiveness of controls.Headway control has been proposed as one way to improve the reliability of transit service. By reliabilitywe mean the ability of transit to adhere to schedule orto maintain regular headways and a consistent traveltime. This ability is important to both the transit userand the transit operator. To the user, nonadherence toschedule results in increased wait time, makes transferring more difficult, and creates uncertainty aboutarrival time at the destination. To the operator, unreliability results in less effective utilization of equipmentand personnel and reflects itself in reduced productivityand increased cost in the system's operations.A study of the potential effectiveness of various strategies for control of unreliability in transit services isthus a vital element in the search for ways to improvetransit productivity and efficiency. Such control strategies have important implications for both planning andmanagement of transit systems. Control strategies maybe divided into two basic groups: planning and real time.In general, the distinction is that planning strategies involve changes of a persistent nature. Examples includerestructuring of routes and schedules, changes in thenumber and location of stops, or provision of exclusiverights-of-way. On the other hand, real-time controlmeasures are designed to act quickly to remedy specificproblems. These actions have immediate effects butseldom exert any influence on the general nature of operations over a longer time period.Several real-time strategies for correcting servicedisruptions have been discussed in the literature. Agood summary of the state of current knowledge in thisarea has been provided by Abkowitz and others (1). Onecommonly considered control strategy is the holding ofselected vehicles at control points along a route to regularize headways between successive vehicles. That is,a vehicle that arrives at the control stop too close behind the preceding vehicle would be deliberately delayedto make the headway between these vehicles more nearlyequal to the scheduled headway.The major incentive for making headways more regular is to reduce waiting time of passengers who board ator beyond the control point. If passengers arrive at astop without regard to the schedule of service (i.e., 1·andomly), a well-known formula [see Welding (2) ] givesthe average wait time asE(W) [E(H)/2] [V(H)/2E(H)]whereE(W) average wait time,(I)E(H) average headway between vehicles, andy(H) variance of headway.Thus, making headways more regular (i.e., reducing thevariance) serves to reduce average wait.On the other hand, the major costs of such a policyare borne by passengers who are already on the vehicle,since they are delayed when the bus is held up. Thus,the implementation of a holding control strategy involvesmaking some passengers better off at the expense ofothers. At a minimum, if control is to be effective, itmust reduce aggregate waiting time by mo1·e than it increases aggregate in-vehicle time (possibly allowing forsome differential weighting of these two elements of totaltrip time).The purpose of this paper is to provide some basicrules of thumb to indicate the conditions under which aholding strategy might be effective. By implication, wealso wish to describe those situations in which such astrategy is not likely to be effective. These rules ofthumb are based on relatively modest data requirementsabout the route and, hence, should be useful in makingbasic planning decisions about whether or not to implement such a control strategy on a given route.PREVIOUS ANALYSISAn article by Barnett (3) has provided several importantideas for the work contained here. He formulated amodel based on a simple discrete approximation to theprobability distribution of vehicle arrival ti.mes at busstops. Based on this simple model, an optimal holdingstrategy can be derived to minimize the total delay to allpassengers who use the route . Tbe resulting strategydepends on (a) the mean and variance (or standard deviation) of the headway distribution, (b) tbe ratio of averagevehicle load at the control point to average number ofboa1·ding passengers at subsequent stops, and (c) thecorrelation between successive vehicle arrival times atthe control stop. This last information is a measure ofthe degree of bunching or pairing of vehicles on the route:A route on which vehicles have bunched in pairs wouldhave a large negative correlation between successiveheadways because a very s hort one (between two pairedvehicles) will be followed by a very long one (betweenbunches). Statistical estimation of this correlation isdifficult, however, because of the small sample sizesavailable and the notorious unreliability of the estimatorsof covariance.The objectives of this paper are to analyze holdingstrategies by using a more general probability model ofvehicle arrival times at the control stop and to shedsome additional light on the question, Under what conditions is control likely to be of value? Specifically, wewish to allow a transit operator to address this questionwithout detailed knowledge of the covariances betweensuccessive vehicle arrival times at stops, as this information is seldom available.Our approach is to use a general model of the probability distribution of headways between successive vehi-
26Transportation Research Record 746cles and then examine two simple cases that provide approximate upper and lower bounds on the potential benefits of a holding strategy. By doing this, basicconclusions can be reached regarding situations inwhich control is likely to be beneficial and those inwhich it is not.We will examine a holding strategy that holds eachearly vehicle (i.e., each vehicle preceded by a shortheadway) until the headway preceding it reaches a minimum allowable value (hm;n). The structure of the analysis is to find the value of hmin that minimizes total delayto passengers (including both wait time and in-vehicledelay). This optimal value of hmin will be denoted h;t;n Once h.1;;n is found, those situations for which control isadvantageous can be identified.UPPER BOUND ON EFFECTIVENESSOF HOLDINGControl of headways will make the greatest reduction intotal delay when headways alternate (i.e., short, long,short, long). This llappens on routes where vehiclesare influenced substantially by the operation of the vehicle in front of them. For example, this would tend tobe the case where loading delays are relatively moreimportant than traffic congestion in determining overallvehicle operating speed. Routes in which pairing isprevalent would be of this type. In such a situation,holding a vehicle to lengthen a short headway also servesto reduce the long one that follows. Thus, the varianceof headways is reduced by a greater amount for a givendelay to the held vehicle than if short headways might befollowed by another short headway.The extreme case is when the observed sequence ofheadways alternates between two discrete values. Inthis case, the sum of any two consecutive headways is aconstant. That is, if one headway is 2 min too short,the next one must be 2 min too long. By the same argument, if the second headway is 2 min too long, the thirdmust be 2 min too shorl, and so on. In a statisticalsense, successive headways are perfectly correlated,so that knowledge of one headway implies knowledge ofthe entire set. For this case, headway control will havemaximum benefits.If we denote the scheduled headway by H and the magnitude of the deviation by x, the marginal probability density function for headways before control is given by p(H):p(H) 0.5H H- x(2a)0.5H H x(2b)IFor the probability distribution of headways described byEquations 2a and 2b, the expected headway is H and thevariance is x 2 The control action lengthens the shortheadways to a value hmin H - px, where 0 ,;; p ,;; 1.We will define an optimal holding strategy to be onethat minimizes total delay to passengers. Total delayis expressed asT -yE(D) (I -1)E(W)(3)whereT total delay to all passengers,E(D) expected delay to passengers already on boardthe vehicle,E(W) expected wait time for passengers arriving ator beyond the control stop, andy weighting constant to reflect the relative number of passengers already on board to thosewaiting to board at subsequent stops.The expected delay to passengers already on the vehicleis simply the average length of time a vehicle will beheld. If we assume that passengers arrive at stops atrandom times, Equation 1 can be used to determine expected wait time.A holding strategy that minimizes T will be definedby the value h;t;n. Because hmin H - px, We can find hit;nby finding the optimal value of p. Note that after controlthe headway distribution is given byp'(H') 0.5H' H-px(4a)0.5H' H px(4b)lThis distribution has expected value still equal to H, buthas variance p 2 This reduces wait time for passengers yet to board toE(W') (H/2) (p 2 x2 /2H)(5)The delay to passengers already on the vehicle isequal to (1 - p)x if the vehicle is held. Since the probability of·a short headway is 0.5, the expected in-vehicledelay isE(D) 0.5(1-p)x(6)By substituting Equations 5 and 6 into Equation 3, we obtain total expected delay asT 0.51(1 -p)x (1 --y)[(H/2) (p 2 x 2 /2 H) ](7)To find the optimal value of p, we can differentiatethe expression for T with respect to p, and set the result equal to zero.dT/dp -0.51x (1--y)p (x 2 /H) 0(8)This implies an optimal value for p:p 0.5/x H/(l- /) x 2 [0. 51/0- 1)] (H/x)(9)The resulting value for h t;n is then1h*min [(l - 1.51)/ l -/]H(10)For control to be effective, we must have htun H - x;that is, the optimal minimum headway after control mustbe greater than the short headways before control, or itdoes not pay to control at all. This means th: .t we musthave p 1, which implies that we must satisfy the condition x/H 0.5y/l - y. "However, recall that the variance of the headway distribution before control was x 2 Thus, the quantity x/H is simply the coefficient of variation (standard deviation divided by mean) of the headwaydistribution. Thus, for control to be effective, the coefficient of variation of the headway distribution mustexceed 0.5y/1 - y. If it does not, the optimal value ofp is 1, which implies no control.This condition, then, provides a simple test for potential effectiveness of a control policy. It is based ontwo simple pieces of information: (a) the coefficient ofvariation in the headway distribution and (b) the relativeproportion of riders who are already on board the vehicle to those who are yet to board at subsequent stops.It must be kept in mind that this condition is derivedfor the best possible case (i.e., when successive headways are perfectly correlated). Thus, if the conditionis not met, we can be confident that control will not beeffective. However, we must look more closely at situations for which the condition is met because the actual
Transportation Research Record 746Figure 1. Headway distribution (a) before controland (b) after control.f ( h)(al27tribution function [G(h) ] as shown in Equation 11:G(h) 0P(Hj "'h) P(H;-1 hm;n)P(H; .;; h) P{H;-J .;; hmin)Xheadway, hg (h)(b)hheadway, hsituation may be less favorable to control than is reflected in this model.P{H;-1 H;.;; h hmin IH;-1 ;;hm;n)hC hmin{1 J)where P( ·) denotes the probability of the event describedby .( ·l. From the distribution function in Equation 11,we can obtain (at least in theory) the probability densityfunction [g(h)] for h ;;: hm1n, shown in Figure l b, by differentiating with respect to h. The probability thath hmin is given by G(h111;,J.As in the previous case of perfectly correlated headways, our analysis proceeds by solving for the optimalvalue of h 01 ;n and then using this to describe the conditions for which control is potentially beneficial. Theprocess of finding the optimal hmin involves trading offreductions in wait time (due to reduced headway variance)against in-vehicle delays due to holding of vehicles.The variance of headways after control can be writtenas shown in Equation 12:V(H')In order to establish a lower bound on the effectivenessof holding, we will examine the opposite extreme case,which corresponds to the situtation in which headwaysbetween successive vehicles are statistically independent. This means that knowledge that a given headwayis short gives us no additional information about theprobable values for the next headway. Such a situationwould arise, for example, when traffic conditions havea much greater effect on vehicle operations than doesthe loading time at stops. In this case control will beless effective because we have no guarantee that bylengthening a short headway we are also reducing a longheadway. We might be simply reducing another, alreadyshort, headway. This case of independent headways thusprovides a lower bound on the effectiveness of controlstrategies, which will allow us to further refine ourevaluation of situations likely to be favorable for control.We assume that the distribution of headways beforecontrol is applied is descr ibed by a cumulative distribution function [F(h) J with a density function [;f(h) ]. Theeffect of the control strategy is to make all headwaysless than some value (hm;n) equa l to that value. The distribution of headways before and after control is shownin Figure 1. There is a nonzero probability that theheadway will take on the discrete value hmin, and forvalues of h hmin, there is a continuous density function.The expression for the distribution of headways aftercontrol is applied can be derived by considering a sequence of two successive headways after control, whichwe will denote H; 1 and H;. The probability that H; ,;; hdepends o·n both the headways H l - l and H 1 before co ntrolof vehicle i (if any) , as well as the value of the minimumallowable headway 01rnrnl . On one hand , H; ,; h if H1 1 hn,;n (and thus not changed by the control st r ategyJ andH 1 ,; h. If H 1 1 ,;; hrnin, it becomes H; 1 hrnin (after control), and the i th headway is shortened. In this caseH; ,;; h if the sum of H 1 1 and H 1 before control of i wasless than hmm h. Of course, because the control policyenforces a minimum headway, the probability is thatH; ,; h will be 0 for h hmin· These statements can besummarized in the form of a cumulative probability dis-The rate of change of this variance with changes in h111in is [hmin - E(H')]2 G(hm;n) f LOWER BOUND ON THE EFFECTIVENESSOF HOLDING[h - E(H')J 2 g(h)dh(12)hmin(d/dhm;n)V(H') 2G(hminHhmin - E(H')J[I-(d/dhm;n)E(H')](13)For rel ativel y small values of h111;n, we can argue (to afirst-o rder app roxima t ion) that changes in hniin will notaffect the mean headway. Thus, (d/dhm;n)E(H') 0.While this approximation is not strictly accurate, agood case can be made that an operator is unlikely toimplement a control policy that increases mean headwaysignificantly. This would have negative impacts on vehicle productivity and also on passenger wait and traveltime. Thus, the magnitude of control delays applied islikely to be small, and hence the approximation is areasonable o ne . For small values of co ntrol delay, wecan a lso approximate G(hrn1n) by F(hmin). T hese two approximations allow us to obtain the result in Equation 14:{d/dhm;n)E(W') (d/dhm;n) { [E(H')/2] [V(H')/2E(H')J} Chm;n - H/H) F(hm;J(14)This provides the ability to evaluate (approximately) themarginal rate of reduction in waiting t ime as the minimum allowable headway increases. Total delay (T) willbe minimized when the marginal rate of reduction inwaiting time is just equal to the marginal rate of increasein in-vehicle delay.The delay incurred by passengers already on a vehiclethat is held is given by:D Ihmin-H(l 5a)0(I Sb)From this, we can derive the expected delay, as shownin Equation 16:E{D) ih .mm(hmin - h) f{h)dh0 hmin F(hmin) -fhmin0hf{h)dh(16)
28Transportation Research Record 746 -!Figure 2. Areas of potential usefulness for headway control.I1.00.8[(1-2-y)/(l--y)JHO c -y 0.5(I 9a)0 S c -y c l(19b)hmin -Control may produce benefitsin this regionThe condition for which nonzero holding is beneficial(O s y 0.5) provides important information on the situations in which holding strategies are potentially useful,in the same way that the condition x/ff 0.5y/l - 'Y fromEquation 9 did.Definite benefitsfrom control inthis regionIMPLICATIONS OF THE ANALYSIS.--.8c0.6No benefits fromcontrol in this regionu0.4"".," " 0.2 -0.20.60.40.81.0Proportion of passengers delayedFigure 3. Trajectory of stops along a bus route.1.0c:00.80--.0 0c:-.0.6·;:;0u/, ''0.4 -9-',3""lO,.,".":i::,',.o4I'I0.2II,. 200" '10.20.40.60.81.0Proportion of passerrgers delayedThe marginal change in expected delay is thenBy using the expressions in Equations 14 and 17 wecan then solve for an optimal value of hrrun by settingdT/dhmin O.(18)Since y O, hmin s H. Because we must also have hmin O,this solution is only valid if y 0,5, We can summarizethis as shown in Equations 19a and 19b, which give theexpression for the optimum value of hrrun, denoted hil:J:.These two pieces of information can be combined, as illustrated in Figure 2, to yield a convenient representation of situations for which headway control is likely toproduce benefits for passengers-those for which it isunlikely to be worthwhile and those for which more careful analysis is required. By analyzing the two extremecases of independent headways and perfectly correlatedheadways in detail, we can bound the regions of effectiveness for a class of headway control strategies, as shownin Figure 2. For situations in which control producesbenefits under both extremes, we can be fairly confidentthat it will be beneficial. On the other hand, there aresituations in which control does not appear to be desirable under the best of circumstances; hence, controlin these situations is unlikely to be useful. There remains one reasonably small region in which controlwould probably produce benefits on routes where vehiclesare substantially influenced by the vehicles in front ofthem but not on routes where vehicles move relativelyindependently of one another. For situations in this region, more detailed and specialized analysis is required.A major implication of the result shown in Figure 2is that it is wise to control a route at a point where relatively few people are on the vehicle and relativelymany are waiting to board at subsequent stops, in orderthat the value of y be small. Generally, this means thatthe control point should be located as early along the vehicle's route as possible. However, reliability problems worsen as one proceeds along a route. If dispatching at the route origin is effective, the headways will bereasonably regular at the early stops along the route,which implies that the coefficient of variation will besmall. At stops further along the route, however, thecoefficient of variation in headways will tend to be larger.Thus, the decision of whether or not to implement a control strategy is tied to identification of a logical controlpoint along the route.Each stop along a route will have a particular headwaydistribution (with implied coefficient of variation) andvalue of y associated with it. Thus, each stop could beplotted as
ment, if the second headway is 2 min too long, the third must be 2 min too shorl, and so on. In a statistical sense, successive headways are perfectly correlated, so that knowledge of one headway implies knowledge of the entire set. For this case, headway control will have maximum benefits. If we de
TIME HEADWAY (HEADWAY) Time headway (h): is the difference between the time the front of a vehicle arrives at a point on the highway and the time the front of the next vehicle arrives at that same point. Time headway is usually expressed in seconds The microscopic character related to volume is the time headway
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