A proof of concept simulator including dynamic interactions in dense suburban railway context

Abstract

8 Dense suburban railway systems are subject to various types of hazards whose effects on supply 9 and demand interact at short time scales: dynamic interactions. The combination of hazards and 10 dynamic interactions may lead delays to propagate and worsen, which significantly deteriorates the 11 system’s performance. As such, it is crucial to simulate these effects to design robust schedules. 12 However, existing simulators for dense suburban railway systems do not fully include demand-to13 supply and supply-to-demand dynamic interactions. In this paper, a proof of concept simulator is 14 presented to show the importance of considering demand-to-supply and supply-to-demand dynamic 15 interactions. Compared to a benchmark simulator that neglects these dynamic interactions, on 16 the presented case study, our simulator is able to highlight additional phenomena resulting from 17 demand-to-supply and supply-to-demand dynamic interactions. In practice, the implementation 18 of all dynamic interactions in a ready-to-use simulator will significantly improve future scheduling. 19 This work is a first step toward this goal in which a proof of concept is provided. 20 Keywords: Demand, Disruption, Public Transport, Railway, Simulation, Supply

21 1 Introduction

22 Urban sprawl in big cities has challenged existing transport modes for commuting between the 23 suburbs and the city center. People need high-performance public transport systems to travel 24 large distances twice a day. Metros, generally underground, are often confined to the city center 25 and nearby suburbs. Differing from metros, suburban trains are often used to connect more distant 26 suburbs to the city center. However, these transport systems also need to be robust. A robust 27 system is a system that has the ability to maintain or recover, by itself, a decent performance 28 after a small hazard (Goverde & Hansen, 2013; Jensen, 2015; Lusby et al., 2018). The robustness 29 of a transport system is generally assessed using simulators considering the specificities of the 30 underlying transport mode. 31 Suburban trains and metros are close transport systems: both depend on railways, and both have 32 high capacities. However, they differ in four main points (that may not be true for all networks, 33 as the separation between train and metro is rather fluid). 34 1. Train infrastructure is generally shared between many train lines and can also be shared 35 between several train types. Metro lines circulate in segregated infrastructures. 36 2. Train lines may be composed of several branches. Services may also vary: omnibus services 37 (serving all stations), semi-direct services (serving some stations but not all), and direct 38 services (serving only the termini). Metro lines have a restricted number of branches and 39 services, usually only one. 40 3. Train operations are generally schedule-based, which means that a train is not allowed to 41 depart from a station before its scheduled departure time. Metro operations are headway42 based, which means that metros leave the origin terminus with a fixed headway and run 43 without schedules. 44 4. Trains are often run with a lower frequency than metros, and with higher differences in 45 services during peak periods and off-peak periods.

Headway

Suburb 1

Suburb 2 City center

Suburb 3

Demand

Figure 1: Archetype of a dense suburban train line with a terminus in the city center. Headway decreases while approaching the city center due to the merging of the branches. Demand increases while approaching the city center due to an increasing urban density.

46 Dense suburban railway systems have similarities to the metros (Altazin, 2018). Their infrastruc47 ture is open to other trains, but in nominal situations, they are essentially dedicated to a single 48 train line. The train line is generally made up of several branches that run between a single city 49 center and several parts of the suburbs. The demand at the branches is generally lower, so the 50 frequency is less, and this justifies the operation to be schedule-based to minimize the waiting time 51 spent by passengers for a given train. The demand in the central section is generally higher due 52 to higher urban densities. At the same time, trains from the different branches merge, resulting in 53 higher frequencies, frequencies generally as high as that of a metro. Figure 1 shows an archetype 54 of a dense suburban train line. 55 In such a line, fluctuations related to operations (running times or dwell times) (Yuan et al., 2010; 56 Buchmüller et al., 2008) or passenger demand (Pan et al., 2017) are numerous and dynamically 57 interact between them. This leads delays to appear more frequently and to propagate more easily 58 compared to other types of train systems. 59 Interactions between supply and demand appear at short-term scale (dynamic interactions): for 60 example, between one vehicle and the vehicle directly after, between one station and the next 61 station, or between one vehicle and its demand. Dynamic interactions are of four types: 62 • Supply-to-supply dynamic interactions, related to signaling (Zieger et al., 2018; Huang et 63 al., 2024); 64 • Demand-to-demand dynamic interactions, related to boarding failure (Leurent et al., 2014; 65 Hänseler et al., 2020; Cornet, 2020) ; 66 • Demand-to-supply dynamic interactions, related to dwell time delays that may appear when 67 the alighting and boarding volumes are too large (Cornet et al., 2019; Palmqvist et al., 2020; 68 Coulaud et al., 2023; Baali, Buisson, et al., 2025); 69 • Supply-to-demand dynamic interactions, related to passenger arrival behaviors that can be 70 split into headway-based passengers (who arrive uniformly at the station, knowing a train 71 will arrive shortly) and schedule-based passengers (aiming for a train scheduled at a specific 72 time) (Baali et al., 2026). 73 These dynamic interactions (illustrated in Figure 2) have been extensively studied in the literature. 74 However, understanding them is not sufficient to solve operational performance and robustness 75 issues. They need to be addressed through simulation tools. Existing simulators are limited to other 76 transport modes, such as buses or metros (Toledo et al., 2010; Fonzone et al., 2015; Srikukenthiran 77 & Shalaby, 2017; D’Acierno et al., 2017), or do not consider all dynamic interactions (Landex 78 & Nielsen, 2006; Cornet, 2020; Othman et al., 2023). To our knowledge, no simulator for dense 79 suburban train systems that integrates all dynamic interactions exists so far. 80 Therefore, the goal of this work is to provide a proof of concept simulator for dense suburban 81 trains. The proof of concept will help highlight several phenomena related to supply-to-demand 82 and demand-to-supply dynamic interactions that cannot be spotted without considering them. 83 The paper is structured as follows. First, a simulator integrating all dynamic interactions is pre84 sented along with a benchmark simulator. Then, a comparison is provided between both simulator 85 results for a case study in Paris. This paper ends with a conclusion.

Alighting and boarding process

Signaling Boarding failure

Supply Demand Demand distribution between trains Transfer sequencing

Figure 2: Scheme of the dynamic interactions between supply and demand in the case of Mass Transit train systems. Some interactions are not specific to Mass Transit train systems.

Departure Departure time Demand-to-supply time dynamic interaction

Alighting and boarding Headway time

Supply-to-demand dynamic interaction

Number of Number of Number of Number of alightings boardings alightings boardings

Figure 3: Framework of the benchmark simulator (left) and our simulator (right) and dependencies between the variables. The dependencies lead to cyclic equations.

86 2 Methodology

87 In the literature, railway simulator commonly do not consider all dynamic interactions. Supply-to88 supply dynamic interactions are integrated but demand and supply are considered separately. A 89 simplified version of such a simulator will serve as a benchmark in this work (Figure 3 right). By 90 contrast, we present a simplified simulator considering all the dynamic interactions (Figure 3 right). 91 This simplified simulator is only a proof-of-concept The comparison between both simulators will 92 serve to demonstrate the ability of our simulator to simulate phenomena that typically result from 93 supply-to-demand and demand-to-supply dynamic interactions.

94 A simplified benchmark simulator

95 To illustrate the relevance of considering dynamic interactions in the simulator, we implement a 96 simulator that only considers supply-to-supply dynamic interactions, similarly to most existing 97 railway simulators.

98 Number of alightings First, the number of alightings A from each train k at each station 99 s, is obtained from passengers who have previously boarded B using an origin-destination matrix 100 mOD . The origin-destination matrix is assumed to be deterministic and independent of the date 101 d. The perimeter will focus only on the morning peak hours, and then the matrix is assumed to 102 be independent of the time of day. The matrix depends on the train index k, but only via the 103 service. If a station is skipped by train index k, no one will alight there. All trains running the 104 same service are assumed to have the same origin-destination matrix. X Ãk,s,d = mODs′ ,s,k B̃k,s′ ,d . (1) s′ <s

105 Number of boardings The number of boardings is proportional to the total number of entries 106 Es,d . This number of entries is distributed between the different trains with a rate βk which only 107 depends on the train index. B̃k,s,d = βk Es,d . (2)

108 Departure time The observed departure time only depends on the scheduled departure time, 109 the train ahead (supply-to-supply dynamic interaction), and some constant potential margin con110 sumption in case of delay. This latter term is independent of alighting and boarding volumes, 111 meaning that no demand-to-supply dynamic interaction is considered contrary to our simulator.

t̃obs sch obs min obs , t̃k,s−1,d + (tsch sch (3)
k,s,d = max tk,s , t̃k−1,s,d + h k,s − tk,s−1 ) − M .

112 A simplified simulator with dynamic interactions

113 To demonstrate the importance of the dynamic interactions, we built a simplified simulator con114 sidering all dynamic interactions to be compared with the benchmark simulator.

115 Number of alightings The number of alightings is computed similarly to what is done with 116 the benchmark simulator. X Ak,s,d = mOD s′ ,s,k Bk,s′ ,d (4) s′ <s

117 Number of boardings The number of boardings B is computed in two steps. In a first step, 118 the boarding demand Bk,s,d O (number of passengers who want to board train k at station s on date 119 d) is obtained using the behavior-based model from Baali et al. (2026) which accounts for supply120 to-demand dynamic interactions. In this model, the boarding demand is split into schedule-based 121 (SB) passengers and headway-based (HB) passengers. Schedule-based passengers aim for a specific 122 train and arrive close to the scheduled departure time. Headway-based passengers arrive uniformly 123 at the station within the headway, knowing that the frequency is high enough to avoid waiting 124 too long. Headway-based passengers generate an additional boarding demand when the headway 125 increases, which is why this distinction is crucial. Similarly to what is done for the benchmark 126 simulator, the boarding demand (Bk,s,dO ) is assumed to be linear with the total number of entries 127 Es,d in the station, allowing for variations in demand from one date to another. qk,s SB represents 128 the rate of schedule-based passengers, and qk,s the rate of headway-based passengers. HB

O SB HB obs (5)
Bk,s,d = qk,s + qk,s hk,s,d Es,d 129 In a second step, to get the effective number of boardings (Bk,s,d ) from the boarding demand 130 (Bk,s,d O ), one has to consider the number of boarding failures. Boarding failures (∆Bk,s,d ) corre131 spond to passengers who cannot board the train k for which they arrived, because the load (Lk,s,d ) 132 of the train cannot exceed the train passengers capacity (C). This illustrates demand-to-demand 133 dynamic interactions. O (6)
Bk,s,d = min Bk,s,d + ∆Bk−1,s,d , C − Lk,s,d − Ak,s,d

O ∆Bk,s,d = Lk,s,d + Ak,s,d + Bk,s,d + ∆Bk−1,s,d − C (7) 134 Notice that, in the case study, the train capacity (2,592 P) is high enough (compared to the 135 boarding demands) to completely prevent boarding failures from appearing.

136 k,s,d is the observed headway, i.e., the difference between the observed departure Headway hobs 137 time tobs k,s,d of the train and the observed departure time of the previous train.

hobs obs obs k,s,d = tk,s,d − tk−1,s,d . (8)

Quantity Name Type Unit Value A Number of alightings Variable [P] B Number of boardings Variable [P] h Headway between two trains Variable [s] tobs Observed departure time Variable [s] ABT Alighting and boarding time Variable [s] BO Boarding demand Variable [P] ∆B Number of boarding failures Variable [P] L Train load Variable [P] mOD Origin-destination matrix Parameter Without unit Context dependent q SB Schedule-based passenger rate Parameter Without unit Context dependent q HB Headway-based passenger rate Parameter [s−1 ] Context dependent C Train passengers capacity Parameter [P] 2,592P hmin Minimal headway (due to signaling) Parameter [s] 120s TT Technical time Parameter [s] 7.5s ζ Critical door ratio Parameter 0.15 ψ Flow size Parameter [P · s−1 ] 4.9P · s−1 1 1 γ Density calibration parameter Parameter [P− 2 ] 0.09P− 2 M Constant margin (benchmark) Parameter [s] 8s tsch Schedule departure time Parameter [s] Context dependent E Total number of entry Parameter [P] Context dependent RT Running time Parameter [s] Context dependent

Table 1: Quantities considered in the simulation.

138 Departure time The observed departure time tobs k,s,d is determined by three constraints: either 139 the train has to wait for its scheduled departure time tsch k,s,d before leaving (nominal situation), or 140 it has to wait until the previous train is far enough according to a minimal headway hmin due to 141 signaling (which is a simplified modeling of supply-to-supply dynamic interactions), or it is limited 142 by the observed running time RT obs and the observed alighting and boarding time ABT obs (plus a 143 constant technical time corresponding to the time for opening/closing the doors and dispatching).

tobs sch obs min obs (9)
k,s,d = max tk,s , tk−1,s,d + h , tk,s−1,d + RTk,s,d + ABTk,s,d + T T

144 Alighting and boarding time Finally, the alighting and boarding time ABT , is assumed 145 to only depend on the number of alightings Ak,s,d,j and the number of boardings Bk,s,d,j at the 146 critical door (door j where the number of alightings and boardings is maximal). This corresponds 147 to demand-to-supply dynamic interactions. The ratio between the total number of movements at max(Ak,s,d,j +Bk,s,d,j ) 148 the critical door and over the entire train (ζs = Ak,s,d +Bk,s,d ) is assumed to be constant 149 for each station and each train. From there, the alighting and boarding time is computed using 150 the deterministic model of Baali, Buisson, et al. (2025) and the method of Baali, Kuipers, et al. 151 (2025).

(10)

R 0.8ζs (Ak,s,d +Bk,s,d ) 1  ABTk,s,d = 0.8 0  √ √  dN. ψ 1−exp(−γ ζs (Ak,s,d +Bk,s,d )−N ) exp(−γ ζs (Ak,s,d +Bk,s,d )−N )

152 Notice that there exist dependencies due to other lines or branches. Trains from other lines or 153 branches may hinder trains from the studied line due to signaling (supply-to-supply dynamic 154 interactions), but we will neglect them in this work. 155 The above-presented equations are cyclic because they depend on each other. As a reminder, 156 Figure 3 illustrates the dependencies between the variables. To solve this, the computation for any 157 (k, s, d) is done twice iteratively. First, the demand-to-supply dynamic interaction is neglected (the 158 alighting and boarding time is assumed to be zero). Then, the observed departure time, headway, 159 and number of boardings are computed from there. Based on the computed number of boardings, 160 the observed departure time is computed again with the demand-to-supply dynamic interaction 161 (influence of the alighting and boarding time). Finally, all quantities are computed again based on 162 this updated value of the observed departure time. 163 Among all above-presented quantities: mOD , q SB , q HB , C, hmin , T T , tsch , E, and RT are constant 164 parameters. The other quantities are variables determined by the equations (see Table 1 for a 165 summary of the notations).