On the Existence and Uniqueness of Equilibrium in the Bottleneck Model with Atomic Users

Transcription

1 On the Exitence and Uniquene of Equilibrium in the Bottleneck Model with Atomic Uer Hugo E. Silva a,, Robin Lindey b, André de Palma c, Vincent A.C. van den Berg d,e a Departamento de Ingeniería de Tranporte y Logítica e Intituto de Economía, Pontificia Univeridad Católica de Chile, Santiago, Chile. b Sauder School of Buine, Univerity of Britih Columbia, Canada. c Ecole Normale Supérieure de Cachan, Département Economie et Getion, Cachan, France. d Department of Spatial Economic, VU Univerity Amterdam, Amterdam, The Netherland. e Tinbergen Intitute, Amterdam, The Netherland. Forthcoming in Tranportation Science Abtract Thi paper invetigate the exitence and uniquene of equilibrium in the Vickrey bottleneck model when each uer control a poitive fraction of total traffic. Uer imultaneouly chooe departure chedule for their vehicle fleet. Each uer internalize the congetion cot that each of it vehicle impoe on other vehicle in it fleet. We etablih three reult. Firt, a pure trategy Nah equilibrium PSNE) may not exit. Second, if a PSNE doe exit, identical uer may incur appreciably different equilibrium cot. Finally, a multiplicity of PSNE can exit in which no queuing occur but departure begin earlier or later than in the ytem optimum. The order in which uer depart can be uboptimal a well. Neverthele, by internalizing elf-impoed congetion cot individual uer can realize much, and poibly all, of the potential cot aving from either centralized traffic control or time-varying congetion toll. Keyword: Bottleneck model, Large uer, Atomic uer, Exitence of Equilibrium, Uniquene of Equilibrium Introduction Individual uer of road and other tranportation facilitie are uually aumed to be mall in the ene that they control a negligible fraction of total traffic. Yet large uer are prevalent in many etting. Commercial airline and rail companie often account for a izable fraction of total traffic at airport and on rail network. Potal ervice and major freight hipper operate large vehicle fleet that travel long ditance each day. For Correponding author addree: huilva@uc.cl Hugo E. Silva), robin.lindey@auder.ubc.ca Robin Lindey), andre.depalma@en-cachan.fr André de Palma), vberg@feweb.vu.nl Vincent A.C. van den Berg)

2 example, FedEx handle about 150 daily flight out of Memphi International Airport and it air-cargo operation upport ten of thouand of job. UPS operate on a imilar cale out of Louiville, Kentucky The Economit, 2013). Major employer uch a government department and large corporation can add ubtantially to traffic on certain road at peak time. Large uer uch a thee uffer from the congetion delay their own aircraft, train, truck, or other vehicle impoe on each other. Thu, at airport, on rail network, on congeted road, and on other tranportation infratructure network, one would expect large uer to internalize their elf-impoed delay, and therefore to make different triprelated deciion than mall uer controlling the ame aggregate traffic. Following the terminology of game theory we will refer to mall uer a non-atomic, and large uer that control a poitive fraction of traffic a atomic. Thi terminology contrat with the terminology ued in the literature on airport congetion, beginning with Daniel 1995), in which uer that control a negligible fraction of traffic and treat the congetion level a parametric are called atomitic uer. Somewhat confuingly, atomitic uer are therefore non-atomic, and non-atomitic uer are atomic. There are everal branche of literature on congetion with atomic uer. In the aviation literature, Daniel 1995) wa the firt to recognize that airline with market power and large hare of total traffic could internalize the delay their aircraft impoe on each other. Brueckner 2002) howed that under Cournot competition airline fully internalize elfimpoed congetion. Further contribution in thi line have been made by Pel and Verhoef 2004), Brueckner 2005), Zhang and Zhang 2006), Bao and Zhang 2007), Brueckner and Van Dender 2008), and Silva and Verhoef 2013). In the context of road tranportation, route-choice deciion by atomic uer have been tudied e.g., Devarajan, 1981; Haurie and Marcotte, 1985; Marcotte, 1987; Harker, 1988; Catoni and Pallottino, 1991; Miller et al., 1991; Yang et al., 2007; Yang and Zhang, 2008; Cominetti et al., 2009). Yang and Zhang 2008) alo tudied optimal pricing with atomic and non-atomic uer, and He et al. 2013) analyzed tradable credit cheme. There i alo a literature in operation reearch and computer cience on atomic congetion game e.g., Fotaki et al., 2008; Hoefer and Skopalik, 2009). The above-mentioned tudie have ued tatic model of congetion except for Daniel 1995) who ue a tochatic queuing model empirically. Other tudie that ue a dynamic tochatic model of congetion with atomic uer include Daniel and Harback 2008) and Molnar 2013). Except for a few tudie decribed below, traffic congetion with atomic uer ha not been tudied with determinitic dynamic model. Thi i urpriing becaue the timing of trip matter a great deal for both paenger and freight tranportation, and congetion i largely a conequence of peak-period load. The goal of thi paper i to invetigate the fundamental quetion of exitence and uniquene of equilibrium in trip-timing deciion with atomic uer. To focu the analy- 2

3 i on fundamental while minimizing mathematical complication we ue Vickrey 1969) bottleneck model. The eence of the bottleneck model i that uer trade off the cot of queuing delay at the bottleneck with the cot of chedule delay i.e., arriving earlier or later than deired). The bottleneck model ha been ued to tudy many apect of trip-timing deciion with congetion including: congetion pricing, route choice on imple road network, mode choice, trip chaining, parking congetion, taggered work hour, and flextime. Exitence and uniquene of equilibrium in the bottleneck model have alo been etablihed under relatively general aumption about trip-timing preference and heterogeneity of non-atomic uer e.g., Newell, 1987; Lindey, 2004). However, very little conideration ha been given to atomic uer in tudie that ue either the bottleneck model or other dynamic model. A few tudie have employed a variant of the bottleneck model in which time i dicretized, and the number of uer i finite, o that each uer control a poitive meaure of traffic e.g., Levinon, 2005; Zou and Levinon, 2006; Otubo and Rapoport, 2008; Werth et al., 2014). However, thee tudie aume that each uer control only one vehicle o that elf-internalization of congetion doe not come into play. To the bet of our knowledge, only Daniel 2009) and Silva et al. 2014) have explored the cheduling deciion of atomic agent in the tandard, continuou-time bottleneck model. Thee tudie conider, in the context of aviation, a equential competition between a Stackelberg leader with market power and a group of perfectly competitive airline non-atomic uer). Both tudie how that, when uer have homogeneou preference, non-atomic uer chedule all their flight during the peak period when paenger prefer to arrive. The Stackelberg leader chedule a fraction of it flight during the peak a well. Queuing time at the bottleneck evolve at the ame rate a in the tandard model of non-atomic player. The leader chedule it remaining flight earlier and later in the off-peak and le popular) period and limit it departure rate to bottleneck capacity o that no queue develop. The exitence of a unique equilibrium in Daniel 2009) and Silva et al. 2014) hinge on the equential nature of the game they conider, and the aumption that there i only one atomic agent. By contrat, we focu in thi paper on etting with two atomic uer who make cheduling deciion imultaneouly. The olution concept we employ i pure trategy Nah equilibrium PSNE). We etablih three major reult. Firt, we how that a PSNE may not exit. We demontrate thi for an example featuring two identical atomic uer who each control half of the total traffic. The trip-timing preference of each vehicle in each fleet are decribed by parameter {α,β,γ,t }, where α i the cot of travel time, β i the cot of chedule delay early, γ i the cot of chedule delay late, and t i the deired arrival time. We how that if γ > α, a PSNE doe not exit. Second, for the ame example, we how that if γ α a multiplicity of PSNE exit in which no queuing occur and the timing of departure i ytem optimal. The PSNE 3

4 differ according to the departure rate of individual uer and the equilibrium cot they incur. Depending on parameter value, one of the two uer can incur up to three quarter of total cot. The cae γ > α and γ α are both of theoretical interet, and each may be relevant in particular etting. Mot empirical tudie of cheduling preference for automobile driver have obtained etimate that atify γ > α e.g., Small, 1982; Wardman, 2001; Aenio and Mata, 2008). In addition, de Palma and Fontan 2001) lit etimate from eleven tudie and, including their own etimate, there are ten cae with γ > α, and two cae with γ < α. By contrat, Daniel and Harback 2008) find that γ < α hold for mot airline at major US airport. Third, we conider a variant of the example in which the two uer differ in their deired arrival time t and can have fleet of different ize. We how that independent of the relative ize of α and γ a multiplicity of PSNE can exit in which no queuing occur but the timing of departure i not optimal. Depending on parameter value, the PSNE may begin earlier than, later than, or at the ame time a the ytem optimum. The order in which uer depart can be uboptimal a well. Neverthele, by internalizing elf-impoed congetion cot the two uer realize much, and poibly all, of the potential cot aving from either centralized traffic control or time-varying congetion toll. Thee example demontrate that neither the exitence of equilibrium nor the uniquene of an equilibrium if one exit) i guaranteed under condition where a unique PSNE doe exit if all the traffic were controlled by non-atomic uer. Given the central role of equilibrium model in the analyi of tranportation ytem, thee reult are troubling and highlight the need for further reearch. The paper i organized a follow. Section 1 review the no-toll equilibrium and the ytem optimum in the tandard bottleneck model with non-atomic uer. Section 2 demontrate the poible non-exitence of PSNE, and the non-uniquene of individual departure rate and cot where a PSNE doe exit. Section 3 demontrate the poible nonuniquene of PSNE in the timing of departure when no queuing occur, and the degree of inefficiency relative to the ytem optimum. Section 4 conclude. 1. The bottleneck model with homogeneou non-atomic uer The bottleneck model wa developed by Vickrey 1969) and extended by Arnott et al. 1990, 1993). It i reviewed in Arnott et al. 1998) and de Palma and Fogerau 2011), and the ummary here i brief. In the model, all uer travel from a common origin to a common detination along a ingle link that ha a bottleneck with fixed flow capacity,. Without lo of generality, travel time from the origin to the bottleneck and from the bottleneck to the detination are normalized to zero. If there i no queue uptream of the bottleneck, travel time through the bottleneck i alo zero and departure time from the origin coincide 4

5 with arrival time at the detination. If the departure rate exceed, a queue develop. Let t be the mot recent time at which there wa no queue, and rt) the aggregate departure rate from the origin at time t. The number of vehicle in the queue i then: Qt) = t t ru) )du. Travel time through the bottleneck i T t) = Qt)/, and a traveler who depart at time t arrive at time t a = t + T t). Following Small 1982) and de Palma et al. 1983), uer are aumed to have a deired arrival time t. They incur a unit cot of β > 0 for arriving early, and a unit cot of γ > 0 for arriving late. Travel time i valued at α, with α > β. The cot of chedule delay and travel time are additive o that the generalized cot of a trip, ct), i: { β t t T t)), t + T t) t ct) = α T t) + γ t + T t) t ), t + T t) t. 1) The number or meaure) of uer, N, i aumed to be exogenou i.e., independent of trip cot). Each uer decide when to depart from the origin by trading off chedule delay againt travel delay. A pure trategy Nah equilibrium PSNE) i a et of departure time uch that no uer can benefit i.e., reduce trip cot) by unilaterally changing departure time while taking the departure time of all other uer a given No-toll equilibrium Let upercript n denote the no-toll non-atomic PSNE, and t n and t n e denote the tart and end of the travel period. Let t be the departure time for which a uer arrive on time i.e., t + T t) = t ). In a PSNE with no toll, ct) mut be contant during the travel period [t n, t n e ]. Uer who arrive cloer to t mut incur longer queuing delay in order to offet their lower chedule delay cot. The equilibrium aggregate departure rate i derived by differentiating Eq. 1) and etting the derivative to zero: { α r n α β t) =, t t n, t ) α α+γ, t t, t n ). 2) e The aumption α > β aure that the departure rate for early arrival i poitive and finite. Thi condition i plauible ince a uer who i detined to arrive early i likely to prefer arriving early to prolonging the trip by making a detour. The condition i alo upported by Small 1982) etimate for automobile commuting trip. For eae of reference we will ometime call the departure rate for early arrival the early departure rate, and the departure rate for late arrival the late departure rate. There are two further equilibrium condition. One i that the firt and lat uer to depart, who encounter no queue, mut incur equal chedule delay cot: β t t n ) = γ t n e t ). 3) 5

6 The other condition i that the travel period lat for N/: Together, equilibrium condition 2), 3) and 4) yield: t n e t n = N. 4) t n = t γ β + γ N, t n e = t + β β + γ N, t =t β α γ β + γ N, c n t) = δ N, t [tn, t n e ] with δ β γ β + γ. 5a) 5b) 5c) 5d) Total cot are: T C n = δ N 2. 6) 1.2. Sytem optimum Queuing delay at the bottleneck i a deadweight lo. The ytem optimum therefore avoid queuing and minimize total chedule delay cot. The departure rate i maintained at over a continuou time interval choen o that the firt and lat uer incur the ame chedule delay cot. The departure period i therefore the ame a in the laiez-faire PSNE cf. Eq. 5a) and 5b)). Uing upercript o to denote the ytem optimum, thee reult are recorded for future reference a: r o t) =, t t o, t o e), r o t) = 0 otherwie, 7a) t o =t γ β + γ N, t o e =t + β β + γ N. Total ytem cot are only half a large a in Eq. 6) for the no-toll equilibrium: T C o = δ 2 N 2 7b) 7c) = 1 2 T Cn. 8) The difference between total cot in the no-toll equilibrium and ytem optimum, T C n T C o, erve a an upper bound on the benefit from elf-internalization of congetion by atomic agent. 2. Exitence and non-exitence of equilibrium with homogeneou atomic uer In thi ection we tudy an example featuring two identical atomic uer. We how that if γ > α, a PSNE in departure chedule doe not exit. We then how that if γ α, a 6

7 PSNE doe exit that entail no queuing and coincide with the ytem optimum. During early departure the two uer can depart at omewhat different rate that add up to. If γ < α, their late departure rate can alo differ. Moreover, with γ α the uer can incur appreciably different total cot for their fleet. At the end of the ection we briefly dicu how thee reult extend to more than two uer. Conider two atomic uer, A and B. Each uer control a fleet of N/2 vehicle. Triptiming preference for each vehicle are defined by the ame {α,β,γ,t } parameter value. Thu, the total cot of a uer fleet i imply the um of the cot of the individual vehicle. Thi aumption eem realitic for delivery van carrying merchandie or parcel to different cutomer. It may be inappropriate for vehicle in a military convoy or emergency vehicle traveling to an accident. Uer A and B imultaneouly chooe departure chedule for their fleet. The chedule for uer i i a departure rate function, r i t) 0. The function can be een a a ditribution function of the N/2 vehicle over ome extended time interval uch a a day. Thi function i not retricted to be continuou, and the poibility of ma departure will be conidered. Each uer recognize that dipatching a vehicle at time t may delay vehicle in it fleet that depart after t. A delay occur if there i a queue at time t that perit when the later vehicle depart. A delay alo occur if there i no queue prior to t, but the bottleneck i at capacity o that adding a vehicle to the departure chedule at t create a mall) queue. A noted above, the exitence of a PSNE in thi example depend on whether γ > α, or γ α. The two cae are conidered in the following two ubection Non-exitence of PSNE with γ > α When γ > α, a PSNE doe not exit. Thi reult i formalized in the following propoition. Propoition 1. Conider two identical atomic uer who each imultaneouly chedule N/2 vehicle with trip-timing preference for each vehicle defined by the ame {α,β,γ,t } parameter value. If γ > α, a PSNE in departure chedule doe not exit. We prove Propoition 1 in four tep. Firt, we prove that a PSNE without queuing doe not exit Lemma 1). Second, we prove that ma departure cannot arie in equilibrium Lemma 2). Hence, we can retrict attention to cae in which there i queuing, but no ma departure. Third, we how that there i a unique departure pattern with queuing uch that a uer cannot reduce it fleet cot by recheduling a ingle vehicle Lemma 3). Finally, we how that thi departure pattern i not a PSNE becaue a uer can reduce it fleet cot by recheduling a poitive meaure of vehicle in it fleet Lemma 4). Many uch deviation are gainful. They all entail potponing departure in order to reduce queuing delay. We demontrate thi graphically, and then identify the mot gainful deviation in 7

8 which the deviant uer avoid queuing altogether. Becaue deviation i gainful, a PSNE with queuing doe not exit. Lemma 1. Conider two identical atomic uer who each imultaneouly chedule N/2 vehicle with trip-timing preference for each vehicle defined by the ame {α,β,γ,t } parameter value. When γ > α, a PSNE without queuing doe not exit. Proof: Conider a pair of departure chedule, {r A ), r B )}, uch that no queuing occur. Some vehicle mut arrive late ince otherwie a uer could reduce it fleet cot by recheduling ome vehicle to jut after t. Conider a period t 1, t 2 ) of late arrival and aume that both uer depart during thi period the cae where only one uer depart i conidered later). The bottleneck mut be ued to capacity ince otherwie a uer could exploit the reidual capacity by advancing departure for vehicle that are cheduled to depart later. A, by aumption, in Lemma 1 there i no queue, if uer i remove a vehicle from the departure chedule at time t, it ave a cot of Ci t) = γ t t ). 9) Removing the vehicle ave the late-arrival cot incurred by the vehicle itelf, but it ha no effect on the ret of the fleet becaue there i no queue. If uer i intead add a vehicle to the departure chedule, it now caue a queue and increae it fleet cot by a marginal private cot MPC) of C + i t) = γ t t ) + α + γ t t r i u) du, 10) where t i the time when the queue created by the additional vehicle diappear. The firt term on the right-hand ide of Eq. 10) matche the right-hand ide of Eq. 9). The econd term i the delay cot impoed on uer i other vehicle that depart from t to t. Each of them uffer an increae in travel time of 1/ valued at α, and an increae in late arrival of 1/ valued at γ. The difference between the cot aved by removing a vehicle given by Eq. 9), and the cot of adding a vehicle to the ame lot given by Eq. 10), arie when the bottleneck i at capacity but there i no queue. Note that Eq. 9) and 10) are valid only for a etting without queuing. Later on we will conider etting with queuing. Suppoe uer i advance the departure of a vehicle from t to t where t 1 t < t t 2. Then, from the aumed etting without queuing, uer i fleet cot change by: C i = C i t) + C + i t ) = γ t t ) + γ t t ) + α + γ t r i u) du. 11) t 8

9 The queue induced by adding the vehicle at t vanihe at t becaue a departure time lot opened up at t when the vehicle wa removed then. Let λ i t,t = t t r i u) du/ t t )) [0, 1] be an auxiliary variable denoting the average fraction of capacity occupied by uer i during the period [t, t ]. The change in uer i fleet cot can then be written a: C i = α + γ λ i t,t t t ) γ t t ) = α + γ) λ i t,t γ) t t ). For a PSNE to exit, C i mut be nonnegative for both uer which require: λ i t,t γ α + γ, t 1 t < t t 2, i = A, B. 12) Since the bottleneck i fully utilized, λ A t,t + λb t,t = 1. Thi condition i leat retrictive if λ A t,t = λb t,t = 1/2 in which cae it reduce to γ α which i inconitent with the aumption γ > α. Now conider the poibility that only one uer, ay uer A, depart during t 1, t 2 ). Thi i not a PSNE if uer B depart after t 2 ince uer B could reduce it cot by recheduling ome of it later vehicle into t 1, t 2 ). Doing o reduce their late-arrival cot, and the queue they create diappear during the time lot they vacated. Suppoe uer B doe not depart after t 2. If t 1 = t, uer B doe not depart late at all. But thi cannot be a PSNE becaue uer B could gain by recheduling ome of it early-arriving vehicle to jut after t, thereby reducing their chedule delay cot without impoing any delay on it other vehicle. If t 1 > t, there mut exit a late arrival period t 0, t 1 ) during which both uer depart. But thi cae ha already been conidered, and hown to be inconitent with a PSNE when γ > α. QED To thi point we have aumed that uer depart at a finite rate. In theory, a uer could chedule a poitive meaure of vehicle to depart at a given moment. In practice, thi might be achieved by aembling a convoy of vehicle on a link that ha right-of-way over other link. Moreover, in the bottleneck model with non-atomic uer and tep toll a PSNE may exit only if ma departure of non-cooperating vehicle) are poible Arnott et al., 1990; Lindey et al., 2012). We now how that in the model with atomic uer ma departure cannot occur in a PSNE, regardle of whether γ > α or γ α. Lemma 2. Conider two identical atomic uer who each imultaneouly chedule N/2 vehicle with trip-timing preference for each vehicle defined by the ame {α,β,γ,t } parameter value. A PSNE cannot exhibit ma departure. Proof: See Appendix A. We now turn to the final poibility for a PSNE with γ > α in which departure rate remain finite and queuing occur. 9

10 Lemma 3. Conider two identical atomic uer who each imultaneouly chedule N/2 vehicle with trip-timing preference for each vehicle defined by the ame {α,β,γ,t } parameter value. When γ > α, there i a unique departure pattern with queuing in which a uer cannot reduce it fleet cot by recheduling a ingle vehicle. Proof: Conider a pair of departure chedule, {r A ), r B )}, and let t denote the departure time for which a uer arrive on time i.e. t = t T t)). Aume that a queue exit during a late-departure period t l, t qe ), where t l i an arbitrary time that atifie t l > t and t qe i the time when the queue diappear. Since uer A and B are identical, it uffice to conider the bet repone of uer A to r B ). The MPC to uer A of cheduling a vehicle at time t t l, t qe ) i C A t) = α T t) + γ t + T t) t ) + α + γ tqe t r A u) du. 13) Eq. 13) ha a imilar interpretation to Eq. 10). The firt two term on the right-hand ide comprie the cot incurred by the vehicle itelf, and the third term i the delay cot impoed on uer A other vehicle that depart from t to t qe. Uer A could rechedule a vehicle from t to another time t t l, t qe ). Thi would leave t qe unchanged becaue the additional queuing time caued by inerting the vehicle at t i offet by the reduction in queuing time due to removing the vehicle at t. Eq. 13) therefore hold if t i replaced by any t t l, t qe ). Hence, a neceary condition for r A ) to be a bet repone to r B ) i that C A t) in Eq. 13) i contant during the interval t l, t qe ). Differentiating Eq. 13) with repect to t, and etting the derivative to zero, yield C A t) t = γ + α + γ) T t) t α + γ r A t) = 0. Uing the relationhip T t)/ t = r A t) + r B t) ) /, thi condition implifie to r B t) = α α + γ, t t l, t qe ). 14) According to Eq. 14), uer A i willing to chedule a vehicle for late arrival when there i a queue if, and only if, uer B i departing at exactly the rate α / α + γ). Thi i none other than the equilibrium aggregate departure rate for the model with non-atomic uer cf. Eq. 2)). Eq. 14) alo hold for uer B with r A t) in place of r B t). The aggregate departure rate during an interval of late arrival mut therefore be r t) = r A t) + r B t) = 2 α / α + γ). With γ > α, r t) < and the queue mut be hrinking for late arrival. Conequently, a queue mut exit at time t and it i poible to et t l = t, where t+t t) = t. Thi in turn implie r A t) = r B t) = α α + γ, t t, t qe ). 10

11 Since a queue exit at time t, it mut have built up during a period of early arrival before t. Let t q be the time at which queuing begin. The MPC to uer A of cheduling a vehicle at any time t t q, t ) i C A t) = α T t) + β t t T t)) + α β t t r A u) du + α + γ tqe t r A u) du. 15) Again, the firt two term on the right-hand ide of Eq. 15) comprie the cot borne by the vehicle itelf. The third term i the cot impoed on uer A other vehicle that depart after t but till arrive early. Each of them uffer an increae in travel time of 1/ valued at α, and benefit from a reduction in early arrival of 1/ valued at β. The lat term in Eq. 15) i the cot impoed on uer A other vehicle that arrive late. A neceary condition for r A ) to be a bet repone to r B ) i for C A t) to be contant during t q, t ). Setting the derivative of C A t) to zero, one obtain a counterpart to Eq. 14): r B t) = α α β, t t q, t ). 16) An analogou neceary condition applie for uer B. Hence, the aggregate early departure rate mut be r t) = r A t) + r B t) = 2 α / α β). In ummary, the unique departure rate of the candidate PSNE during the full period of queuing i where r A t) = r B t) = r t) 2 = { α α β, t t q, t ) α α+γ, t t, t qe ), 17) t + T t) = t. 18) Becaue queuing begin at t q, and end at t qe, cumulative departure during the period t q, t qe ) match cumulative arrival: tqe t q r u) ) du = 0. 19) Eq. 17), 18), and 19) define evolution of the queue for the candidate PSNE with queuing. By Lemma 1, no vehicle can depart without queuing after t qe, o departure end at time t e = t qe. However, the cot of a vehicle trip at the beginning of the queuing period, c t q ), i le than the cot at the end of the period, c t e ), becaue the trip at t q impoe a private delay cot on ubequent vehicle wherea the trip at t e doe not if ct q ) = ct e ), C + t q ) > C + t e ) would hold and the uer equilibrium condition would be violated). Departure mut therefore occur during ome time interval [t, t q ) preceding t q. Thi interval i defined by two condition. Firt, vehicle trip cot mut be the ame at t and t e ince otherwie a uer could rechedule vehicle from the time with higher cot to the time with lower cot and reduce it overall fleet cot without cauing any queuing. Second, the full departure 11

12 period [t, t e ] mut be long enough for all N vehicle to pa the bottleneck. Eq. 3) and 4) for the non-atomic PSNE therefore hold for the candidate PSNE: β t t ) = γ t e t ), 20) t e t = N. 21) The two uer each control N/2 vehicle and chedule the ame number of vehicle during the queuing period. Therefore, they mut alo chedule the ame number during [t, t q ]: where tq t r A u) du = tq t r B u) du, 22) r A u) + r B u) =, u t, t q ). 23) The early departure chedule defined by Eq. 22) and 23) i conitent with a PSNE a far a trip timing by individual vehicle. The firt vehicle cheduled at t create the ame MPC a all vehicle cheduled during [t q, t e ]. Vehicle cheduled during t, t q ) create a lower MPC, o that recheduling them to any time outide t, t q ) would increae total fleet cot. Recheduling any vehicle into t, t q ) would alo increae fleet cot becaue it would impoe a queuing delay on all vehicle departing later until t e, and would therefore create a higher MPC than a vehicle departing at t q. QED Together, Eq. 17) 23) define the candidate PSNE with queuing. Uing upercript c to denote thi candidate, the critical time are: t c =t γ β + γ N, t c e =t + β β + γ N, t c = t β γ α) 2 α β + γ) N, t c q = t β γ α) α + β) β + γ) N. 24a) 24b) 24c) 24d) The candidate PSNE i depicted in Figure 1. Cumulative departure during the whole travel period are hown by the piecewie linear chedule OABC. During the firt interval t c, t c ) q, the two uer depart early at rate conitent with Eq. 22). During the remaining interval t c q, t c e), uer B contribute to cumulative departure the portion between chedule ADGE and the horizontal line AF. Uer A contribute the equally big portion between chedule ABC and chedule ADGE. During the interval t c q, t c), each uer depart early at rate α / α β), and during t c, t c e) each uer depart late at rate α / α + γ). At time ˆt, each uer cumulative departure over the interval t c q, ˆt ) match cumulative bottleneck throughput over the ame interval. 12

13 C Cumulative departure and arrival lope = Cumulative departure lope = α α β 2 α α β B D lope = 2 α α + γ G Uer B departure lope = lope = α α + γ E A F O t c t c q t c ˆt t t c e t Figure 1 We now how that the candidate PSNE jut derived i not a PSNE becaue either uer can reduce it fleet cot by recheduling a poitive fraction of it vehicle. Thi reult i formalized a: Lemma 4. Conider two identical atomic uer who each imultaneouly chedule N/2 vehicle with trip-timing preference for each vehicle defined by the ame {α,β,γ,t } parameter value. When γ > α, the unique candidate departure pattern with queuing in which uer cannot reduce cot by recheduling a ingle vehicle i not a PSNE with repect to recheduling a poitive fraction of the fleet. Proof: Figure 2 how the portion of the candidate PSNE in Figure 1 during which queuing occur. Point A, C, and E have the ame label a in Figure 1. Suppoe uer A deviate from the candidate PSNE by departing at a rate below α/ α β) during the interval t H, t c), and a rate above α/ α + γ) during the interval t c ), t K o that at time t K cumulative departure have caught up to the candidate PSNE departure. The cumulative departure curve with the deviant chedule i hown by the thick dahed line paing through point H, J, K, L, and M. It lie to the right of the candidate PSNE during the interval t H, t K ) o that the queue i horter, and the on-time departure time i potponed from t c to t d. 13

14 Aume without lo of generality that uer A chedule it vehicle in the ame order a in the candidate PSNE. When the vehicle originally cheduled to depart at point J depart, JJ other vehicle in uer A fleet have already left ince time t c q recall that the thick dahdot line paing through point A, J, and E depict uer B cumulative departure). In the deviant chedule thi vehicle therefore depart at point J where ditance J J equal ditance JJ. Similarly, the vehicle originally cheduled to depart at point K now depart at point K where ditance K K equal ditance KK. And the vehicle originally cheduled to depart at point L now depart at point L where ditance L L equal ditance LJ. Retiming of departure for other vehicle can be determined in the ame way. We will how that all vehicle originally cheduled to depart after point J and before point L incur lower cot with the deviant chedule than in the candidate PSNE. All of uer A other vehicle experience no change in cot o that uer A total fleet cot fall. Candidate PSNE Deviant chedule M C Cumulative departure and arrival Recheduling: J to J K to K L to L H J J K K L J N K J lope = α α β K L lope = L α α + γ E A t c q t c t d t t c e t t H t K Figure 2 Firt conider the vehicle that i recheduled from point J to point J. The dahed line from J to J run parallel to the egment JJ of uer B cumulative departure curve. The lope of thi curve, α/ α β), matche the equilibrium departure rate for non-atomic uer cf. Eq. 2)), for which travel cot of individual vehicle remain contant over time. The recheduled vehicle thu incur no change in cot a the decreae in the early arrival cot exactly offet the increae in queuing cot. For the ame reaon, all vehicle that were 14

15 cheduled to depart between H and J incur the ame cot a they did before. Next conider the vehicle that i recheduled from point L to point L. The dahed line from L to L run parallel to the egment J L of uer B cumulative departure curve. Since the lope of thi curve, α/ α + γ), again matche the equilibrium departure rate for non-atomic uer, the vehicle incur the ame cot a it did at point L. Thi time it doe o by reducing the queuing cot and increaing the late arrival cot by the ame amount. Similarly, all vehicle originally cheduled to depart after L incur the ame cot a before. Finally, conider the vehicle originally cheduled to depart at point K which i repreentative of the vehicle cheduled after J and before L. Thi vehicle i recheduled to point K. The hift from K to K can be divided into two part: from K to N, and from N to K. The hift from K to N leave the vehicle cot unchanged ince the dahed line from K to N run parallel to egment KJ of uer B cumulative departure curve. However, the hift from N to K caue the vehicle cot to drop becaue egment NK ha a lope α/ α + γ), wherea the vehicle till arrive early when it depart at K. The vehicle benefit both from a reduction in early arrival time and a reduction in queuing delay. For the ame reaon, all vehicle originally cheduled between point J and L experience a drop in cot. Uer A can gain from deviation becaue uer B departure rate i not contant, but rather drop when vehicle tart arriving late in the candidate PSNE. Uer A take advantage of thi by potponing departure to the period when uer B departure rate i lower. Note that uer A would not benefit if all the vehicle that deviate arrived early both before and after the deviation. Similarly, uer A would not benefit if all thee vehicle arrived late both before and after deviation. A further point to note i that the gain from deviation increae more than proportionally with the number of vehicle that deviate. The candidate PSNE i contructed o that recheduling a ingle vehicle i not gainful at all. If uer A were to rechedule vehicle within a hort interval panning t c, few vehicle would benefit and the benefit per vehicle would be mall becaue queuing delay would drop only marginally. Recheduling more vehicle would increae both the number of vehicle that benefit and the benefit per vehicle. In fact, uer A mot profitable deviation trategy i to avoid queuing altogether by recheduling all it vehicle from period t c q, ˆt ) to period ˆt, t c e) where ˆt i hown in Figure 1. Uer A top departing until the queue created by uer B ha vanihed at time ˆt, and then maintain a departure rate of γ / α + γ) during ˆt, t c e). Since uer B depart at rate α / α + γ) during ˆt, t c e), the total departure rate during thi period equal and the bottleneck i fully utilized without queuing. Moreover, ince uer B keep the bottleneck fully utilized during t c q, ˆt ), all vehicle in both fleet complete their trip by t c e. Conequently, with the deviant chedule, all of uer A vehicle can complete their trip within the ame time period a in the candidate PSNE, but without queuing. 15

16 Another way to ee why deviation i gainful i to reconider the marginal private cot function with queuing in Eq. 15): C A t) = α T t) + β t t T t)) + α β t t r A u) du + α + γ tqe t r A u) du. The candidate PSNE i derived while treating C A t) a given. Neverthele, C A t) clearly depend on uer A choice of departure chedule. The derivative with repect to T t) and t are C At) T t) = α β > 0, and C At) = β+γ t r A t ) < 0 repectively. Thi ugget that uer A can lower it fleet cot by reducing queuing delay, T t), and potponing the time at which vehicle tart arriving late, t. demontration in Figure 2. QED Both obervation are conitent with the Propoition 1 etablihe that a PSNE doe not exit when γ > α. A noted in the introduction, mot empirical tudie of cheduling preference for automobile driver have obtained etimate that atify thi inequality. However, there i little evidence either on the trip-timing preference of uer that may control large hare of road traffic e.g., freight hipper), or on preference for travel by other mode of tranportation. Daniel and Harback 2008), without making explicit the role of the paenger valuation of time, etimate that γ < α hold for many US airline. Thu, it i of interet to tudy the exitence and nature of PSNE when γ α Exitence and Nature of PSNE with γ α In thi ection we etablih two reult for the cae γ α. Firt, we how that there exit a unique PSNE in the aggregate departure chedule that coincide with the ytem optimum Propoition 2). Second, we how that the two uer individual departure rate are not uniquely defined in the PSNE Propoition 3), and the uer can incur different fleet cot Section 2.2.3) Exitence of PSNE with γ α Propoition 2. Conider two identical atomic uer who each imultaneouly chedule N/2 vehicle with trip-timing preference for each vehicle defined by the ame {α,β,γ,t } parameter value. If γ α, multiple PSNE exit. All of them have an aggregate departure chedule that coincide with the ytem optimum i.e. r A t) + r B t) =, t t o, t o e)). During the late-arrival period, each uer equilibrium departure rate i bounded in the [ γ range r i t) α+γ, α α+γ ], t t, t o e), i = A, B. During the early-arrival period, each uer equilibrium departure rate i bounded below: r i t) r E, t t o, t ), i = A, B, where r E 2 α β+γ)+γ β+2 γ) β 2 γ 2 +4 α β+γ) 2 α+γ) γ α+γ) 2 0, γ α+γ ). 16

17 exit. Proof: The proof entail etablihing four reult: 1) A PSNE with queuing doe not 2) All PSNE without queuing mut coincide with the ytem-optimal departure pattern given by Eq. 7a), 7b), and 7c). 3) Given the lower bound on individual departure rate for late arrival, neither uer can gain by recheduling a ingle vehicle. 4) Given the lower bound on individual departure rate for early and late arrival, neither uer can gain by recheduling part of it fleet. Reult 1. A PSNE with queuing doe not exit. By Lemma 2, ma departure cannot be part of a PSNE. Any candidate PSNE with queuing mut atify condition 17), 18), and 19). With γ α, thee condition cannot all be atified ince the aggregate early departure rate exceed capacity, and the aggregate late departure rate i no le than capacity. Hence any queue cannot diipate while uer are departing, which i inconitent with a PSNE. Reult 2. The ytem-optimal departure pattern i the only poible PSNE in the aggregate departure chedule. Given Reult 1, in equilibrium, the aggregate departure rate cannot exceed the bottleneck capacity when γ α. Therefore, equilibrium departure mut occur at rate over a connected time interval ince otherwie either uer could reduce it fleet cot by recheduling vehicle into gap in the departure chedule when bottleneck capacity i not fully utilized. The departure period, [t, t e ], mut be a given by Eq. 7b) and 7c) ince otherwie trip cot at t and t e would differ, and at leat one uer could reduce it fleet cot by recheduling vehicle from the higher-cot endpoint to the lower-cot endpoint. Reult 3. A uer cannot gain by recheduling a ingle vehicle if each uer departure rate atifie the condition in Propoition 2. In the candidate PSNE with a ytem-optimal aggregate departure pattern, there i no queuing but the bottleneck i ued to capacity. A in the proof of Lemma 1, it i therefore neceary to ditinguih between the cot aved by removing a vehicle from the departure chedule which doe not affect other vehicle cot) and the cot of adding a vehicle which create a queue unle the vehicle i added at t o e). The repective cot are: { β t Ci t), t [t o t) =, t ], γ t t ), t [t, t o e] t t o β t t) + α β C i + r i u) du + α+γ e r i u) du, t [t o, t ] t) = t t t o γ t t ) + α+γ e r i u) du, t [t, t o e] A vehicle can be recheduled in four way: i) late to late, ii) late to early, iii) early to late, and iv) early to early. Conider each poibility in turn. i. Recheduling late to late: Recheduling a late vehicle to a later time i never beneficial becaue the vehicle trip cot increae, and other vehicle do not gain. Suppoe a vehicle 17 t.

18 i recheduled earlier from t to t where t t < t. The change in fleet cot i given by Eq. 11): C i = Ci t) + C i + t ) = γ t t ) + α + γ = λ i t,t γ α + γ t r i u) du t where = mean identical in ign. Given r i t) γ / α + γ) for t t, t o e), λ i t,t γ/ α + γ), C i 0, and the deviation i not beneficial. ii. Recheduling late to early: Recheduling a late vehicle to an early time i clearly inferior to recheduling it to t becaue the vehicle incur an early-arrival cot and create a queue for a longer period. But recheduling it to t i not beneficial a per cae i. iii. Recheduling early to late: The bet option in thi cae i to rechedule a vehicle from t o. However, the gain i the ame a for recheduling a vehicle from t o e, and thi i not beneficial a per cae i. Recheduling early to late therefore cannot be beneficial. iv. Recheduling early to early: The bet option in thi cae i to rechedule a vehicle from t o to t, but thi, too, i not beneficial for the ame reaon a in cae iii. Thi etablihe that the candidate PSNE in Propoition 2 i robut to deviation in which a ingle vehicle i recheduled. Reult 4. A uer cannot gain by recheduling a poitive meaure of it fleet. If uer A rechedule vehicle to depart outide [t o, t o e], it fleet cot necearily increae ince the recheduled vehicle experience greater chedule delay cot without benefiting the ret of the fleet. If uer A intead rechedule vehicle to depart inide [t o, t o e], queuing occur. The optimal departure rate in the preence of a queue wa derived in Section 2.1 and, although with γ α a PSNE cannot exhibit queuing, we need to check if a deviation from the candidate to a etting with queuing i cot reducing. For early arrival, uer A i willing to depart at a poitive and finite rate only if condition 16) i atified; i.e. r B t) = α / α β) >. Since uer B depart at a rate le than in the candidate PSNE, uer A i better off cheduling the vehicle later. For late arrival, uer A i willing to depart at a poitive and finite rate only if condition 14) i atified o that r B t) = α / α + γ). Propoition 2 tipulate that r B t) α / α + γ). If r B t) = α / α + γ), and therefore r A t) = γ / α + γ), uer A i indifferent between departing or not o that the deviation doe not reduce it fleet cot. If r B t) < α / α + γ), uer A i better off cheduling the vehicle later. The only remaining poibility for gainful deviation i one that involve ma departure. In Appendix B we prove that a deviation with ma departure i not gainful if the lower bound on early and late departure rate tated in Propoition 2 are both atified. QED The intuition for Propoition 2 i a follow. The only way for uer A to gainfully deviate 18

19 from the candidate PSNE by recheduling a ingle vehicle i to advance it departure during the late arrival period. Doing o reduce the vehicle late-arrival cot, but impoe queuing delay on uer A vehicle that depart later. The tradeoff i not worthwhile if enough vehicle in A fleet have yet to depart, and queuing i ufficiently cotly relative to late arrival. The lower bound on the late departure rate tated in Propoition 2 aure thi condition i met. If uer A deviate by recheduling vehicle in ma departure and impoe queuing delay on it other vehicle, the ame argument hold. The tradeoff i not worthwhile becaue the lower bound on the late departure rate enure that there are enough vehicle in the fleet yet to depart that will be negatively affected. To avoid delaying other vehicle in it fleet, uer A mut advance departure for vehicle that participate in a ma. Since vehicle in the ma uffer queuing delay, uer A can benefit from uch a deviation only if the vehicle chedule delay cot are reduced enough. Thi require that the vehicle were departing late over a period longer than the time they take to pa the bottleneck when in the ma. Thi, in turn, i poible only if uer B occupie a large enough hare of bottleneck capacity during the candidate PSNE. The lower bound on the late departure rate identified in Propoition 2 enure that thi condition i not met. To ee why a lower bound on the early departure rate i alo required, uppoe that uer A doe not depart during ome time interval t, t ). Uer A can then rechedule it vehicle departing in ome interval t, t ) by launching them in a ma at a time t m t, t ). None of it fleet departing after t will be delayed by the ma. If t m i choen to minimize the total chedule delay cot of vehicle in the ma, their total cot will fall. However, if uer A i cheduling enough departure during t m, t ), the deviation will not be gainful ince the ma departure either impoe queuing delay on it other vehicle, or it ha to include vehicle that were arriving early, which will uffer higher early arrival cot and queuing cot. If the early departure rate i high enough, uch a ma departure i unfavorable. Propoition 2 identifie a minimum early departure rate to guarantee thi when the late departure rate i held fixed at it minimum value: γ /α + γ) Non-uniquene of PNSE with γ α Although the aggregate departure chedule in the PSNE decribed in Propoition 2 i unique when γ α, many pair of departure chedule {r A ), r B )} are conitent with the aggregate pattern. Thu, the PSNE i not unique in term of individual departure rate. Thi reult i formalized in the following propoition: Propoition 3. Conider two identical atomic uer who each imultaneouly chedule N/2 vehicle with trip-timing preference for each vehicle defined by the ame {α,β,γ,t } parameter value. During the late departure period t t, t o e), the two uer depart at the ame rate r A t) = r B t) = /2 if α = γ, but they can depart at different rate if 19

20 γ < α. During the early departure period t t o, t ), a continuum of departure chedule {r A ), r B )} are conitent with the PSNE. Proof: By Propoition 2, for t t, t o e), γ/α + γ) r A t)/ α/α + γ) and r B t) = r A t). If γ = α, γ/α + γ) = 1/2 and therefore r A t) = r B t) = /2. If γ < α, there i a continuum of pair of {r A t), r B t)} that atify the equilibrium condition while auring that all vehicle depart during the interval [t o, t o e]. For t t o, t ), the contraint r i t) r E i le trict becaue r E < /2. Any pair of departure chedule {r A ), r B )} that atifie r E r A t) r E and r B t) = r A t) atifie the aggregate PSNE condition. QED The non-uniquene of individual departure rate in a PSNE implie that, unlike in the model for identical non-atomic uer, uer can experience different fleet cot in a PSNE. Thi propect i examined further in the next ubection Aymmetric equilibrium cot with γ α Clearly, if uer A and B depart at the ame rate throughout the departure period they incur the ame fleet cot. In addition, there are many aymmetric PSNE departure chedule that reult in the ame average chedule delay cot for the two uer and hence the ame fleet cot. However, there are alo many aymmetric departure chedule that reult in different fleet cot. Figure 3 depict an illutrative example of a PSNE in which uer A incur lower fleet cot than uer B. In the example, uer A departure are concentrated near t o that uer A ha lower average chedule delay cot than B. Uer A departure rate i hown by olid line and uer B by broken line. During the interval t o, t BA ), uer A depart at the minimum rate r E defined in Propoition 2. During the next interval t BA, t ), uer B depart at rate r E and uer A at the complementary rate r E. During the firt part t, t AB ) of the late-arrival period, uer A depart at the maximum rate conitent with a PSNE for uer B, α / α + γ). During the remaining part t AB, t o e) of the departure period, uer A depart at the minimum rate conitent with a PSNE for itelf, γ / α + γ). The tranition time t BA and t AB are uch that each uer dipatche a total of N/2 vehicle. Expreed in term of uer A fleet, the requiite condition i r E t BA t o )+ r E ) t t BA )+ α α + γ t AB t )+ γ α + γ to e t AB ) = N 2. 25) Total cot in the PSNE, T C o, are a given; i.e. independent of uer A and B individual departure rate. The maximum difference between the uer cot, T C B T C A, can therefore be found by minimizing T C A with repect to t BA and t AB. A decribed in Appendix C, uer A cot a a fraction of total cot work out to 20

21 f T C A T C o = 3 + z 1 α 4 2 α + γ + z2, where z β γ/ α + γ) β + γ)). The fraction f depend on α, β, and γ only through the ratio β/α < 1 and γ/α 1. It i readily hown that f i a monotonically increaing function of β/α and γ/α. At the upper limit with β/α = 1 and γ/α = 1, f = A β/α and γ/α approach their lower limit of 0, f approache 1/4. Departure rate r Bt) = r E r At) = r E r At) = α α + γ rbt) = α α + γ r At) = r E r Bt) = r E r Bt) = γ α + γ rat) = γ α + γ t 0 t BA t t AB t o e t Figure 3 Thi example illutrate that although there exit a unique PSNE in term of the aggregate departure rate, individual uer departure rate can differ ubtantially and o can their cot. If the unit chedule delay cot parameter, β and γ, are mall compared to the cot of travel time, α, one uer fleet cot can be a little a one third the other uer cot. Thi i becaue one uer can concentrate it departure around t without the other uer wanting to rechedule it fleet becaue the gain from reducing chedule delay cot would be outweighed by the high cot of queuing delay. Thi ugget that equity of acce to a bottleneck can be an iue with atomic uer Extenion to multiple uer The analyi in thi ection can be generalized to m > 2 homogeneou uer. Propoition 1 and 2 can be extended in a traightforward manner by following imilar line of reaoning. For example, condition 12) for exitence of a PSNE without queuing, λ i t,t γ/ α + γ), till hold. In the leat retrictive cae in which all m uer depart at equal rate, thi condition implie that 1/m γ/ α + γ), or m 1) γ α. The intuition i imilar to the cae with m = 2. In a ymmetric candidate PSNE, each uer depart 21