Strategies for Mitigating Impacts of Near-Side Bus Stops on Cars. Weihua Gu, Michael J. Cassidy, Vikash V. Gayah, Yanfeng Ouyang

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1 INSTITUTE OF TRANSPORTATION STUDIES UNIVERSITY OF CALIFORNIA, BERKELEY Strategies for Mitigating Impacts of Near-Sie Bus Stops on Cars Weihua u, Michael J. Cassiy, Vikash V. ayah, Yanfeng Ouyang WORKIN PAPER UCB-ITS-VWP January 2012

2 Abstract We consier a bus stop that is locate a short istance upstream of a signalize intersection. A bus that wells at this so-calle near-sie stop can impee queue cars upstream as they ischarge uring their green time at the intersection. Ae car elays an resiual queues can result. All else equal, the closer the stop s location to the intersection, the greater the potential amage to car. We have formulate moels for locating these near-sie stops to achieve target levels of resiual queueing among cars. Kinematic wave theory was use to this en. This same approach was also use to evelop a strategy for further mitigating resiual car queues by temporarily etaining some buses from reaching the stop. This bus-holing strategy can be applie selectively, so that the times that hel buses epart from the stop are not affecte. The strategy therefore will not elay buses over the longer run. Our assessments inicate that this holing strategy can significantly reuce instances of car elays an resiual queueing, especially for stops that are locate very close to their intersections. Keywors: near-sie bus stops, bus holing, kinematic wave theory, car queues 1 Introuction Bus stops on city streets are often locate short istances from signalize intersections. This is one to improve accessibility; e.g., by enabling bus-users to reaily transfer between ifferent bus lines (TRB, 1996). Often, the choice is mae to place the bus stops upstream of their nearest intersections (Kim an Rilett, 2005), an buses that use these so-calle near-sie stops often well in a travel lane when loaing an unloaing their passengers. A welling bus will therefore constrain any queue cars upstream when they ischarge into the intersection uring green times. This can create ae car elays an resiual car queues at the intersection ownstream. Despite these concerns, much of the literature in this realm has focuse only on how the stops affect the operation of buses (e.g., ibson, 1996; Furth an SanClemente, 2006; Kim an Rilett, 2005; Zhou an an, 2005). There are instances in which computer simulation was use to stuy the impacts to car for select sets of inputs (Joyce an Yagar, 1990; Zhao et al., 2007). However, we are unaware of any previous work to formulate analytical moels for preicting these impacts for more general cases. In light of this, we explore strategies for eploying near-sie bus stops that limit the amage one to cars. The strategies are evelope using kinematic wave theory (Lighthill an Whitham, 1955; Richars, 1956; Newell, 1993). It is assume that the car eman for the intersection approach is fixe an sufficiently low that, in the absence of a welling bus, car queues are fully serve in each signal cycle. When the stop is occupie, it is further assume that cars can maneuver aroun the welling bus without elay whenever the stop is not engulfe by a car queue from the ownstream intersection. A problem will arise when the queue has expane beyon the stop: as note above, a welling bus will thereafter constrain the ischarging car flow from the queue upstream. The following section unveils the conitions uner which a welling bus will not prevent a car queue from fully issipating within a single green perio, such that resiual queues o not form. We call these the first-best conitions. The less stringent conitions uner which resiual car queues form, but 1

3 issipate within the next signal cycle, the so-calle secon-best conitions, are unveile as well. We then explore how these first- an secon-best conitions can be satisfie by juiciously selecting the location of a near-sie stop relative to the intersection ownstream. In certain instances, these locations may be prohibitively far from the intersection. Section 3 therefore examines a strategy for managing buses so that first- an secon-best conitions can sometimes be achieve even when a stop is locate in close proximity to its ownstream intersection. The strategy entails etaining select buses in reaching the stop, without postponing the buses epartures from that stop. This bus holing strategy can greatly iminish car elays, as is emonstrate in Section 4 via some illustrative examples. The strategies for locating near-sie stops an for ispatching buses to them are of practical value. These matters are iscusse in Section 5, as are our ieas for enhancing the strategies in future work. 2 Strategies for Locating Stops The framework of our theoretical analysis is presente in Section 2.1. First- an secon-best conitions are evelope in Sections 2.2 an 2.3, respectively. The bus-stop locations require to inuce these conitions are explore parametrically in Section Moeling Framework Consier a near-sie bus stop that is locate a istance,, from the ownstream intersection; see Fig. 1. We enote: as the fixe car inflow in the subject irection 1 ; as the fixe signal cycle length; as the fixe green time; as the green ratio; as the capacity of the approach, an the capacity of the bottleneck create by a welling bus. We assume that the buses well times are of ranom uration that never excee ; an that their heaways are ranom an large enough that each bus arrival can be treate inepenently. Recall too our assumption that the approach is uner-saturate, i.e., ; (1) an that a welling bus will not restrict the inflow of un-queue cars to the intersection, 2 i.e.,. (2) 1 Left-turn can probably be exclue from when these left-turners enjoy a protecte signal phase an their own turn lane(s). 2 If, then a bus welling in a travel lane will create a bottleneck for cars as they approach the intersection, regarless of when an where the bus wells. In this case, the remey coul entail having buses well offline in bus bays. 2

4 q welling bus boaring area Fig. 1 Near-sie bus stop signal Traffic states on the subject approach are moele using a triangular funamental iagram in a movingtime coorinate system, whereby time travels forwar over at a free-flow vehicle pace (see Newell, 1993). For illustration, suppose that point A in Fig. 2 enotes the state of freely flowing cars, C the state when cars ischarge at the approach capacity, B an D the states ownstream an upstream of the bottleneck create by a welling bus, respectively, an J the jam state of cars. We enote as the backwar wave spee in a car queue;,, an as the spees of the waves that separate the other car states shown in Fig. 2. We obtain: ; (3) ; (4). (5) Q flow C w Q B B q A w AD w AJ D w BJ 2.2 First-Best Conitions ensity Fig. 2 Funamental iagram for the approach in moving-time coorinates Following the above framework, the states of car upstream of an intersection can be shown in a time- iagram. For example, Fig. 3 shows the states in the absence of a welling bus, where the circle letters, A, J, an C, enote the car states that correspon to those labele on the funamental J Q /w 3

5 iagram in Fig The horizontal line at istance from the intersection marks the bus-stop location. Note that a bus can only arrive at the stop in one of the perioic time winows that are encircle by otte ellipses in Fig. 3. At other times, the bus woul be blocke from reaching the stop by a queue of (stoppe) cars in state J. We enote as the time interval between the bus arrival at the stop an the last signal transition from re to green; an as the time require of the bus to serve boaring an alighting passengers at the stop. We assume the value of is boune in an interval,. As an example, the soli, bol horizontal line segment in Fig. 3 represents the time that a welling bus spens serving its passengers. We have the following proposition: signal bus stop cycle length, L C re perio green perio, C A J C A J C Fig. 3 Time- iagram in the absence of a welling bus Proposition 1. No resiual car queue will occur at the en of any green signal if an only if one of the following three conitions hols: (a) The bus stop is locate no less than (see Fig. 4) from the intersection, i.e.,, where ; (6) (b), (i.e., the bus arrives to the left of line U 1 in Fig. 4), an, where ; (7) or (c), (i.e., the bus arrives to the right of U 1 in Fig. 4), an The proof of this proposition is furnishe in Appenix A.. (8) Note that if, then we have from (6), an from (7). This means that a welling bus may impose limite elays to cars as they ischarge into the intersection, but no resiual car queue will exist at the en of a green time. This woul be true regarless of where the stop is locate an the perios of time when buses serve passengers there. t a S 3 The waves that separate states C an A are epicte in the iagram as if they travelle at infinite spee; i.e., these waves are shown as vertical lines. This is ue to our use of a moving coorinate system for time. 4

6 re perio signal U 1 w 1 Fig. 4 Illustration of Proposition Secon-Best Conitions In many cases, however, the first-best conitions are not feasible, e.g., because of limite street block length upstream of the intersection, as we shall see in Section 2.4. Thus, in this section we examine the secon-best conitions, uner which some queue cars might have to stop twice at the intersection ue to the constraints on ischarge flow impose by a welling bus. However, uner these conitions, the resiual queue will clear within two signal cycles. Proposition 2. No resiual car queue will persist for more than one signal cycle if an only if one of the following four conitions is satisfie (see Fig. 5): (a) The bus stop is locate no less than from the intersection, where ; (9) (b), (i.e., the bus arrives to the left of U 2 V 2 ), an ; (c), (i.e., the bus arrives between U 2 V 2 an U 1 ), an ; (10) or (), (i.e., the bus arrives to the right of U 1 ), an The proof of this proposition is furnishe in Appenix B.. (11) re perio signal U 2 U 1 w 2 V 2 Fig. 5 Illustration of Proposition 2 Note that if, we have an ; i.e., the secon-best conitions will always be achieve, regarless of when an where a bus wells to serve its passengers. 5

7 These secon-best conitions give transit planners more flexibility to choose the location of the bus stop, as shown next. 2.4 Parametric Analysis We now use (6) an (9) to examine the istance requirements for some typical cases. Figs. 6a an b present values of (ashe curves) an (otte curves) for a range of inflow-to-capacity ratios for cars,. These figures were constructe for an. Note that cases are explore for capacity ratios an in Fig. 6a an b, respectively; an for an in both figures. Further note the soli curves in both figures. These curves isplay values of, efine as the maximum istance that a car queue will reach when not impacte by a welling bus. Thus, a near-sie stop locate from the intersection will not impose any elays on cars. This istance is given by:. (12) The figures show that ashe lines often lie well below their soli counterparts. Thus, we see the greater flexibility one has in locating a stop by forgoing efforts to eliminate car elays entirely an choosing firstbest conitions instea. Interestingly, even greater vertical isplacements can occur between a otte line an its ashe counterpart. Hence we see that still greater flexibility that can come by accepting seconbest conitions. 4 However, the figures also show how both an can approach as increases. Note too that require istances in these instances can be quite long relative to the length of a typical city block (m) g = q/q (a) max 1 2 g = 0.6 Fig. 6 Comparison of,, an 50 q/q It thus seems that realizing even secon-best conitions is not always feasible by the juicious choice of stop location alone. Fortunately, first- an secon-best conitions can still be realize in many instances by using the strategy escribe next (m) g = 0.4 (b) max 1 2 g = We shall also note that,, an all increase monotonically with car eman. While selecting the location of a near-sie bus stop uner variable eman (e.g., in a typical ay), it might be suitable to consier the highest eman. 6

8 3 A Strategy for Holing Buses The propositions of the previous section show that resiual car queues can be inuce not only by the choice of, but by other factors as well, incluing a bus arrival time at the stop, enote. Fortunately, it is sometimes possible to alter a bus arrival to mitigate the amage that woul otherwise be one to car, an to o so without elaying the bus in the longer run. The action woul be simple: when a bus preicte arrival time at a stop woul be amaging to cars, this arrival woul, in some instances, be postpone to a more opportune time. Drivers of these hel buses woul be instructe to temporarily halt (or slow-own) in their lane in avance of the stop. The strategy is technically feasible: bus arrival times can be reliably preicte over the short run (e.g., when buses are equippe with PS evices); an the holing instruction coul be elivere in automate fashion via wireless communication technology. Recall that a bus that is halte or slowe in its lane at a istance of at least from the intersection will not, in itself, impee cars given our assumption of a sufficiently low car inflow,. An all passengers woul soon thereafter be serve at the stop. Very importantly, the holing strategy coul be impose selectively, so that buses are never elaye in eparting from the stop. Thus, the only stake-holers who woul be mae worse-off by the strategy are those passengers on a hel bus who alight at the coming stop. As we shall see, their elays woul be moest; i.e., less than a signal cycle length. Details are offere below. Consier the time- iagram in Fig. 7a. By virtue of Proposition 1, we fin that resiual car queues will occur at the intersection if any portion of a bus well time at the stop falls within the shae areas of the figure. The soli horizontal line represents a bus that arrives at the stop locate istance from the ownstream intersection. Left to its own evices, suppose that the bus woul arrive at a time after the start of the previous green phase, an the perio that it woul spen serving boaring an alighting passengers there is enote by the soli horizontal line. Note that the bus arrival lies within a shae area, an woul thus be amaging to car. Further note that the bus finishes serving passengers at a time that lies to the left of point P in the figure. Yet, the bus cannot epart the stop immeiately thereafter. It must wait instea for part of the car queue to issipate uring the next green time. The bus therefore eparts the stop at the time enote by point P, an eparts the ownstream intersection at the time enote by P (shown in moving time). Note now the bol, ashe horizontal line in Fig. 7a. It is merely the soli line shifte forwar in time. This shifte ashe line oes not lie within a shae area of the figure. Moreover, the perio spent serving passengers at the stop ens by point P. The bus still eparts the stop at the time coinciing with P, an then eparts the intersection at the time coinciing with P. Thus, by eferring the bus arrival at the stop to a time that lies to the right of the vertical line U 1 in the figure, interruptions to ischarging cars woul be iminishe or even eliminate. 5 Yet, the bus can still serve all of its passengers an then epart the stop without encountering elay in the longer run. 5 Consieration of Fig. 7a shows that if the bus arrival can be postpone to a time that coincies with the line U 1, then the isruptions to cars will be iminishe an resiual queues will not form at the intersection. An if the bus arrival can be further postpone to a time that lies at or to the right of the vertical line U 0 V 0 as in the figure, then car interruptions woul be eliminate entirely. 7

9 re perio signal I E U 1 w 1 a welling bus without holing re perio signal I E U 1 w 1 a welling bus without holing t a H (a) An example that is suitable for holing (b) An example that is unsuitable for holing U 0 V 0 F F H bus eparture time from the stop (without holing) the welling bus after holing K P bus eparture time from the stop (with an without holing) the welling bus after holing K P P P bus eparture time from the stop (with holing) L L re perio signal I E U 1 w 1 H F a welling bus without holing K P P L re perio signal I E U 1 w 1 (c) A secon example unsuitable for holing a welling bus without holing H () A thir example unsuitable for holing Fig. 7 Illustration of amaging bus arrivals an their remeies F bus trajectory (without holing) the welling bus after holing K P P L Regrettably, not all amaging bus arrivals can be so easily remeie. Imagine a bus that arrives at the stop at an earlier time that lies to the left of line IH, an with a well time that extens into the shae area boune by line E (see the soli, horizontal line in Fig. 7b). In this case, postponing the bus might be 8

10 objectionable: Fig. 7b shows that a postponement woul elay the early bus eparture from the stop until the time corresponing to point P. Avancing the bus arrival at the stop might preclue the well time from penetrating a shae region in the figure. Avancing a bus arrival is probably infeasible, however. Fortunately, bus well times in cases like this latter one might often persist for only short initial perios of car ischarge; i.e., note how the soli horizontal line in Fig. 7b penetrates a shae region for a limite time. This occurs, in part, because bus passengers can be serve uring the signal s re phases (Newell, 1989). All else equal, the shorter the perios that buses constrain car queues, the lower the amages. Thus, we focus now on the cases like the one in Fig. 7a, an explore the range of bus arrival times for which our holing strategy can be use without imposing long-run elays on buses. Since caniate arrivals for this strategy fall in shae areas like EU 1, we have the constraints:, (13) an. (14) Further, a bus woul not qualify for holing if its well time (in the absence of holing) coul possibly exten into the shae area to the right of line KL. In this instance, the bus eparture time from the stop woul fall beyon point P; see the example shown again with a soli, horizontal line in Fig. 7c. If this bus were to have been hel, its eparture from the stop woul only be further elaye in time. Thus, we have the constraint:. (15) Similarly, a bus woul not qualify for holing if its (un-hel) well time coul possibly en before the queue of jamme cars expane beyon the bus stop; see the soli, horizontal line in Fig. 7. In this case, the bus woul epart the stop to join the tail of the car queue somewhere ownstream, an eventually epart the stop at a time in avance of point P. An example trajectory for this bus is shown in Fig. 7 with ashe lines. Note that holing this bus woul elay what woul otherwise have been an early eparture from the stop. Thus we have:, (16) where is the time interval from the start of the previous green phase (labele point E in Figs. 7a~) to the time when the tail of the car queue expans to the bus stop. Two cases are to be consiere when etermining. These are shown in Figs. 8a an b. In the first case (Fig. 8a), the bus arrives at the stop at the point labele W 1, which is relatively soon after the start of the green. The resulting queue of ischarging cars that forms upstream of the welling bus (a queue of state D) persists when the car queue from the following re perio (of state J) eventually reaches the bus-stop. This meeting of the two queue car states is labele Y in the figure. The bus woul be a caniate for holing only if the time require to serve its passengers extens to Y or beyon, but not beyon P. A bus 9

11 trajectory for this case is exemplifie by short, ashe lines in Fig. 8a. Note how the eparture time from the stop in this case (point P) coul be unaltere by temporarily holing the bus arrival. The secon case (Fig. 8b) is like the first, except that the bus arrival at the stop (point W 2 ) occurs later in time. As a result, the car queue of state D issipates prior to the next arrival of the jamme queue at the bus stop, again labele Y. As in the previous case, the bus is a caniate for holing if its time spent serving passengers extens at least to Y, but not beyon P. re perio signal E C U 1 W 1 t q D B w BJ Y A J w AJ P bus trajectory (a) The first case re perio t q signal E U 1 w BJ C B w AJ W 2 A D A Y J P bus trajectory (b) The secon case Fig. 8 Determining From the geometries of Figs. 8a an b, we fin the values of that rener buses caniates for holing:. (17) The top an bottom equations of (17) correspon to the cases in Figs. 8a an b, respectively. Constraints (13) ~ (17) collectively boun a region, like the shae one in Fig. 9. The bounary line RT is from (14); OR is from (15); MN an TM are etermine from (16), where the values of come from (17); an (13) is a slack constraint in this example. The slope of TM is non-negative if, an negative otherwise; see (16) an the bottom equation of (17). An example of this latter case is shown with otte line MT in the figure. Buses with arrivals that fall within this shae region are suitable for holing. An example arrival of this kin is shown in Fig. 9 with an X. The arrow originating from the X epicts the maximum uration over which this bus arrival can be hel. 10

12 The shae region unveils useful insights. For example, moving the bus arrival to the right of line U 2 V 2 (such that the X lies between that line an bounary OR) woul, by virtue of Proposition 2, prouce secon-best conitions. Recall that moving the arrival all the way to the bounary OR woul mean lower elays for cars (though greater elay for bus passengers who alight at the stop). Thus, we see from the figure that the range of arrivals suitable for holing expans as iminishes. This means that the holing strategy can be especially beneficial to cars when bus stops lie close to their ownstream intersections. As we saw in Section 2, obtaining secon-best conitions can, in the absence of bus-holing, be impossible for a stop place close to the intersection. Holing buses can also benefit cars when the stop is locate well upstream of the intersection. For example, if a bus stop location were to fall within the spatial range that correspons to bounary RT in Fig. 9, then first-best conitions can be obtaine by pushing bus arrivals to that bounary. This is true by virtue of Proposition 1. re perio bus arrival signal E N U 2 O U 1 U 0 I time (moving) R T M T V 2 1 V 0 Fig. 9 The ranges of bus arrivals suitable for holing Finally, if the shae region in Fig. 9 can be enlarge, an if bus arrivals that fall within the region can be pushe to the line U 0 V 0, we fin that a welling bus woul not inuce any elays to cars at all. Interestingly, assessments of (13) ~ (17) show that the size of the shae region increases when ecreases. Thus, cars can benefit more from the bus-holing strategy if the variation in the time require to serve bus passengers is somehow kept small. This an other properties of the holing strategy are illustrate next by means of examples. 4 Effects of Bus Holing on Car Delays We evaluate impacts of bus holing via the simulation of select scenarios. The well-known Cell Transmission Moel (Daganzo, 1994) is use for this purpose. Each simulation epicts operation over a number of consecutive signal cycles that is sufficient to capture the full effects of a welling bus. 11

13 Each scenario reflects istinct,, or istribution of, but all scenarios feature a bus that seeks to arrive an well at a near-sie stop at a time that is uniformly istribute within the first cycle. For each scenario, 10,000 simulations were performe. From these 10,000 realizations, we obtaine the average aitional car elay create by the welling bus, both with an without bus holing. We take as fixe inputs for all scenarios:,,,, an. For our first battery of simulations, was uniformly istribute with an. Fig. 10 presents the expecte aitional car elays for the range of shown an for an. The latter ratio is just below 0.5, which is the upper boun ue to our choices of, an ; see (1) an (2). The ashe curves in the figure isplay the aitional car elays when the bus holing strategy is use. Their soli counterparts isplay these ae elays in the absence of bus holing. Note from Fig. 10 that the vertical eviation between a ashe curve an its soli counterpart unveils the car elay that is save by the holing strategy. Further note how these eviations: are maximum when ; iminish rapily as increases; an isappear for sufficiently large. All this is consistent with the finings reporte in Section 3. Fig.10 also shows how bus holing can provie greater benefit to cars when car eman,, is relatively high. For example, when an, the car elay save via bus holing is 1022 car-sec. This is a large amount consiering that in the absence of a welling bus, the car elay is only 898 car-sec per cycle Expecte aitional car elay (car-secon) q/q = 0.47 without holing with holing 1000 q/q = Fig. 10 Expecte aitional car elay versus (m) Moreover, the simulation also shows that the expecte bus holing times for the two cases of Fig. 10 are 7.4 sec an 5.6 sec respectively. Note how marginal they are, compare to the corresponing savings in car elays (1022 car-sec an 215 car-sec respectively; see Fig. 10). Thus, the holing strategy can significantly reuce the total elay of both car an bus passengers even if the number of bus passengers alighting at the coming stop is large. 12

14 We next explore how the impacts of the bus holing strategy epen upon the variations in bus well time. To this en, we fix an the average bus well time as 45 sec, but allow the variation in bus well time,, to range from 0 to 60 sec. Outcomes are shown in Fig. 11. Note from this figure that the car elay save by bus holing is greatest when the bus well time is eterministic; i.e., when. Note too how these savings: iminish as increase; an eventually vanish when the ifference is sufficiently large. Hence we see the value of limiting the variation in bus well time Expecte aitional car elay (car-secon) 2000 q/q = q/q = 0.4 without holing with holing 0 S max S min (secon) Fig. 11 Expecte aitional car elay versus Aitional consierations for selecting bus-stop locations an for implementing the bus holing strategy are iscusse next. 5 Conclusions When a car queue spills-over beyon a near-sie bus stop, a bus welling there in its lane will constrain the queue cars upstream uring the green time. Kinematic wave theory was use to explore ways of mitigating this amage. Our moels can be use to etermine a stop s location to meet a specifie objective. For example, one might choose to eliminate car elays completely by placing the stop upstream of the fully-expane car queue. Or, one can limit these elays by locating the stop so that a resiual car queue create by a welling bus will not arise (first-best conitions), or will arise, but issipate within two cycles (secon-best conitions). Of course, the choice of a stop s location entails a trae-off between limiting the amage to car an maintaining accessibility to bus users, e.g., to those who transfer between bus lines. This latter consieration favors placing a stop a short istance from an intersection, making the istances require even to achieve secon-best conitions unesirable. One might therefore choose to iminish the require istances between a stop an its intersection by either increasing the signal s green ratio or ecreasing its cycle length (see again (6) an (9)). Or, 13

15 one can jointly choose a stop location an employ a strategy for eferring some bus arrivals at that stop. A bus that halts (or slows-own) in its lane in avance of the stop woul be harmless uner the low to moerate car emans for which our strategies are appropriate; an the holing strategy woul be impose selectively, so that buses woul not fall behin their long-run scheules. Simulations inicate that the favorable impacts of bus holing can be consierable, particularly when car eman is relatively high or when the variation in bus well time can be kept small. As a practical matter, bus passengers might by-an-large object to temporarily halting upstream of a stop. This coul be the case even though the only passengers mae worse-off woul be those who alight at that coming stop; an even though the holing strategy can make it possible to locate stops closer to intersections to enhance bus-user access. These objections might be lessene by having select buses slow, rather than halt, in avance of the stop. In some cases (e.g., when buses carry relatively few onboar occupants), a transport authority might elect to impose small elays on buses in the interest of mitigating car congestion. Caniate buses for this policy woul inclue those that might finish serving their passengers before the car queue expans beyon the stop. Left to their own evices, some of these buses woul epart the stop an join the tail of the expaning car queue somewhere ownstream. Each such bus woul eventually ischarge through the intersection uring the following green time. Shoul these buses instea be hel upstream, this coul be one in such way that each still ischarges from the intersection uring the following green. But some of these buses woul o so from a position that is further back in the car queue; i.e., these buses woul suffer elay, but the elay woul be moest. We note that our strategies are suitable for only a select range of car eman. Moreover, the choice of stop location will be base on a single choice for this eman. Unfortunately, our assessments inicate that stop location is sensitive to the magnitue of the car eman use as input. Thus, the strategies, which were evelope by eterministic means, might not be robust to increases in this eman over time or shorter-run ranom fluctuations in eman. This matter will be explore in future work. We also inten to explore cases when car arrivals to an intersection are batche, e.g., ue to the effects of another signalize intersection upstream. Acknowlegement Funing for this work was provie by the National Science Founation, the Volvo Research an Eucational Founations, an the University of California Transportation Center. Appenix A Proof of Proposition 1 in Section 2.2 The well time of a bus can be ivie into a few segments (see Fig. A1 for an example), some of which are insie the otte ellipses (we term these segments blocking segments ), while others are outsie. Since we assume, a well time can inclue at most two blocking segments as in Fig. A1. To prove Proposition 1, we first prove the following lemma in regar to blocking segments. Lemma 1. No resiual queue will occur at the en of any green signal if an only if each blocking segment of the bus well time either: 14

16 (a) starts to the left of U 1 (see Fig. A2), an is no longer than (b) starts to the right of U 1. ; or These conitions are illustrate in Fig. A2, where each bol line represents a typical blocking segment that satisfies the above conitions. signal bus stop re perio C A J C A J C S blocking segment 1 blocking segment 2 Fig. A1 Blocking segments signal re perio U 1 Conition (a) Conition (b) Fig. A2 Lemma 1 Proof of Lemma 1: First consier the case when the blocking segment starts to the left of U 1, as shown in Fig. A3. re perio signal U 1 θ J C B D C w AD A Fig. A3 Conition (a) of Lemma 1 From Fig. A3, it can be seen that a segment of the green perio is blocke by the welling bus, uring which time cars can ischarge only at rate. We enote the length of that segment as. To guarantee that all queue cars can ischarge by the en of the green perio, must satisfy: 15

17 , i.e.,. (A.1) Equation (A.1) implies that is the maximum portion of the green perio that can be blocke by a welling bus if no resiual queue is inuce. Since the blocking segment starts to the left of U 1, will excee if the blocking segment is greater than (note that the uration between U 1 an the en of the green perio is ). Hence in this case, no resiual queue is left at the en of green if an only if conition (a) of Lemma 1 hols. In the other case, i.e., when the blocking segment starts to the right of U 1 (Fig. A4), state B will persist for at most, thus (A.1) is satisfie with certainty, an no resiual queue will persist at the en of the green perio. Note that after the queue issipates, the welling bus will no longer be an active bottleneck (recall Equation (2) in Section 2.1). Thus the length of the blocking segment is irrelevant. This proves conition (b). re perio signal U C 1 J B D A 1 θ w AD Fig. A4 Conition (b) of Lemma 1 Now we can prove Proposition 1. To prove conition (a) of Proposition 1, by conition (b) of Lemma 1, it suffices to show that the length of U 1 is. From Fig. 4 in Section 2.2, we have:. When, an if the bus arrives to the left of U 1, then conition (b) of Proposition 1 immeiately follows from conition (a) of Lemma 1. If the bus arrives to the right of U 1, its well time can contain at most two blocking segments; as explaine below. The first blocking segment, in Fig. A5, satisfies conition (b) of Lemma 1, an thus will not inuce any resiual queue. In the mean time, the secon blocking segment,, has to satisfy conition (a) of Lemma 1. So,. 16

18 This proves conition (c) of Proposition 1. re perio /w signal U 1 K L N M P t a R 1 Fig. A5 Conition (c) of Proposition 1 S Appenix B Proof of Proposition 2 in Section 2.3 We first prove the following lemma: Lemma 2. If the bus arrives between U 2 V 2 an U 1 (see Fig. 5 in Section 2.3), then the secon-best conition is achieve if an only if. signal re perio J θ 1 U 2 U 1 X W C V 2 B D A S C B J Y θ 2 Z C D Fig. B1 Lemma 2 Proof of Lemma 2: A time- iagram of this case is shown in Fig. B1. From this figure, it can be seen that the portion of green perios uring which cars can only ischarge at, enote as, shoul be no greater than. From this we have:. Thus,. Now we prove the conitions of Proposition 2 one by one. 17

19 For conition (a), we first verify that :. We examine three cases: Case I: the bus arrives between U 2 V 2 an U 1, an. Since we assume, we have. So from Lemma 2, we know that the secon-best conition can be achieve. Case II: the bus arrives to the right of U 1, an conition (a) of Proposition 1) an is therefore trivial.. This case satisfies the first-best conitions (see Case III: the bus arrives to the right of U 1, an. The bus well time in this case can contain at most two blocking segments (see the efinition of blocking segments in Appenix A). Due to conition (b) of Lemma 1 (also given in Appenix A), the first blocking segment will not inuce any resiual queue at the en of the green perio. The secon blocking segment itself can be seen as a (shortene) well time which starts between the counterparts of U 2 V 2 an U 1 in the next cycle (see Fig. B2). Hence, it follows from Case I above that the resiual queue inuce by the secon blocking segment will never persist for more than two signal cycles. This completes the proof of conition (a). Conition (b) of Proposition 2 becomes obvious by noting that is the maximum portion of two consecutive green perios that can be blocke by a welling bus, shoul no resiual queue be present at the en of the secon green perio. Conition (c) immeiately follows from Lemma 2. re perio signal U 2 U V 2 Fig. B2 Case III of conition (a) of Proposition 2 S For conition (), since the bus arrives to the right of U 1, the first blocking segment will not cause any resiual queue. Thus by conition (b) of Proposition 2, the secon-best conitions can be achieve in this case if the secon blocking segment is no greater than (see Fig. B3). Hence,. 18

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21 Zhou,., an, A., Performance of transit signal priority with queue jumper lanes. Transportation Research Recor: Journal of Transportation Research Boar 1925, Transportation Research Boar of the National Acaemies, Washington D.C.,