Morning commute in congested urban rail transit system: A macroscopic model for equilibrium distribution of passenger arrivals
MMorning commute in congested urban rail transit system: Amacroscopic model for equilibrium distribution of passengerarrivals
Jiahua Zhang a , ∗ , Kentaro Wada b and Takashi Oguchi a a Institute of Industrial Science, The University of Tokyo, Tokyo, Japan b Faculty of Engineering, Information and Systems, University of Tsukuba, Ibaraki, Japan
A R T I C L E I N F O
Keywords :Rail transit systemTransit congestionFundamental diagramDeparture time choice equilibriumTimetable optimization
A B S T R A C T
This paper proposes a macroscopic model to describe the equilibrium distribution of passengerarrivals for the morning commute problem in a congested urban rail transit system. We employ amacroscopic train operation sub-model developed by Seo et al. (2017a,b) to express the interac-tion between dynamics of passengers and trains in a simplified manner while maintaining theiressential physical relations. We derive the equilibrium conditions of the proposed model anddiscuss the existence of equilibrium. The characteristics of the equilibrium are then examinedthrough numerical examples under different passenger demand settings. As an application ofthe proposed model, we finally analyze a simple time-dependent timetable optimization prob-lem with equilibrium constraints and show that there exists a “capacity increasing paradox” inwhich a higher dispatch frequency can increase the equilibrium cost. Further insights into thedesign of the timetable and its influence on passengers’ equilibrium travel costs are also obtained.
1. Introduction
Urban rail transit, with its high capacity and punctuality, serves as a typical solution to commuters’ travel demandduring rush hours in most metropolises worldwide (Vuchic, 2017). However, the travel experience of commuting by railtransit frequently deteriorates owing to severe congestion and unexpected delays. In many metropolises, the congestionand delay of rail transit have brought about tremendous psychological stress to commuters and considerable economicloss to society. For example, according to the report by the Ministry of Land, Infrastructure, Transport and Tourismof Japan, the delay of trains (more than 5 minutes) for 45 railway lines in the Tokyo metropolitan area averagely occurin 11.7 days of 20 weekdays in a month, and more than half of the short delays (within 10 minutes) are caused byextended dwell time (MLIT, 2020). Kariyazaki et al. (2015) estimated that the social cost owing to the train delays inJapan exceeded 1.8 billion dollars per year.In a high-frequency operated rail transit system, once the delay of a train occurs owing to either an accident orextended dwell time, the following trains will be forced to decelerate or stop between stations to maintain a safetyclearance, which is a “knock-on delay” on the rail track (Carey and Kwieciński, 1994). Meanwhile, more passengersaccumulate on the platform when trains decelerate or stop (because headways of trains are extended), which requiresa longer dwell time of trains. This is a typical vicious circle of passenger concentration and on-track congestiondeveloped during rush hours (Kato et al., 2012; Tirachini et al., 2013; Kariyazaki et al., 2015).To mitigate congestion and prevent the occurrence of a delay, management strategies have long been investigated.As an important supply-side management strategy, the optimization of train timetables has received considerable atten-tion during the past decades (e.g., Carey, 1994; Zhou and Zhong, 2007; Niu and Zhou, 2013; Barrena et al., 2014; Niuet al., 2015; Wang et al., 2015; Robenek et al., 2016; Cats et al., 2016; Shi et al., 2018). Although most of these studiesconsider the time-dependent passenger demand in their optimization, the demand is treated as the given information.Therefore, the optimized timetable based on the given demand may not be optimal because passengers will changetheir departure time (i.e., the demand distribution changes) according to the new timetable. More importantly, demandmanagement strategies cannot be examined by such studies because the dynamic interaction between the departuretime decisions of passengers and rail transit operation is not described. ∗ Corresponding author
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Page 1 of 14 a r X i v : . [ ee ss . S Y ] F e b orning commute in congested urban rail transit system To understand the relationship between commuters’ decisions and congestion during morning rush hours, Vickrey(1969) proposed a departure-time choice equilibrium problem (morning commute problem) at a single bottleneck roadnetwork. The importance of this problem in transportation planning and demand management strategies has led tovarious extensions of the basic model (see Li et al., 2020, for a comprehensive review). However, the models for roadtraffic may not be readily applicable to rail transit because the mechanisms of congestion and delay are quite differentbetween these two systems. Several studies have addressed the problem in public transit systems (e.g., Kraus andYoshida, 2002; Tian et al., 2007; de Palma et al., 2015, 2017; Yang and Tang, 2018). They analyzed travel decisions oftransit users and fare optimization issues under the premise that the in-vehicle crowding and/or waiting time at stationsis the primary congestion cost of traveling . However, no studies have dealt with passengers’ departure time choicebehavior when considering the above-mentioned vicious circle of the demand concentration and congestion on-trackin high-frequency operated rail transit system.The purpose of this study is to develop a macroscopic model that describes the equilibrium distribution of pas-senger arrivals for the morning commute problem in a congested urban rail transit system. In the model, we employa macroscopic train operation sub-model (train-FD model by Seo et al., 2017a,b) to express the interaction betweenthe dynamics of passengers and trains in a simplified manner while maintaining their essential physical relations. Wederive the equilibrium conditions of the proposed model and discuss the existence of equilibrium. The characteristicsof the equilibrium are then examined through numerical examples under different passenger demand settings. Finally,by employing the proposed model, we analyze a simple time-dependent timetable optimization problem with equi-librium constraints and show that there exists a “capacity increasing paradox" in which a higher dispatch frequencycan increase the equilibrium cost. Further insights into the design of timetables and their influence on passengers’equilibrium travel costs are also obtained. Owing to its simplicity and comprehensiveness, the proposed macroscopicmodel is expected to contribute to revealing big-picture policy implications for time-dependent demand and supplymanagement strategies of congested rail transit systems.The remainder of this paper is organized as follows. Section 2 introduces the model for the morning commuteproblem in rail transit. Section 3 derives the user equilibrium and provides a solution method and existence conditionsof the equilibrium. Section 4 describes the characteristics of the equilibrium through several numerical examples. Sec-tion 5 applies the proposed model to a simple time-dependent timetable optimization problem. Finally, the conclusionsand future works are discussed in Section 6.
2. Macroscopic model for morning commute problem in rail transit
In this section, we formulate a model for the morning commute problem in rail transit. In Section 2.1, we present anoverview of the macroscopic train operation model proposed by Seo et al. (2017a,b), which is a supply side sub-modelof the proposed model. In Section 2.2, we describe behavioral assumptions of users’ departure time choice, which isa demand side sub-model.
Consider a railway system on a single-line track, where stations are homogeneously located along the line. Alltrains stop at every station, and thus first-in-first-out (FIFO) service is assumed to be satisfied along the railway track.In the following part of this subsection, we first show the microscopic operation assumptions on passenger boarding and train cruising to obtain a macroscopic model.Passenger boarding behavior is described using a queuing model (Wada et al., 2012). Specifically, the train dwelltime 𝑡 𝑏 at each station is given by 𝑡 𝑏 = 𝑡 𝑏 + 𝑎 𝑝 ℎ ∕ 𝜇, (1)where 𝑡 𝑏 is the buffer time, including the time needed for door opening and closing, 𝜇 is the maximum flow rate ofpassenger boarding, 𝑎 𝑝 is the passengers’ arrival rate at the platform, and ℎ is the headway of two succeeding trains.Note here that we assume that passengers can always board the next arriving train. In these studies, the delay of trains is not considered. Even in such a situation, a significant waiting (queuing) time of passengers (and also thein-vehicle crowding) can occur, for instance, in an oversaturated railway system (Shi et al., 2018; Xu et al., 2019). Throughout this paper, we do not consider the costs and revenue of the transit operator/agency. Thus, we treat the train operation exogenouslyexcept for Section 5 in which an optimal timetable setting is discussed from the passenger’s viewpoint.
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Rail transit system
Train operation based on Eq. (1) and Eq. (2)Internal average train flow: 𝑞 = 𝑄 𝑘, 𝑎 𝑝 Train inflow Passenger inflow Train outflow Passenger outflow
Figure 1:
Rail transit system as an input-output system.
The cruising behavior of the trains is assumed to be described by Newell’s simplified car-following model (Newell,2002). In this model, a vehicle either travels at its desired speed or follows the preceding vehicle while maintainingsafety clearance . More specifically, the position of train 𝑛 at time 𝑡 is described as 𝑥 𝑛 ( 𝑡 ) = min{ 𝑥 𝑛 ( 𝑡 − 𝜏 ) + 𝑣 𝑓 𝜏, 𝑥 𝑛 −1 ( 𝑡 − 𝜏 ) − 𝛿 } , (2)where 𝑛 −1 refers to the preceding train of train 𝑛 , 𝜏 is the reaction time of the train, and 𝛿 is the minimum spacing. Thefirst term represents the free-flow regime, where the train cruises at its desired speed 𝑣 𝑓 . The second term representsthe congested regime where the train decreases its speed to maintain minimum spacing.Now, let us show a train fundamental diagram (train-FD), 𝑞 = 𝑄 ( 𝑘, 𝑎 𝑝 ) , which describes the steady-state relationamong train flow 𝑞 ( 𝑞 = 1∕ ℎ ), train density 𝑘 , and passenger arrival rate 𝑎 𝑝 in a homogeneously congested transitsystem. Specifically, based on the operating principles described in Eqs. (1) and (2), train-FD can be analyticallyexpressed as follows (see Seo et al., 2017a,b, for a derivation). 𝑄 ( 𝑘, 𝑎 𝑝 ) = ⎧⎪⎪⎨⎪⎪⎩ 𝑘𝑙 − 𝑎 𝑝 ∕ 𝜇𝑡 𝑏 + 𝑙 ∕ 𝑣 𝑓 , if 𝑘 < 𝑘 ∗ ( 𝑎 𝑝 ) , − 𝛿𝑙 ( 𝑙 − 𝛿 ) 𝑡 𝑏 + 𝜏𝑙 ( 𝑘 − 𝑘 ∗ ( 𝑎 𝑝 )) + 𝑞 ∗ ( 𝑎 𝑝 ) , if 𝑘 ≥ 𝑘 ∗ ( 𝑎 𝑝 ) , (3)where 𝑙 is the (average) distance between adjacent stations, and 𝑞 ∗ ( 𝑎 𝑝 ) and 𝑘 ∗ ( 𝑎 𝑝 ) are the critical train flow and traindensity, respectively: 𝑞 ∗ ( 𝑎 𝑝 ) = 1 − 𝑎 𝑝 ∕ 𝜇𝑡 𝑏 + 𝛿 ∕ 𝑣 𝑓 + 𝜏 , (4) 𝑘 ∗ ( 𝑎 𝑝 ) = (1 − 𝑎 𝑝 ∕ 𝜇 )( 𝑡 𝑏 + 𝑙 ∕ 𝑣 𝑓 )( 𝑡 𝑏 + 𝛿 ∕ 𝑣 𝑓 + 𝜏 ) 𝑙 + 𝑎 𝑝 𝜇𝑙 . (5)The train-FD was inspired by the macroscopic fundamental diagram (MFD) for road networks (Geroliminis andDaganzo, 2007; Daganzo, 2007), and they are similar in the following two senses. First, they both describe the trafficstates in a homogeneously congested area using system-wide aggregate variables. Second, they both have unimodalrelations between the density (accumulation) and flow (throughput) of the system, which yields two different regimes,that is, the free-flow and congested regimes. One essential difference between the train-FD and MFD is that the train-FD has an additional dimension of passenger flow. Introducing this new dimension enables simplified modeling of railtransit operations in which passenger concentration is considered .To describe the rail transit system behavior when the demand (i.e., passenger flow) and supply (i.e., train density)change dynamically, we consider it as an input-output system with the train-FD, as illustrated in Fig. 1. There are twotypes of inputs: train inflow (equivalent to timetable information), and passenger inflow (i.e., passenger arrival rate 𝑎 𝑝 ). Accordingly, there are two outputs: train outflow and passenger outflow. Within the system, trains operate basedon the rules in Eqs. (1) and (2), whereas passenger arrival in the system based on their assessment of the travel costintroduced in the next subsection. As in existing MFD applications for morning commute problems (e.g., Geroliminisand Levinson, 2009; Geroliminis et al., 2013; Fosgerau, 2015), it is expected that this simplified model will provideinsight into the time-dependent characteristics of the rail transit system despite its inability to capture spatial dynamicsor heterogeneity within the system. This assumption is more appropriate for moving block rather than fixed block railway signaling system. Empirical investigations of the train-FD can be found in Fukuda et al. (2019) and Zhang and Wada (2019).
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Space Time
𝐿𝑇 𝑛 ℎ 𝑎 (𝑛) ℎ 𝑑 (𝑛) Figure 2:
Example of trajectories of trains and definition of variables.
Consider a fixed number, 𝑁 𝑝 , of passengers that take the train system during the morning rush period. The lengthof their trip in the system is common for all passengers and is denoted by 𝐿 . Passengers choose their departure timefrom home to minimize their travel costs. The travel time from leaving home to arriving at the nearest station for anypassenger is assumed to be constant; thus, without a loss of generality, it is set to be zero. We also assume that thedeparture time from the system is the arrival time at the destination (i.e., the workplace). For the sake of clarity, if notparticularly indicated, we refer to “passenger/train departure” as the departure (or exit) from the rail transit system.The travel cost (TC) is assumed to consist of the travel delay cost (TDC) in the train system and schedule delaycost (SDC) . Specifically, the TC of a passenger 𝑖 departing from the system at time 𝑡 is defined as 𝑇 𝐶 ( 𝑡, 𝑡 ∗ 𝑖 ) = 𝛼 ( 𝑇 ( 𝑡 ) − 𝑇 ) + 𝑠 ( 𝑡, 𝑡 ∗ 𝑖 ) , (6)where 𝑡 ∗ 𝑖 is the desired departure time, 𝛼 is the time value for a travel delay, 𝑇 ( 𝑡 ) is the travel time for a passengerdeparting from the rail transit system at time 𝑡 , 𝑇 is the minimum travel time before the morning rush starts, and 𝑠 ( 𝑡, 𝑡 ∗ 𝑖 ) is the schedule delay cost. Here, we employ the following piecewise linear schedule delay cost function 𝑠 ( 𝑡, 𝑡 ∗ 𝑖 ) that has been widely used in previous studies (e.g., Hendrickson and Kocur, 1981; Tian et al., 2007; Geroliminis andLevinson, 2009; de Palma et al., 2017; Yang and Tang, 2018). 𝑠 ( 𝑡, 𝑡 ∗ 𝑖 ) = { 𝛽 ( 𝑡 ∗ 𝑖 − 𝑡 ) , if 𝑡 < 𝑡 ∗ 𝑖 ,𝛾 ( 𝑡 − 𝑡 ∗ 𝑖 ) , if 𝑡 ≥ 𝑡 ∗ 𝑖 , (7)where 𝛽 and 𝛾 are the values of time for earliness and lateness, respectively. For simplicity, we assume that all passen-gers have the same cost parameters 𝛼 , 𝛽 and 𝛾 . We specify the desired departure time distribution in a later section.The travel time for a passenger departing from the rail transit system at time 𝑡 is equal to that of a train departingfrom the system at the same time. Let 𝑛 be train number departing from the system at time 𝑡 , and 𝑇 ( 𝑛 ) (= 𝑇 ( 𝑡 )) beits travel time. The service (or average traveling) speed of train 𝑛 is 𝐿 ∕ 𝑇 ( 𝑛 ) . We also denote the headway of train 𝑛 when arriving at the system by ℎ 𝑎 ( 𝑛 ) and that when departing from the system by ℎ 𝑑 ( 𝑛 ) . If we approximate time-spacetrajectories of the trains as straight lines whose slopes are their service speeds (see Fig. 2), the average spacing of train 𝑛 , 𝑠 ( 𝑛 ) , can be defined as 𝑠 ( 𝑛 ) ≡ 𝐿𝑇 ( 𝑛 ) ℎ ( 𝑛 ) (8)where ℎ ( 𝑛 ) = ℎ 𝑎 ( 𝑛 ) + ℎ 𝑑 ( 𝑛 )2 . This is consistent with Edie’s generalized definition of traffic variables (Edie, 1963). This paper does not consider dynamic pricing. Thus, a constant fare is excluded from the cost function. Because we treat trains as a continuum or fluid, the number of trains can be a non-integer value.
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Here, we introduce the main assumption in this paper: the (average) train flow 𝑞 ( 𝑛 ) = 1∕ ℎ ( 𝑛 ) and (average) traindensity 𝑘 ( 𝑛 ) = 1∕ 𝑠 ( 𝑛 ) of the system with respect to train 𝑛 satisfy the train-FD, that is, 𝑞 ( 𝑛 ) = 𝑄 ( 𝑘 ( 𝑛 ) , 𝑎 𝑝 ( 𝑛 )) ⇔ ℎ ( 𝑛 ) = 𝑄 ( 𝑠 ( 𝑛 ) , 𝑎 𝑝 ( 𝑛 ) ) . (9)where 𝑎 𝑝 ( 𝑛 ) is the average passenger arrival rate for train 𝑛 at the stations along the line. If the system is in a steadystate, the relation in (9) must hold. We can also expect that the relation in (9) approximately holds if the (average)values of the state variables vary gradually.This assumption enables us to link the time-dependent (more precisely, train-dependent ) passenger demand { 𝑎 𝑝 ( 𝑛 )} to travel time { 𝑇 ( 𝑛 )} in a simplified manner while maintaining their essential physical relationships. More specifically,as we will show in the next section, the train traffic state variables are determined by the departure-time choice equi-librium conditions first, and the equilibrium passenger arrival rates { 𝑎 𝑝 ( 𝑛 )} can then be estimated using the relation in(9). It should be noted that Eq. (9) does not represent macroscopic train system dynamics. The train system dynamicsusing the train-FD (i.e., an exit-function model) can be found in Seo et al. (2017a,b).
3. User equilibrium
Under the setting described in the previous section, the user equilibrium is defined as the state in which no transituser can reduce his/her travel cost by changing his/her departure time from the system unilaterally. In this section,we first derive the equilibrium conditions. We then present a solution method. Finally, we discuss the existence ofequilibrium.
Because each passenger 𝑖 chooses his/her departure time 𝑡 𝑖 from the system to minimize the travel cost at equilib-rium, the following condition is satisfied at time 𝑡 = 𝑡 𝑖 : 𝜕𝑇 𝐶 ( 𝑡 𝑖 , 𝑡 ∗ 𝑖 ) 𝜕𝑡 = 𝛼 d 𝑇 ( 𝑡 𝑖 )d 𝑡 + 𝜕𝑠 ( 𝑡 𝑖 , 𝑡 ∗ 𝑖 ) 𝜕𝑡 = 0 . (10)The derivative of the travel time 𝑇 ( 𝑡 ) is obtained by substituting Eqs. (6) and (7) into Eq. (10) as 𝑑𝑇 ( 𝑡 𝑖 ) 𝑑𝑡 = { 𝛽 ∕ 𝛼, if 𝑡 𝑖 < 𝑡 ∗ 𝑖 , − 𝛾 ∕ 𝛼, if 𝑡 𝑖 ≥ 𝑡 ∗ 𝑖 . (11)Furthermore, with the first-in-first-work assumption (Daganzo, 1985), the travel time 𝑇 ( 𝑡 ) is maximized when theschedule delay is zero (we refer to this time as 𝑡 𝑚 ). Consequently, the travel time 𝑇 ( 𝑡 ) under equilibrium is given by 𝑇 ( 𝑡 ) = { 𝑇 + 𝛽𝛼 ( 𝑡 − 𝑡 ) , if 𝑡 ≤ 𝑡 < 𝑡 𝑚 ,𝑇 + 𝛽𝛼 ( 𝑡 𝑚 − 𝑡 ) − 𝛾𝛼 ( 𝑡 − 𝑡 𝑚 ) , if 𝑡 𝑚 ≤ 𝑡 ≤ 𝑡 𝑒𝑑 , . (12)where 𝑡 and 𝑡 𝑒𝑑 represent the start and end of the morning rush period, respectively.As mentioned in the previous section, the equilibrium passenger arrivals are estimated using the train traffic statevariables (i.e., 𝑇 ( 𝑛 ) , ℎ 𝑎 ( 𝑛 ) , ℎ 𝑑 ( 𝑛 ) ). Because we have already showed the travel time under the equilibrium 𝑇 ( 𝑛 ) = 𝑇 ( 𝐷 −1 ( 𝑛 )) through Eq. (12), the headways for all dispatched trains are derived next. Let 𝐴 ( 𝑡 ) be the cumulativenumber of train arrivals at the system at time 𝑡 . Then, the FIFO condition is written as 𝐷 ( 𝑡 ) = 𝐴 ( 𝑡 − 𝑇 ( 𝑡 )) or in itsderivative form as 𝑑 ( 𝑡 ) = 𝑎 ( 𝑡 − 𝑇 ( 𝑡 )) ( 𝑑𝑇 ( 𝑡 ) 𝑑𝑡 ) , (13) Because the cumulative number of train departures from the system at time 𝑡 , 𝐷 ( 𝑡 ) , is an increasing function of 𝑡 , there is a one-to-onecorrespondence between the number of trains and their departure time, that is, 𝑛 = 𝐷 ( 𝑡 ) ⇔ 𝑡 = 𝐷 −1 ( 𝑛 ) . Jiahua Zhang et al.:
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Time P a ss e ng e r nu m b e r 𝑊 𝑝 (𝑡)𝐴 𝑝 (𝑡) 𝐷 𝑝 (𝑡) 𝑡 𝑚 = 𝑡 ∗ 𝑡 𝑡 𝑒𝑑 𝑁 𝑝 𝑇 𝑇(𝑡 𝑒𝑑 ) P a ss e ng e r nu m b e r Time 𝑊 𝑝 (𝑡) 𝐴 𝑝 (𝑡) 𝐷 𝑝 (𝑡)𝑡 𝑚 𝑡 𝑡 𝑒𝑑 𝑁 𝑝 𝑇(𝑡 𝑒𝑑 )𝑇 (a) Case C Time P a ss e ng e r nu m b e r 𝑊 𝑝 (𝑡)𝐴 𝑝 (𝑡) 𝐷 𝑝 (𝑡) 𝑡 𝑚 = 𝑡 ∗ 𝑡 𝑡 𝑒𝑑 𝑁 𝑝 𝑇 𝑇(𝑡 𝑒𝑑 ) P a ss e ng e r nu m b e r Time 𝑊 𝑝 (𝑡) 𝐴 𝑝 (𝑡) 𝐷 𝑝 (𝑡)𝑡 𝑚 𝑡 𝑡 𝑒𝑑 𝑁 𝑝 𝑇(𝑡 𝑒𝑑 )𝑇 (b) Case D Figure 3:
An illustration of cumulative curves of passengers. where 𝑎 ( 𝑡 ) and 𝑑 ( 𝑡 ) are the inflow and outflow of the trains, respectively. Because 𝐴 ( 𝑡 ) is the given information (i.e.,timetable), 𝐷 ( 𝑡 ) can be obtained from this FIFO condition. From the definition, ℎ 𝑎 ( 𝑛 ) and ℎ 𝑑 ( 𝑛 ) are thus derived as: ℎ 𝑎 ( 𝑛 ) = d( 𝑡 − 𝑇 ( 𝑡 ))d 𝑛 = 1 𝑎 ( 𝑡 − 𝑇 ( 𝑡 )) , ℎ 𝑑 ( 𝑛 ) = d 𝑡 d 𝑛 = 1 𝑑 ( 𝑡 ) . (14)We can also calculate the average headway ℎ ( 𝑛 ) and spacing 𝑠 ( 𝑛 ) from these variables.Now, we are in a position to estimate the passenger arrivals under equilibrium. For a given train density, the train-FD provides a one-to-one correspondence between the train and passenger flows, that is, 𝑞 = ̂𝑄 ( 𝑎 𝑝 | 𝑘 ) = 𝑄 ( 𝑎 𝑝 , 𝑘 ) .Therefore, from our main assumption (9), we have 𝑎 𝑝 ( 𝑛 ) = ̂𝑄 −1 ( ℎ ( 𝑛 ) ||| 𝑠 ( 𝑛 ) ) (15)where we use the following inverse function 𝑎 𝑝 = ̂𝑄 −1 ( 𝑞 | 𝑘 ) .To obtain a complete equilibrium solution (i.e., to determine 𝑡 , 𝑡 𝑚 and 𝑡 𝑒𝑑 ), we need to specify the desired departuretime distribution. In this study, we consider two types of distributions: Cases C and D. For Case C, a fixed number 𝑁 𝑝 of passengers has a common desired departure time 𝑡 ∗ (or work start time). For Case D, the cumulative numberof passengers who want to depart by time 𝑡 is given by a Z-shaped function, 𝑊 𝑝 ( 𝑡 ) , with 𝑁 𝑝 passengers and a positiveconstant slope (i.e., demand rate) (e.g., Gonzales and Daganzo, 2012). An illustration of the cumulative curves ofpassengers for these two cases is shown in Fig. 3.For Case C, the first condition is 𝑡 𝑚 = 𝑡 ∗ . As the second condition, the last user experiences only the scheduledelay cost, that is, 𝑇 ( 𝑡 𝑒𝑑 ) = 𝑇 = 𝐿 ∕ 𝑙 ( 𝑡 𝑏 + 𝑙 ∕ 𝑣 𝑓 ) . (16)The last condition is the conservation of the number of users, that is, 𝐷 𝑝 ( 𝑡 𝑒𝑑 ) = ∫ 𝐷 ( 𝑡 𝑒𝑑 ) 𝐷 ( 𝑡 ) 𝑎 𝑝 ( 𝑛 )d 𝑛 = 𝑁 𝑝 (17)where 𝐷 𝑝 ( 𝑡 ) is the cumulative number of passengers departing from the system at time 𝑡 , and 𝐷 𝑝 ( 𝑡 ) = 0 . By solvingthe latter two conditions simultaneously, 𝑡 and 𝑡 𝑒𝑑 are determined.For Case D, we assume that there is a unique time instant 𝑡 𝑚 when the schedule delay becomes zero, as in thestandard morning commute problem for road traffic (Smith, 1984; Daganzo, 1985). We then have 𝐷 𝑝 ( 𝑡 𝑚 ) = 𝑊 𝑝 ( 𝑡 𝑚 ) . (18) Jiahua Zhang et al.:
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Algorithm 1
Solution to Case C
Input:
Operational parameters, 𝑙 , 𝐿 , 𝑡 𝑏 , 𝜇 , 𝑣 𝑓 , 𝛿 , 𝜏 ; cost parameters, 𝛼 , 𝛽 , 𝛾 , 𝑡 ∗ ; train inflow, 𝑎 ( 𝑡 ) , and total traveldemand, 𝑁 𝑝 . Output:
Train flow 𝑞 ( 𝑛 ) , train density 𝑘 ( 𝑛 ) , and passenger arrival rate 𝑎 𝑝 ( 𝑛 ) . Set an initial 𝑡 . Calculate 𝑇 ( 𝑡 ) and 𝑡 𝑒𝑑 by Eqs. (12) and (16). Calculate 𝑑 ( 𝑡 ) using Eq. (13). Calculate 𝑎 𝑝 ( 𝑛 ) by Eq. (15), together with Eqs. (8) and (14). Calculate the LHS − RHS of the discrete version of Eq. (17) (with unit Δ 𝑛 ), denoted as an 𝑒𝑟𝑟𝑜𝑟 . if 𝑒𝑟𝑟𝑜𝑟 < − 𝜖 𝑝 , then 𝑡 = 𝑡 − Δ 𝑡 , repeat lines 2-5. else if 𝑒𝑟𝑟𝑜𝑟 > 𝜖 𝑝 , then 𝑡 = 𝑡 + Δ 𝑡 , repeat lines 2-5. else Calculation converges, 𝑡 and 𝑡 𝑒𝑑 are determined. end if Outputs are obtained from line 4 when the calculation converges.By solving the three conditions (16), (17) and (18) simultaneously, 𝑡 , 𝑡 𝑚 and 𝑡 𝑒𝑑 are determined.A solution method for Case C is presented in Algorithm 1, where Δ 𝑡 is the step size of time, Δ 𝑛 is the discreteunit of train, and 𝜖 𝑝 is the tolerance of error in the number of passengers. The solution method for Case D is verysimilar to Algorithm 1, i.e., another step is simply added to calculate 𝑡 𝑚 that satisfies Eq. (18) after line 1. Note that anequilibrium solution might not exist (i.e., the solution method can produce a physically infeasible result). We addressthis issue in the next subsection. Several conditions must be satisfied to ensure the existence of equilibrium. As the first condition, the train outflowshould be positive . That is, 𝑑 ( 𝑡 ) = 𝑎 ( 𝑡 − 𝑇 ( 𝑡 )) ( 𝑑𝑇 ( 𝑡 ) 𝑑𝑡 ) > , ∀ 𝑡. (19)Since the train inflow 𝑎 ( 𝑡 ) is a positive given input, 𝑑𝑇 ( 𝑡 )∕ 𝑑𝑡 > should hold. By substituting Eq. (11) into thiscondition, we obtain 𝛼 > 𝛽. (20)This condition is consistent with that of the equilibrium models for road traffic (e.g., Hendrickson and Kocur, 1981;Arnott et al., 1990).As the second condition, the passenger arrival rate, 𝑎 𝑝 ( 𝑛 ) , calculated from Eq. (15) should not be negative, that is, 𝑎 𝑝 ( 𝑛 ) ≥ , ∀ 𝑛. (21)This is equivalent to the condition that the equilibrium traffic state ( 𝑘 ( 𝑛 ) , 𝑞 ( 𝑛 )) should not be outside the train-FD.Obtaining an explicit expression of such a condition may be difficult in general , and we therefore consider a constanttrain inflow case 𝑎 ( 𝑡 ) = 𝑎 𝑐 that is simple and practical. As we will see in Section 4, the evolution of ( 𝑘 ( 𝑛 ) , 𝑞 ( 𝑛 )) lies inside the train-FD, as long as the upper-right corner of the ( 𝑘 ( 𝑛 ) , 𝑞 ( 𝑛 )) loop is not outside the train-FD. Further,according to the equilibrium condition, this point corresponds to the state when the travel time is maximized and juststarts to decrease. Therefore, the underlying point ( 𝑘 𝑟𝑐 ( 𝑛 ) , 𝑞 𝑟𝑐 ( 𝑛 ) ) can be explicitly written as: 𝑘 𝑟𝑐 ( 𝑛 ) = 𝑇 ( 𝑡 𝑚 ) ( 𝑎 𝑐 + 𝑎 𝑐 (1+ 𝛾 ∕ 𝛼 ) ) 𝐿 ∕2 , 𝑞 𝑟𝑐 ( 𝑛 ) = 1 ( 𝑎 𝑐 + 𝑎 𝑐 (1+ 𝛾 ∕ 𝛼 ) ) ∕2 . (22) We do not consider the condition when trains are stopped by an accident or when the rail transit is not in operation. The evolution of ( 𝑘 ( 𝑛 ) , 𝑞 ( 𝑛 )) depends not only on the operational parameters of the rail transit system but also on the settings of the train inflowand total travel demand. Jiahua Zhang et al.:
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Table 1
Parameter settings for numerical example.
Parameter Value Parameter Value 𝑙 𝛼 $∕ h 𝐿
18 km 𝛽 $∕ h 𝑣 𝑓
40 km/h 𝛾 $∕ h 𝑡 𝑏
20 sec 𝑡 ∗
240 min 𝜇 𝑎 ( 𝑡 )
12 tr/h 𝛿 𝑤 𝑝 𝜏 𝑁 𝑝 Δ 𝑡 𝜖 𝑝
100 pax Δ 𝑛 The equivalent condition to Eq. (21) is obtained by substituting 𝑘 𝑟𝑐 ( 𝑛 ) and 𝑎 𝑝 = 0 into the second line of Eq. (3), andcompared with 𝑞 𝑟𝑐 ( 𝑛 ) : 𝛼 + 𝛾𝛼 + 𝛾 ∕2 𝑎 𝑐 [ 𝑇 ( 𝑡 𝑚 ) 𝐿 + ( 𝑙 − 𝛿 ) 𝑡 𝑏 + 𝜏𝑙𝛿𝑙 ] ≤ 𝛿 , (23)From this expression, we can see that this condition is violated if either the train supply 𝑎 𝑐 or 𝑇 ( 𝑡 𝑚 ) , which is determinedby the relationship between passenger demand and train supply, is too large.In summary, Eqs. (20) and (23) are the existence conditions of equilibrium when the train inflow is constant. Whenthe train inflow is time-dependent, Eq. (21) should be checked in the solution method.
4. Characteristics of equilibrium
In this section, the basic characteristics of the equilibrium of the proposed model are examined through severalnumerical examples. The parameter settings are presented in Table 1. For simplicity, the train inflow 𝑎 ( 𝑡 ) was set as aconstant. For Case C, the common desired departure time was set to min; for Case D, the slope of the Z-shapedfunction was set to 𝑤 𝑝 = 30000 pax/h, and the time period for the increase in 𝑊 𝑝 ( 𝑡 ) is [210 , min.We first present the dynamics of rail transit for Case C in Fig. 4. Fig. 4(a) shows the cumulative arrival anddeparture curves of trains. Fig. 4(b) shows the headways of the vehicles under user equilibrium. The train dynamicsfor Case D were almost the same as those in these figures. From Fig. 4(a), we can see that 𝐷 ( 𝑡 ) first deviates from 𝐴 ( 𝑡 ) during [ 𝑡 , 𝑡 ∗ ] and again approaches 𝐴 ( 𝑡 ) during [ 𝑡 ∗ , 𝑡 𝑒𝑑 ] . This train system behavior leads to an equilibrium inthe travel cost. Fig. 5 shows the costs for both Cases C and D. We see that the cost pattern is the same as the standardmorning commute problem for road traffic with a piece-wise linear schedule delay cost function.Fig. 6 shows the cumulative arrival and departure curves of passengers. Because we set the same travel demandfor the two cases, the characteristics of 𝐴 𝑝 ( 𝑡 ) and 𝐷 𝑝 ( 𝑡 ) for the two cases are almost the same. However, compared toCase C, the schedule delay in Case D (i.e., the distance between 𝐷 𝑝 ( 𝑡 ) and 𝑊 𝑝 ( 𝑡 ) ) becomes significantly smaller.Another important finding from Fig. 6 is that the passenger arrival rate 𝑎 𝑝 ( 𝑡 ) (i.e., the derivative of 𝐴 𝑝 ( 𝑡 ) ) hastwo peaks. The larger peak occurs around the arrival time of passengers departing from the system just before 𝑡 𝑚 ,and the other one occurs near the end of rush hour. To understand the mechanism behind this observation, we showhow 𝑘 ( 𝑛 ) and 𝑞 ( 𝑛 ) evolve on train-FD in Fig. 7(b). The black line shows that the evolution of ( 𝑘 ( 𝑛 ) , 𝑞 ( 𝑛 )) for thedemand 𝑁 𝑝 = 30 , starts from the left boundary of the train-FD and moves along a counter-clockwise closed loopduring rush hour. The dotted line indicates the sudden change in traffic states owing to the discontinuity of travel timederivatives at 𝑡 , 𝑡 ∗ , and 𝑡 𝑒𝑑 . The lower part of the loop ( 𝑞 ( 𝑛 ) < tr/h) represents the dynamics of trains departingfrom the system during [ 𝑡 , 𝑡 ∗ ], whereas the upper part represents the dynamics during [ 𝑡 ∗ , 𝑡 𝑒𝑑 ]. The maximum of 𝑎 𝑝 ( 𝑡 ) is reached at the lower-right corner of the loop, whereas the other peak occurs at the critical density in the upperpart of the loop. This two-peak phenomenon is a new and interesting characteristic of the equilibrium distribution ofpassenger arrivals for a congested rail transit system, which should be verified through empirical observations.However, when the total travel demand is rather low, 𝑎 𝑝 ( 𝑡 ) may have only one peak, as shown in Fig. 7(a). Themagenta line in Fig. 7(b) shows the evolution of ( 𝑘 ( 𝑛 ) , 𝑞 ( 𝑛 )) for this low-demand case. It can be seen that if ( 𝑘 ( 𝑛 ) , 𝑞 ( 𝑛 )) Jiahua Zhang et al.:
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Time (min) N u m be r o f t r a i n ( t r) (a) Cumulative number of trains Train sequence number H ead w a y ( m i n ) (b) Headways of trains Figure 4:
Dynamics of rail transit system.
Time (min) C o s t ( $ ) (a) Case C Time (min) C o s t ( $ ) (b) Case D with 𝑡 ∗ 𝑖 < 𝑡 𝑚 Figure 5:
Travel cost for two cases. after 𝑡 ∗ (upper-right part of the loop) does not enter the congested regime of the train-FD, 𝑎 𝑝 has only one peak. Thefinal remark is that the passenger departure flow is higher around the end of the rush hour than that at the early timeperiod for both high- and low-demand cases, unlike the passenger arrival flow. This would also be worth investigatingfrom empirical data.
5. A simple time-dependent timetable optimization
This section presents the optimization problem of a time-dependent timetable pattern as an application of theproposed model. The first subsection describes the problem setting, and the second subsection presents the results andprovides their interpretation.
Herein, we consider the following simple time-dependent timetable pattern:1. Two dispatch frequencies (or train inflows) 𝑎 and 𝑎 ( 𝑎 ≥ 𝑎 ) are employed.2. The train inflow is 𝑎 initially; it becomes 𝑎 from the beginning of the rush hour and lasts until the time at whichthe train carries the passenger departing from the system at 𝑡 ∗ on time, and then back to 𝑎 , as shown in Fig. 8.The first condition is widely employed in practice. The second condition may be necessary to avoid degrading thetrain system significantly under user equilibrium (i.e., the timings of the inflow changes may be near-optimal). More Jiahua Zhang et al.:
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Time (min) P a ss enge r nu m be r ( pa x ) (a) Case C Time (min) P a ss enge r nu m be r ( pa x ) (b) Case D Figure 6:
Cumulative number of passengers for two cases.
50 100 150 200 250 300 350
Time (min) P a ss enge r nu m be r ( pa x ) (a) Cumulative number of passengers ( 𝑁 𝑝 = 8000 ) Start Start (b) Dynamics of density and flow on train-FD
Figure 7:
A comparison of rail transit dynamics when travel demand 𝑁 𝑝 = 8000 . specifically, this condition is aimed to prevent train outflow from becoming very low in the first half of the rush hour,and the traffic state from entering congested regime in the second half. According to Fig. 8, the ratio 𝜔 ∈ (0 , of theduration for 𝑎 to the rush hour is given as a constant: 𝜔 = 𝛾 ( 𝛼 − 𝛽 ) 𝛼 ( 𝛽 + 𝛾 ) (24)Thanks to the second condition that exploits the characteristic of the equilibrium, the optimization problem of de-termining ( 𝑎 , 𝑎 ) to minimize passengers’ travel cost can be expressed as the following concise mathematical problemwith equilibrium constraints (MPEC). min 𝑎 ≥ 𝑎 > 𝑇 𝐶 𝑒 ( 𝑎 , 𝑎 | 𝑁 𝑝 ) (25)subject to 𝜔𝑎 + (1 − 𝜔 ) 𝑎 ≤ 𝑎 (26)where 𝑇 𝐶 𝑒 ( 𝑎 , 𝑎 | 𝑁 𝑝 ) is the equilibrium travel cost as a function of the decision variables. Eq. (26) indicates thedispatch capacity constraint, where 𝑎 is the maximum available train inflow during rush hour. We can easily solve theproblem using a brute-force search with Algorithm 1. Because the train inflow is time-dependent, we must check thecondition (21) of the existence of equilibrium while evaluating the objective function. Jiahua Zhang et al.:
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Page 10 of 14orning commute in congested urban rail transit system 𝑎 𝑎 𝑇 𝑡 𝑚 𝑡 𝑒𝑑 𝑡 ∗ 𝑇 𝑇 𝑡 Time C u m u l a ti v e t r a i n nu m b e r 𝑎 𝑎 𝑎 𝛼 𝑎 𝜔 𝑡 𝑒𝑑 − 𝑡 𝑒𝑑 − 𝑡 Figure 8:
A simple time-dependent timetable pattern.
Dispatch capacity constraint
Equilibrium existence constraints 𝑎 = 𝑎 S1 S0 S3S2 𝑇 𝐶 𝑒 ( $ ) Dispatch capacity constraintEquilibrium existence constraints 𝑎 = 𝑎 S1 S0 S3S2
Figure 9:
Counter plot of the objective function.
The optimization results under the parameter settings in Table 1 (and 𝑎 = 18 tr/h) for Case C are shown in Fig. 9.The horizontal and vertical axes represent the high inflow rate 𝑎 and low inflow rate 𝑎 , respectively. The colorrepresents the value of the objective function. We here evaluated the objective function every . tr/h for both inflowrates.From this figure, it can be seen that the objective function is almost convex, and a unique optimal solution (S0) isobtained. We also see that maximizing the dispatch frequency can increase the equilibrium cost. This is a particulartype of “capacity increasing paradox", known to occur in equilibrium transportation systems (e.g., Braess, 1968; Arnottet al., 1993a). To understand the reason for this phenomenon, we show the train dynamics for scenarios S0, S2, andS3 in Fig. 10. The ratio 𝑎 ∕ 𝑎 of the S2 and S3 patterns is the same as the optimal pattern S0, but with differentaverage inflow rates. From the train cumulative curves in Fig. 10(a), we see that the equilibrium rush period for S0is shorter than that of S2 and S3. The reason can be understood from Fig. 10(b). For S0, a high passenger arrivalrate was achieved while maintaining a relatively high train flow. This means that, at the optimum, the intention of thesecond condition in the previous subsection is achieved. By contrast, for S2 and S3, a train flow that is either too lowor too high cannot accommodate the high passenger arrival rate. For scenario S2, on the one hand, the travel time is Jiahua Zhang et al.:
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Page 11 of 14orning commute in congested urban rail transit system (a) Cumulative number of trains (double arrow line: theequilibrium rush period) (b) Density and flow on train-FD
Figure 10:
Dynamics of rail transit system under time-dependent timetable patterns.
Table 2
Comparison of travel costs for different timetable patterns.
Scenario 𝑎 (tr/h) 𝑎 (tr/h) average inflow(tr/h) ∑ 𝑇 𝐷𝐶 ( $ ) ∑ 𝑆𝐷𝐶 ( $ ) ∑ 𝑇 𝐶 ( $ ) 𝑇 𝐶 change (%)S0 18.7 10.1 14.0 29.30 16.13 45.43 -S1 15.0 13.2 14.0 30.68 21.11 51.79 +14.0S2 12.0 6.5 9.0 37.30 18.46 55.76 +22.7S3 23.1 12.5 16.5 29.75 29.59 59.34 +30.6 extended even by the low passenger arrival rate owing to the insufficient dispatch frequency. On the other hand, underequilibrium, the schedule delay cost is compensated by the travel delay cost. As a result, the demand concentrationbecomes more moderate, and the rush period becomes longer. For scenario S3, a similar conclusion is obtained becausea considerable proportion of trains operate in the congested regime of the train-FD (i.e., on-track congestion occurs).In addition, we see that the average train flow 𝑞 ( 𝑛 ) maintains an almost constant level before and after 𝑡 ∗ under theoptimal setting, which can be stated as 𝑎 𝑎 ≈ ( 𝛼 + 𝛾 )( 𝛼 − 𝛽 ∕2)( 𝛼 − 𝛽 )( 𝛼 + 𝛾 ∕2) . (27)From this equation, we see that 𝑎 ∕ 𝑎 should increase with the decrease of 𝛼 and with the increase of 𝛽 or 𝛾 . Althoughthis specific condition relies on the current problem setting, the strategy of flattening the train operation performancecould be a useful guide for a more general case.Finally, Table 2 summarizes the travel costs for the timetable settings S0–S3 in Fig. 9. Scenario S1 representsthe case with the same average inflow as S0, but a rather smaller difference between 𝑎 and 𝑎 . We observed that thetotal travel cost ∑ 𝑇 𝐶 ( ∑ 𝑇 𝐶 = 𝑇 𝐶 𝑒 𝑁 𝑝 ) of scenarios S1–S3 are significantly higher than that of the optimal one S0.More specifically, by comparing S0 and S1, the increase in the total schedule delay cost ∑ 𝑆𝐷𝐶 (31%) is much greaterthan that of the total travel delay cost ∑ 𝑇 𝐷𝐶 (5%). This suggests a primary deficiency of timetable patterns with arelatively low 𝑎 ∕ 𝑎 ratio is that passengers cannot arrive at their desired arrival time 𝑡 ∗ sufficiently. We can obtaina similar property for the scenario with an abundant train supply (S3). However, when the train supply is insufficient(S2), passengers would suffer from a significantly longer travel delay ( ∑ 𝑇 𝐷𝐶 increases by 27% compared to S0).
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6. Conclusions
This paper proposes a macroscopic model to describe the equilibrium distribution of passenger arrivals for themorning commute problem in a congested urban rail transit system. We first developed a model for the morningcommute problem in rail transit based on the train-FD and showed the equilibrium conditions. Further, we discusseda solution method and the existence of an equilibrium. We then examined the characteristics of the proposed modelthrough numerical examples under different passenger demand settings. Finally, by employing the proposed model,we analyzed a simple time-dependent timetable optimization problem with equilibrium constraints.The proposed model is not only mathematically tractable but can also thoroughly consider the relations amongpassenger concentration, on-track congestion, and time-dependent timetable in a congested rail transit system. Thisenables us to investigate the characteristics of the equilibrium and the optimal design of the timetable in a simplemanner. Throughout the numerical experiments, we obtained the following findings: (i) the equilibrium passengerarrival rate can have two peaks depending on whether on-track congestion occurs; (ii) there exists a “capacity increasingparadox" in which a higher dispatch frequency can increase the equilibrium cost; (iii) under equilibrium, an insufficientsupply of rail transit mainly increases the travel delay cost while redundant supply increases the schedule delay cost;(iv) during the rush period, the average train flow maintains an almost constant level under an optimal timetable setting.The straightforward extensions of the proposed model include a consideration of elastic demands and captiveusers (e.g., Gonzales and Daganzo, 2012). For the former, we only need to specify the travel demand 𝑁 𝑝 ( 𝑇 𝐶 𝑒 ) as amonotonically decreasing function of the equilibrium travel cost (e.g., Arnott et al., 1993b; Zhou et al., 2005). Thelatter can be achieved by modifying 𝑎 𝑝 in Eq. (15) as 𝑎 𝑝 ( 𝑛 ) = 𝑎 𝑝𝑐 + 𝑎 𝑝𝑓 ( 𝑛 ) , where 𝑎 𝑝𝑐 is the arrival rate of captiveusers, and 𝑎 𝑝𝑓 ( 𝑛 ) is the arrival rate of flexible users for train 𝑛 .In this study, rail transit is assumed to be a homogeneous system in which both stations and passenger demandare evenly distributed. Thus, we need to develop a train-FD model applicable to a heterogeneous railway system todeal with a more realistic situation. Considerations of heterogeneity in passenger preferences (i.e., the value of time)(Newell, 1987; Akamatsu et al., 2020) and the costs/revenue of the transit agency in the optimization of timetable/faresettings are also important topics. Another fruitful future work would be the design of the pricing schemes. Usingthe proposed model, we would obtain insights into not only the first-best pricing scheme but also the second-best ones(e.g., step tolls in Arnott et al., 1990; Laih, 1994; Lindsey et al., 2012) that are generally formulated as MPEC. Acknowledgements
This study was financially supported by JSPS KAKENHI Grant No. JP17H03320.
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