Mitigating bus bunching with real-time crowding information

A common problem in public transport systems is bus bunching, characterized by a negative feedback loop between service headways, number of boarding passengers and dwell times. In this study, we examine whether providing real-time crowding information (RTCI) at the stop regarding the two next vehicle departures can stimulate passengers to wait for a less-crowded departure, and thus alleviate the bunching effect. To this end, we leverage on results from own stated-preference survey and develop a boarding choice model. The model accounts for the presence of RTCI and is implemented within dynamic public transport simulation framework. Application to the case-study model of a major bus corridor in Warsaw (Poland) reveals that RTCI can induce a significant probability (30–70%) of intentionally skipping an overcrowded bus and waiting for a later departure instead. This behaviour, in turn, results in significantly lower vehicle headway and load variations, without deteriorations in total waiting utility. Overall, journey experience improves by 6%, and crucially—the prevalence of denial-of-boarding and excessive on-board overcrowding is substantially reduced, by ca. 40%. Results of our study indicate that the willingness to wait induced by RTCI can be a potential demand management strategy in counteracting bunching, with benefits already attainable at limited RTCI response rates.

Scheduled dispatching headway of consecutive trips of a line L from the origin stop s 0 h r,e Actual headway between consecutive trips r, r + 1 along the line segment e φ s (r) Dispatching time offset of run r from stop s, with respect to nominal headway h L,0 t r,e Riding time of trip r along line segment e t r,s Dwell time of trip r at stop s t a,k,s Decision time instance of action a by passenger k at decision point s p a,k,s Choice probability of action a by passenger k at decision point s u a,k,s Total (expected) utility of action a considered by passenger k at decision point s u i,k Total (experienced) utility along path i for passenger k u i,k,s Total (expected) utility of path i for passenger k at decision point s v i,k,s Systematic part of (expected) utility of path i for passenger k at decision point s ε k Random error part of (expected) utility for passenger k t e ivt (Expected)  Recorded RTCI level (i.e. 1-to-4 scale) of run r at the departure from stop s β r ,j(t) Generated RTCI value of trip r at the downstream stop j, valid at time t ( s, j ∈ L : j > s) β a WTW WTW crowding multiplier at the boarding action instance a δ a WTW WTW (willingness to wait) decision instance

Introduction
Reliability of public transport (PT) operations is an important aspect for the perceived PT quality of service, which is profoundly influenced by variations in travel times and onboard travel conditions. In this context, variability of passenger flows is one of the key determinants of PT service reliability (Sorratini et al. 2008), especially in urban, congested transport networks. Excessive, unstable demand flows can ignite supply performance perturbations that can quickly spiral from an isolated event into a network-wide disruption (Cats et al. 2011b), and exacerbate service disruptions even without exogenous disturbances, posing major risks in low and high demand conditions (Fonzone et al. 2015).
A clear illustration of this inherent PT system instability is the so-called bus bunching effect (Newell and Potts 1964;Cats et al. 2011b). Bunching is the consequence of a negative feedback loop between service headways, number of arriving passengers and dwell times. The bus bunching pattern is likely to become self-amplified along the service line, as the first bus becomes increasingly overcrowded and delayed at downstream stops, while the second bus is underutilized and 'catches up' with the predecessor, leading to their 'platooning' in extreme cases (Yu et al. 2016). This phenomenon results in unreliable services, with a series of negative implications for passengers and operators-longer journey times, service delays, higher travel discomfort, risk of overcrowding and denied boarding, uneven headway and load distribution and an inefficient capacity utilization . A major contributor to the bunching problem is passengers' inherent preference towards boarding the first available departure, which exacerbates further the crowding-bunching feedback effect. Therefore, strategies aimed at stimulating co-operative boarding behaviour amongst passengers-presumably through the provision of real-time crowding information-shall be explored as a potential measure to counteract the bus bunching (Delgado et al. 2012;Wu et al. 2017).

Literature review
To counteract the undesirable bunching effect, various tactical operational strategies have been conceived. On the supply side, these involve holding control strategies [commonly evaluated as an optimized weighted function of passenger travel (dis)utility and/or the closed-form function of bus arrival times, e.g.: Adamski and Turnau 1998;Cats et al. 2011b; Bartholdi and Eisenstein 2012;Delgado et al. 2012;Berrebi et al. 2015;Berrebi et al. 2018;Laskaris et al. 2019;Gkiotsalitis and van Berkum 2020;Wang and Sun 2020]; allowing for buses to overtake each other (Sun and Schmöcker 2018;Schmöcker et al. 2016;Wu et al. 2017); stop-skipping (expressing) (Fu et al. 2003;Sun and Hickman 2005;Larrain and Muñoz 2020); short-turning (Zhang et al. 2017;Leffler et al. 2017; rescheduling (Gkiotsalitis 2019); speed adjustment (Daganzo and Pilachowski 2011); and robust slack-time planning (Zhao et al. 2016). Another, relatively less-explored stream of research is devoted to impacts of demand-side phenomena in managing the bus bunching problem, albeit with limited empirical underpinning (Wu et al. 2017;Sun and Schmöcker 2018;Wu et al. 2019).
A series of works have dealt with the notion of applying boarding limits to counteract the bunching effect. Imposing boarding limits or a no-boarding policy implies that when specific (in)stability criteria are met, the PT operator constrains the number of passengers allowed to board a PT vehicle, forcing the remaining ones to wait at the stop for a later departure. In the context of reversing the bus bunching mechanism, while holding strategies aim at 'slowing down' the second bus that moves ahead of schedule, the objective of no-boarding strategies is to 'speed up' the first bus that is otherwise increasingly delayed (Saw et al. 2019). Delgado et al. (2012) estimate that boarding limits can foster the benefits of holding strategies when applied simultaneously, particularly in the case of highfrequency and high-demand services, requiring fewer and shorter holding times to restore service regularity and travel comfort. Boarding limits implemented as a sole (isolated) measure were also found to be effective in reducing headway variability and improving travel times (Zhao et al. 2016;Enayatollahi et al. 2019). Saw et al. (2019) conclude that noboarding policies perform favorably in mitigating the bus bunching when compared against holding strategies in busy periods, but contrarily-backfire during low-demand periods (by imposing excessive waiting times) when holding is a more advantageous solution.
Aside from the notion of fixed boarding limits, a number of papers analyse whether shifts in passengers' boarding behaviour can also help mitigate bus bunching. Wu et al. (2017) introduce a dynamic queue swapping behaviour model, where boarding probability is proportional to the available (residual) capacity of next bus arrivals-i.e. passengers at a stop form self-equilibrating boarding queues for the first and second incoming departures. They found that in conjunction with enabling overtaking between buses, this brings substantial performance and travel experience benefits, regardless of network demand levels and without employing any holding controls. Sun and Schmöcker (2018) demonstrate that in the event of bus bunching, a hypothetically high preference towards waiting deliberately for a second bus is in general advantageous for restoring the service regularity and improving travel experience, even without additional control measures.
Studies acknowledge that favourable shifts in passengers' boarding behaviour may be encouraged by providing information on the in-vehicle crowding levels (Delgado et al. 2012;Palma et al. 2015;Schmöcker et al. 2016;Wu et al. 2017). This is an increasingly feasible solution as data on past and current passenger flows in PT networks is gathered via a variety of sources-including Automated Passenger Counts (APC) and Automated Fare Collection (AFC) systems, video recording data, remote sensing, mobile and wireless networks, and most importantly-smart card ticketing systems. These data can be processed and conveyed to passengers in the form of real-time crowding information (RTCI) (Jenelius 2020). Current state-of-the-practice of RTCI systems involves usually pilot and limitedscale projects, ranging from localised crowding information on individual train carriages' loads (e.g. London (Schmitt 2017), Sydney (Susan 2017), Tokyo (EJRC 2019), information on bus occupancy loads (e.g. Seoul (SMC 2017) and train crowding [e.g. Stockholm Metro (Zhang et al. 2017), Dutch Railways (NS 2021, to network-wide information on expected crowding levels of PT routes, available via travel apps [e.g. Google Maps (Google LLC 2021), Moovit (Moovit Inc 2021), JakDojade.pl (City-Nav LLC 2021)]. Recently, RTCI provision has been gaining momentum as operators aim to tackle the ramifications of the (on-going) covid-19 pandemic crisis for perceived travel safety. This has been witnessed in several systems in the US, e.g. in Boston (MBTA 2021), San Jose (VTA 2021) and Washington D. C. (WMATA 2020), where crowding information is disclosed in real-time to reassure travellers about on-board crowding conditions and the possibility to maintain social distancing.
Access to crowding information concerning the next PT trips (departures) at a stop can induce a specific decision pattern among passengers, namely the willingness to wait (WTW) to reduce (or avoid) on-board overcrowding experience-i.e., by skipping the first (overcrowded) trip and waiting at the stop to board a second, less-crowded trip later. This phenomenon has been estimated in a number of stated preference (SP) studies by means of survey questions (Kim et al. 2009;Kroes et al. 2014) or choice experiments (Preston et al. 2017;Kattan and Bai 2018;Drabicki et al. 2022). These studies found that the extent to which passengers are willing to wait is mainly driven by high overcrowding levels in the first departure, where the WTW tends to rise non-linearly above a perceived discomfort threshold, and on the waiting time for the second departure from the same PT stop. Average acceptable waiting times were found to oscillate between: ca. 3-12 [mins]-for urban PT (bus and tram) systems (Drabicki et al. 2022), and ca. 8-25 [mins]-for regional rail trips (Preston et al. 2017). In the case of urban PT trips, a 5-min wait is deemed acceptable for 40% of respondents-if the first bus/tram is moderately crowded (i.e. has 'comfortable' standing space available), and even for up to 80% of them-if it is severely overcrowded (i.e. possible denied-boarding risk). Survey findings from a Paris metro system report analogous figures, ranging from 12% (first train-minor crowding) to 75% (first train-severe overcrowding) (Kroes et al. 2014).
The potential network-wide impact of RTCI upon passengers' travel behaviour has been assessed by several studies using PT simulation models. These studies focus on specific RTCI consequences and spatial (i.e., route choice) or temporal (i.e., departure choice) impact dimensions. Bouman et al. (2016) develop a game-theory based model and analyse crowding information provision in context of the so-called El Farol Bar problem. Toy network simulations underline its potential to enhance capacity utilization, but also the risk of lower payoffs with rising network coverage and responsiveness to crowding information. Nuzzolo et al. (2016) proposed a mesoscopic PT assignment model that simulates the impact of predictive RTCI on long-term (day-to-day) travel decisions. Predictive RTCI and network assignment are achieved as an iterative outcome of in-vehicle passenger loads merging towards choice probabilities. Case study results show that long-term adaptation to such crowding prediction induces substantial departure time shifts in everyday journeys (e.g., AM peak widening), and a certain decrease in waiting times and denied-boarding risk. Noursalehi et al. (2019) introduce a dynamic, mesoscopic PT assignment model for analysing the impacts of localized (station-level) predictive RTCI on instantaneous departure choices of next train departures. Crowding prediction refers here to the probability of being able to board an arriving PT vehicle (i.e., guaranteed/likely/unlikely). Likewise, it is estimated in a rolling horizon approach in 15-to 30-min intervals as an iterative outcome of convergence between passengers' anticipations, disseminated RTCI content and actual travel condition. Application findings reveal that passengers' acceptance of boarding deference (i.e. WTW rate induced by predictive RTCI) eventually leads to much lower overcrowding experience and improved on-board comfort. Drabicki et al. (2021b) develop a modelling framework of instantaneous RTCI effects upon route (path) choices. The model, embedded within BusMezzo, a dynamic transit assignment simulation model, assumes that RTCI is generated based on latest reported crowding levels of downstream line segments, and then utilized by passengers to evaluate route choice probabilities. Model demonstration reveals that such crowding information can improve travel experience-in particular, by reducing the incidence of worst overcrowding experience-but it is also marked by substantial inaccuracy risks, especially on the crowding underestimation side. Finally, a recent study of Wang et al. (2021) presents the toy-network simulation of impacts of passenger boarding shifts in response to RTCI on bus crowding loads. Demonstration of their algorithm suggest improvements of ca. 20% in terms of bus headways and bus run times and its potential to coordinate with holding control strategies. The policy performance is seemingly robust even with low WTW rates among the passengers.
In summary, although state-of-the-art studies have explored the prospects of demandside interventions in managing the bus bunching, they predominantly involved centralized, top-down assumptions on operators' boarding policies (fixed boarding limits) or passengers' boarding choices (fixed splitting rates or self-equilibrating queuing behaviour). In contrast, the potential ramifications of decentralized, individual boarding decision patterns that might be stimulated by provisioning RTCI are not yet fully understood. Arguably, there is still a lack of comprehensive and evidence-based knowledge on collective dynamics, which might emerge from more informed individual decision-making in the context of bus bunching.

Objective and contribution
The objective of this study is to assess the effects of real-time crowding information (RTCI) on travel experience and service reliability in the context of bus bunching problem. To this end, we model the passengers' instantaneous boarding choices with access to RTCI on crowding levels of next two PT departures from the current stop. The choice model is developed based on own stated-preference empirical findings of the willingness to wait (WTW) behaviour with RTCI (Drabicki et al. 2022). It is then implemented within a dynamic, agent-based PT assignment model, reproducing its wider impacts upon PT system performance and journey experience.
The potential of RTCI in mitigating the bunching is demonstrated through an application to a case study of a busy bus corridor in Warsaw (Poland). As shown by our results, the WTW with RTCI can greatly mitigate the evolution of bus bunching, prompting up to 30-70% of passengers to wait voluntarily for a less-crowded departure. Overall, this results in major improvements in terms of service performance, journey experience and eliminated risk of 'full bunching' phenomenon. The supply side witnesses lower headway variations and more uniform vehicle loads' distribution. Benefits for passengers are evident across the majority of bus stops, especially at central and downstream parts of the service, where further evolution of bus bunching is effectively eliminated. Interestingly, WTW does not necessarily result in longer total waiting time. Consequently, RTCI provision leads to global decline in both nominal journey times (2%) and generalized travel disutility (5%), and substantially reduced risks of in-vehicle overcrowding and denial-of-boarding (40%). Travel experience benefits are attainable already with limited responsiveness to RTCI, while higher RTCI utilization among passengers can also foster the service regularity.
This work aims to contribute to the (insofar limited) state-of-the-art stream of research on the prospective role of RTCI in mitigating bus bunching. Initial literature findings (Wang et al. 2021) have illustrated the value of RTCI in regulating bus services for numerical, toy-network simulations. We extend these considerations in our study and develop the PT operations and assignment model, underpinned by empirical WTW sensitivity analysis. Case study application reveals plausible RTCI effects on a real-world bus corridor model, whose overall benefits are captured via multiple metrics. We also investigate the impact of various level of responsiveness to the information provisioned. We conclude this study with discussion on our findings and the prospects of RTCI in real-time service and demand management strategies.
The remainder of this paper is organized as follows. Section "Method" presents the methodology of the proposed WTW choice model with RTCI and its embedment within a dynamic PT simulation model. Section "Application and Results" follows with application results of the proposed simulation model to a bus corridor case study network. We conclude this study with Sect. "Discussion", containing a summary of findings and practical implications.

Method
The following section presents the methodological framework of our research investigation of RTCI impacts on bus bunching effect. We begin with description of modelling inputs, requirements and the dynamic PT assignment model used in this study. Then, we present the passenger departure choice algorithm, simulating the impact of instantaneous RTCI on instantaneous boarding decisions at stops and further ramifications for PT system performance.

Modelling requirements
Capturing the interactions between crowding levels, real-time information and bus bunching requires a simulation framework that models the PT system in an explicit, disaggregate way-with PT demand represented by individual agents (travellers) and PT supply model described by individual vehicles (trips). The PT model shall capture how interactions between the PT demand (i.e. passenger overcrowding) and PT supply (i.e. bus bunching) may evolve dynamically and amplify each other. This pertains especially to the realtime feedback between passenger flows, dwell times and service headways. In addition, several effects of passenger overcrowding need to be properly reproduced: the influence of on-board passenger loads upon rising travel discomfort; the effects of strict capacity constraints and possible denial-of-boarding effect; and resultant consequences in form of uneven load distribution and service regularity.
Another modelling requirement relates to the working principles of RTCI systems in PT networks. Firstly, how the crowding information is recorded from instantaneous PT data on passenger loads in the PT vehicles. Secondly, the mapping of 'raw' crowding data into the provision of a user-friendly RTCI. Thirdly, how the generated RTCI information is instantaneously disseminated at PT stops and stations across the network.
Finally, on the demand side, the modelling framework shall represent the RTCI impacts on passengers' dynamic decision-making process. This includes the acquisition of RTCI at the stop(s); ubiquitous (system-wide) reaction to RTCI content; the incorporation of RTCI input into individuals' boarding choice; and the resultant travel decision, i.e. boarding now vs. willingness to wait for next departure.

Simulation model environment-BusMezzo
For the purposes of our analysis, we will use the BusMezzo mesoscopic, agent-based PT simulation model (Cats 2011). The BusMezzo model assumes an explicit, disaggregate representation of PT system, which enables in-depth analysis of the PT overcrowding ) and PT capacity reductions' (Cats and Jenelius 2018) effects on passengers' travel experience and network performance. It has been also applied for analysing the impacts of real-time information (RTI) concerning subsequent vehicle arrivals on travel times (Cats et al. 2011a;Cats and Jenelius 2014) and real-time crowding information (RTCI) on instantaneous route choices (Drabicki et al. 2021b). Furthermore, BusMezzo is a dynamic assignment model, where PT network performance is an outcome of sequential and mutual interactions between supply side (individual vehicle movements) and demand side components (passengers' travel actions). Consequently, it is also applicable for investigating the bus bunching phenomena ( The PT network model in BusMezzo is represented in form of directed, connected graph G(S,E), comprising a set of nodes (stops) S, and a set of links E. The latter consists of line (trip) segments and walking connections (access, egress and transfer links). The line segment e of a PT line L connects its tail stop e − ∈ S with its head stop e + ∈ S .The PT line L is an ordered sequence of vehicle trips (runs) r, r + 1, ... ∈ L serving a sequence of stops L = (s 0 , s 1 , s 2 , …, s n ), or equivalently, operating along a sequence of line segments L = (e 0 , e 1 , e 2 , …, e n-1 ).
The BusMezzo model enforces strict capacity constraints ; Gavriilidou and Cats 2019). Each vehicle r is characterized by its respective seat capacity ƞ r and total (crush) capacity κ r . This allows to account for three principal categories of (over)crowding effects. First, the impact of increasing volume-capacity ratio that implies higher in-vehicle travel discomfort (disutility), especially for standing passengers. Second, passenger volumes K r,e c that exceed the vehicle crush capacity limit κ r are eventually left behind at stop. This triggers a further chain of events and increases the boarding volume of the next incoming departure r + 1. Third, these changes in passenger flows directly affect the PT service regularity due to flow-dependent dwell times. Vehicle dwell times t r,s at stop s are a function of the total number of boarding and alighting passengers, K r,s b and K r,s a , respectively, plus the number of passengers on-board K r,s c of the vehicle run r (TCRP 2013). In this study, we consider the case where boarding and alighting is allowed via multiple bus doors, and more precisely-two set of doors per bus and uniform split of boarding and alighting flows (i.e. 50% of passengers per door). Boarding times are affected by the on-board crowding conditions. Given such settings, the dwell time t r,s is given by the following formula (Eq. 1): In this study, we assume dwell-time increments of τ r b = 2.0 [s/pass.] for boarding flows and τ r a = 1.5 [s/pass.] for alighting flows, respectively. Moreover, an extra boarding time penalty of τ r b,crowd = 2.0 [s/pass.] applies if on-board passenger volume exceeds the seated capacity, i.e. K r,s c > ƞ r (and equal zero otherwise). This is to reflect the impact of bus overcrowding on rising dwell times (and more often on boarding times), which has been noted in the literature (Tirachini et al. 2013;TCRP 2013;Gentile and Noekel 2016).
Total operating time of a trip r is the sum of running times t r,e (which are assumed constant) along line segments e ∈ L and dwell times t r,s (which are flow-dependent) at line stops s ∈ L . Any fluctuations in dwell times between consecutive trips r, r + 1 influence their actual headway h r,e further downstream. This allows for reproducing the classical bus bunching pattern, i.e. the feedback loop between dwell times t r,s , trip headways h r,e and boarding volumes K r,s b that is likely to become self-amplified at next downstream stops. The PT demand model is represented by the OD (origin-destination) passenger matrix, consisting of K o,d -passenger rates (per hour or other time period) generated at constant inflow rates at the origin stop o and travelling to the destination stop d. While progressing through the PT network, each passenger k makes a sequence of travel decisions, choosing one of the possible travel actions a k,s ∈ A at each decision point s. The choice model is based on the utility maximization principle, and the choice probability p k,s is given by the probabilistic MNL formula (Eq. 2), which evaluates the action utility u a,k,s against the complementary utility of remaining choice (action) set A': Taking an action a is associated with travelling along a path i belonging to path set I a , and action utility u a,k,s is calculated as the logsum of path utilities u i,k,s (Eq. 3): Each path is a sequence of stops (or equivalently, line segments) between the current decision point s and the destination stop d, i.e. i = (s, s 1 , s 2 , …, d). The path set I a contains all the possible and relevant travel alternatives between s and d, aside from paths which do not fulfill logical constraints (e.g. getting off and on the same bus again, for details see Cats et al. (2011a)). The path utility u i.k,s is the sum of systematic path utility v i,k,s , and a random error term ε k (Eq. 4). The latter is described by normal distribution (with mean value equal to zero) and aims to reflect utility perception (taste) differences among passengers k ∈ K: The systematic part of utility, v i,k,s is the sum of expected travel utility of path components x, i. The baseline values for perceived disutility coefficients of wait time β e wt and walk time β e wkt are assumed to be equal twice the perceived disutility coefficient rate of uncrowded in-vehicle travel time, i.e. β e ivt = − 1.0 and β e wt = β e wkt = − 2.0, as suggested in the literature Gentile and Noekel 2016). Perceived transfer disutility coefficient β tr is assumed to be equivalent to extra 5 min of the uncrowded in-vehicle travel time, i.e. every transfer imposes additional disutility equal to the β e ivt = − 1.0 multiplied by 5 min. Since all perceived utility coefficients are negative, the resultant path utility is a negative value. ( Crucially, with access to RTCI, the β e x coefficients become a function of the information provisioned, as passengers evaluate trade-off(s) between waiting time vs. total path (dis) utility based on instantaneous crowding information available at a given stop. In the following subsection, we describe the modelling extension of the BusMezzo framework that allows for capturing the effects of RTCI upon passengers' instantaneous boarding choices.

Model extension-willingness to wait with RTCI
We build on the modelling framework presented in Drabicki et al. (2021b) and proof-ofconcept analysis in Drabicki et al. (2021a), extending them to cover the broad spectrum of corridor-specific dynamics of boarding decisions. We model the RTCI effects on boarding choices as part of a dynamic PT assignment model with a three-step framework (Fig. 1): 1. Recording (observing) the crowding information of PT vehicles as they depart from upstream stop(s). 2. Generating the real-time crowding information (RTCI) from instantaneous crowding data and its dissemination at downstream stop(s). 3. Incorporating RTCI into passengers' boarding decisions.
In line with our empirical survey design (Drabicki et al. 2022), we assume that RTCI system conveys information on the two nearest vehicle departures from a given stop, containing their (expected) arrival times and on-board crowding levels. The key variable affecting the boarding choice model with RTCI is β r,s RTCI -the runspecific in-vehicle passenger crowding rate, valid at a given stop s. This value denotes the information on the in-vehicle (over)crowding conditions, plotted on a 1-to-4 scale (Tables 1  and 2). We use this parameter to describe the working principles of the RTCI system across the following stages: 1. Recording the RTCI At each stop-exit instance t r,s , when a vehicle run r departs from an upstream stop s, its crowding level β r,s is recorded. It is computed as a value on an ordinal scale (1-4) according to the mapping procedure described in Tables 1 and 2 below, as a function of: current on-board passenger load K r,s c , vehicle seat capacity ƞ r and total (crush) capacity κ r (Eq. 6): 2. Generating the RTCI At each stop exit instance t r,s , the newly recorded RTCI is used to update the current RTCI β r,j (t) of vehicle run r that is instantaneously disseminated to passengers at all downstream stops j = s + 1, s + 2, … of line L (Eq. 7): 3. Utilising the RTCI At each boarding decision instance t a,k , when the vehicle run r arrives at the downstream stop j, we assume that passenger k acquires the currently generated RTCI on two subsequent vehicle departures β r,j (t) and β r+1,j (t). We adopt the empirical findings from own stated-preference survey to specify the so-called willingness to wait (WTW) threshold, t r,j WTW . For each combination of crowding levels of these two vehicle departures, we determine the average acceptable waiting time t r+1,j wt for next departure r + 1, in exchange for (expected) reduction in on-board (over)crowding levels between β r,j (t) and β r+1,j (t), as displayed by the RTCI system (Eq. 8). The WTW thresholds from the stated-preference study are presented in Table 2 and used in Eq. 8 as follows: Based on these, each passenger evaluates her/his WTW utility coefficient β r,j WTW (t). Analogous to Preston et al. (2017), we obtain this rate as a function of the WTW threshold t r,j WTW and the remaining (in-vehicle) travel time t e ivt between the current stop j and the downstream alighting stop. Analogous to other path utility coefficients (Eq. 5), the WTW coefficient is evaluated in negative values (Eq. 9): The WTW coefficient acts as an additional travel disutility factor due to (expected) overcrowding in deciding whether to board the first run r or skip it and wait for the second run r + 1. When vehicle run r approaches the downstream stop j, this instantaneously triggers the WTW-based boarding decision process among waiting passengers. Each passenger k, if he/she has access to RTCI, chooses between the boarding utility v i.k,j (r) vs. staying utility v i.k,j (r + 1). Here, we set the staying utility, associated with taking the later departure r + 1, as reference value equal to the (negative of) the remaining in-vehicle time t e ivt , plus the required waiting time t r+1,j wt . Hence, the perceived (dis) utility coefficient β r+1,j ivt (t) = β r+1,j wt (t) = -1.0 and staying utility is evaluated according to (Eq. 10): The utility of boarding the first departure r involves no extra waiting time (t r,j wt = 0) and the same amount of remaining in-vehicle time t e ivt as in [Eq. (10)]. Given that β r+1,j (t) = − 1.0 , the WTW utility coefficient β r,j WTW (t) acts then as a travel time multiplier due to higher (over)crowding disutility on-board the first departure r, which are-on the basis of acquired RTCI -expected upon its arrival at stop j. Finally, the boarding utility is calculated as follows (Eq. 11): The formulations in Eqs. 10 and 11 define how boarding and staying utilities are calculated with access to RTCI concerning next vehicle departures from a given stop j. Essentially, depending on the trade-off between extra waiting time t r+1,j wt vs. RTCI-based crowding disutility factor β r,j WTW (t), the staying utility v i,k,j (r + 1) may become higher than boarding utility v i,k,j (r), resulting with a greater propensity towards staying. In the event that passenger k does not access RTCI or has already chosen to skip the previous vehicle, we assume that no WTW is possible anymore and set β r,j WTW (t) = − 1.0. Passengers will then choose to board the first incoming departure r.
Finally, the choice probability p a,k,j of passenger k boarding the nearest vehicle run r at stop j is calculated. Since this involves two choice alternatives-i.e., boarding u a,k,j (r) vs. staying u a,k,j (r + 1)-the choice model in (Eq. 2) reduces down to the binary logit model (Eq. 12): The queuing process in BusMezzo observes the so-called first-come-first-serve (FCFS) principle, i.e. passengers are served in their arrival order at the stop. This implies that passengers who form the residual queue at stop s after the departure of bus r-either due to denied boarding or voluntary (WTW) decision-have the priority in boarding the next bus r + 1 over those who will arrive first at the same stop s after the departure t r,s of bus r.

System-wide indicators of bus-bunching
Finally, to evaluate how RTCI provision and the induced WTW behaviour influences the effects of bus bunching, we investigate service performance using two following metrics.
The first metric relates to the aggregate journey experience in PT network and is denoted as passenger welfare w. This expresses the total experienced travel utility u i,k of all passengers K o,d weighted by the monetary travel time valuation factor z (Eq. 13). Assuming fixed OD demand volumes, an increasing welfare value w can be interpreted as greater (perceived) travel time savings and therefore an improved overall journey experience.
The second metric relates to the regularity (reliability) of PT service operations, which is measured by means of the coefficient of headway variation cv e h , commonly used in bus bunching analysis [e.g. Cats et al. (2011b)]. Coefficient of headway variation cv e h is calculated as the ratio between the standard deviation σ(h r,e ) of service headways observed for runs r, r + 1, ... ∈ L along a specific line segment e, relative to their mean value (Eq. 14). An increasing value of the cv e h parameter will thus imply a greater magnitude of bunching.

Experimental set-up
To analyse the potential RTCI effects in relation to bus bunching problem, we introduce the following simulation setup (Fig. 2), (Table 3). It is modelled based on one of the busiest bus lines in Warsaw. Line no. 523 is a cross-city bus service, spanning between western and eastern parts of the city with 30 stops, served by 12 buses per hour. We model the westbound direction, which is particularly busy and susceptible to bunching during the morning peak period. Supply and demand data is based on the Greater Warsaw strategic transport model (MTAW 2017 (12) p a k,j = u a k,j (r) u a k,j (r) + u a k,j (r + 1) A key driver of demand-supply variations pertains to the flow-dependent dwell time function as described in previous section by (Eq. 1). To induce bunching in our PT network, we introduce an 'initial' service disturbance, whereby alternate bus trips are either dispatched on-time (i.e. dispatching time offset φ s (r,r + 2,…) = 0 [s]) or with an initial delay of φ s (r + 1,r + 3,…) = + 90 [s]. Therefore, even though the nominal service headway is equal to h L,0 = 5 [mins], the actual headway at origin stop will oscillate alternately between h r,e = 3.5 [mins] and h r+1,e = 6.5 [mins]. Consequently, bus service regularity may become substantially disrupted en-route, due to emergence and amplification of bunching effects, which may still persist in downstream network despite decreasing passenger volumes.
Since we run a stochastic PT simulation model, each scenario outputs are reported as an average of 30 simulation replications with a random seed. This number of replications was found to be statistically significant, yielding a maximum allowable error of 3.8%. PT supply is simulated for a period of 3 h (to account for service warm-up and cool-down periods), while PT demand is generated during an intermediate 60-min period, from the 30th to the 90th min of simulation run time.
We analyse and compare scenarios with identical demand and supply and differing in the RTCI availability: the baseline "no RTCI" case (i.e. default passenger behaviour without crowding information access); and the "RTCI" case (i.e. where passengers utilize crowding information in their boarding decisions). Additionally, in Subsection 3.1.4. below we also introduce intermediate scenarios with variable RTCI penetration rates (i.e. 25%, 50% and 75% of passengers obeying the RTCI). Output passenger welfare estimations are based on monetary time valuations projected according to JASPERS (2015) for the 2021 data. Once weighted across three basic trip categories (business, commuting and leisure trips), these yield an average rate of 34.6 [PLN/h], or ca. 7.5 [EUR/h] (assuming the conversion rate of 4.6 [EUR/PLN]).
The main objective of experiments presented below is to analyse the implications of RTCI provision as a potential mitigation measure against bus bunching, as demonstrated on a real-world bus line model with the proposed RTCI simulation algorithm.

Emergence of the WTW behaviour
Differences between scenarios are principally driven by passengers' boarding decisions. The no RTCI scenario employs the default decision making process, where passengers always choose the first arriving bus trip, yet might nevertheless be denied from boarding in the event that capacity limits become binding. In the RTCI scenario, crowding information induces variable willingness-to-wait (WTW) probability at PT network stops (Fig. 3). Initially, share of travellers that opt to wait deliberately for a later departure stays low at upstream stops, until stop no. 110, where the RTCI indicates noticeable on-board load variations between approaching bus trips. Consequently, the WTW pattern holds true in 30% of all boarding decisions at that stop. The share of WTW-based decision peaks then at stops no. 111 and 112 around 60-70%. This high probability results from the corresponding RTCI content which shows high overcrowding of the first bus trip (which has already left stop no. 110), and simultaneously, seats available on-board the second bus trip (which is yet at stops no. 108-109). The WTW share plunges then to ca. 40% at stops no. 113-114 and further down to 20-30% thereafter. Despite similar average passenger load, the share of WTW decisions from the stop no. 113 onwards becomes successively lower. This is attributable to much more uniform (over)crowding conditions being displayed for individual, consecutive bus trips, which in turn reduces the propensity to wait further at the next stops. Once buses exit the high-demand segment, the WTW share reduces to 20% or lower values. In the following subsections, we summarise the main implications of emerging WTW behaviour for demand and supply performance.

Effects on service performance
Figure 4 presents space-time bus trajectories in our case-study network. These reveal an evident bus bunching effect in the no RTCI scenario, as the initial dispatching delay eventually propagates into a considerable service disruption. The delayed bus trip is boarded by a larger number of waiting passenger at downstream stops than expected and thereby becomes even more delayed, while the next bus picks up fewer passengers and departs earlier than expected. This effect becomes amplified as high-demand conditions emerge further along the line. Major deteriorations take place in effective service headways h r,e , which become almost double the rate of nominal values (h L,0 = 5 [min]). Eventually, the faster bus 'catches up' with the delayed (preceding) bus and 'full bunching' effect occurs, observable as bundled bus trajectories in the Fig. 4 (left). From this point onwards, bus service is unable to recover towards regular service pattern and bus trips operate in such staggered combination for the remainder of their trip.
Conversely, the prevalence of WTW behaviour in the RTCI scenario results in a substantially different bus trajectory pattern. As projected in the Fig. 4 (right), this is especially observable from the beginning of high-demand line segment. It is the 'faster', lesscrowded bus, which undergoes a greater increase in dwell time at stop no. 110, while the 'slower' and overcrowded bus departs quicker. Eventually, though full service regularity cannot be restored, a major improvement is reflected in the effective headway h r,e regularity (Fig. 5). Thus, effective headways between buses are much more favourable in contrast to the no RTCI scenario, and no progression towards 'full' bunching takes place. These service regularity patterns are also reflected in the coefficient of variation (CV) of bus headways, plotted along consecutive stops (Fig. 5). In both scenarios, the initial dispatching delay implies a baseline CV rate of ca. 0.3 at the departure from origin stop no. 101. The CV rate increases then gradually, exceeding 0.5 at stop no. 110 where different patterns emerge for both scenarios. Without access to RTCI, high demand conditions cause a substantial increase in CV rate to values of 0.8-1.0 at the next stops-implying a 'fully bunched' bus service. Although passenger demand decreases, service regularity is still substantially impeded along the rest of the line, and bus trips arrive at final stop with an average headway CV of ca. 0.6. Conversely, in the RTCI scenario, risk of service regularity deterioration is effectively curtailed in the downstream network, notably along the highdemand line segment. The headway CV initially drops down by half from ca. 0.55 at stop no. 109 to 0.27 at stop no. 111 and stabilizes from this point onwards. Henceforth, even despite the rising influence of alighting actions upon dwell times at stops no. 118-130, this does not negatively affect the service regularity. Headway CV hovers around 0.3 and gradually decreases further, down to ca. 0.23 at the final stop. Comparison of both plots (Fig. 5) reveals thus that RTCI availability mitigates the risk of progression towards full bunching and suppresses headway CV at the rates of 0.2-0.3 along the majority of the line, including the most overcrowded conditions.

Effects on trip loads' distribution
Figure 6 depicts passenger load distribution at selected stops where WTW choice behaviour becomes evident with the emergence of high network demand conditions. Passenger loads of bus trips at stop no. 109 exhibit an analogous pattern (albeit with certain variations), as RTCI does not influence much boarding choices at the initial stops. Then, in the no RTCI scenario, it is evident how the feedback between growing bus bunching effect and the default 'take-the-first-available-departure' behaviour fosters discrepancies in trip loads' variations. The alternate bus trips departing from stop no. 110 are either filled up to 90-100% of their capacity limit or have plenty of on-board space available (volumecapacity ratio of 50-60%). At the following stop no. 111, results show a relative increase in on-board loads of the 'less-crowded' buses (ca. 70%), which is-however-a consequence Fig. 3 Share of WTW boarding decisions, plotted against the average passenger load, along the central bus corridor segment in the RTCI scenario of denied-boardings from the fully loaded buses. Such load constellation is yet prevalent along the remaining stops, and even with lower demand in downstream network, variations in bus trip loads are still likely to persist until the final stops.
An opposite pattern is demonstrated in the RTCI scenario. Flexibility in boarding choice behaviour results in more equalized load distribution pattern from the stop no. 110 onwards, since the RTCI raises passengers' awareness of less-crowded buses approaching the stop. In contrast to the no RTCI scenario, on-board volumes oscillate in a uniform manner around 70% at every consecutive bus departure. Likewise, with the influx of additional passenger volumes at the stop no. 111, uniformity in vehicle loads' distribution is preserved, and not a single bus exceeds ca. 90% of its capacity limit. The RTCI-induced WTW choice probability encourages equalization in trip loads at further downstream stops. Its effectiveness becomes somewhat lower, though, with rising impact of alighting actions.  . 120 and 130), yet in a much more limited extent compared to the no RTCI scenario. Consequently, without RTCI access, load factors of bus trips bounce alternately between full and moderate utilization, whereas with RTCI availability they stabilise within a constrained range. Table 4 presents output passenger journey experience results, measured by total travel disutility-i.e., perceived journey time (PJT), as well as by specific PJT components. Globally, RTCI access leads to overall PJT reduction of ca. 5%. Firstly, these changes are mostly attributable to a 6% decrease in the in-vehicle time (IVT) disutility, reflecting more positive on-board comfort experience. Secondly, despite the WTW behaviour encouraging further waiting at the stop, overall waiting time (WT) disutility does not deteriorate in relation to the no RTCI scenario. This is attributable to the counteracting effect of reduced waiting times because of improved service regularity. The decrease in total PJT is also intertwined with a 2% lower mean absolute journey time, reflecting that network-wide WTW choices may contribute positively to real-time service performance as well. On average, the PJT changes induced by RTCI are equivalent to welfare benefits of ca. 0.2 EUR (0.9 PLN) per each passenger trip. Once projected in annual terms, total welfare benefits-attainable for this single bus line during a single PM peak hour only-would amount to ca. 223 k EUR (1.0 m PLN).

Effects on journey experience
In particular, a major benefit of RTCI-induced effects in load distribution and service regularity relates to the substantial, 45% decrease in WT disutility due to denied boarding. Their favourable influence is also reflected in a global 2% decline in excess waiting time (EWT). Both these effects vary locally, though. Passengers boarding at the high-demand stops are more likely to experience longer EWT (up to extra 2-3 [mins] on average), yet in exchange-they gain the most from denial-of-boarding risk lower by as much as 70-100% at selected stops. Otherwise, passengers boarding before the high-demand segment (stops no. 107 to 109) may experience an extra EWT of 1-2 [mins] on average, while those entering downstream of stop no. 120 stand to gain the most from lower WT disutility-being (in turn) an outcome of average EWT shorter by 2-4 [mins] and a wholly mitigated deniedboarding risk.
A further inspection of RTCI impact on passengers' on-board comfort experience is illustrated in Fig. 7. In addition to the aforementioned 6% reduction of total IVT disutility, another major benefit is revealed in total IVT spent by passengers in the highest overcrowding conditions (i.e. RTCI level 4), which is globally lower by 40%. This is accompanied by a modest rise of 2% in journey experience corresponding to seated conditions (RTCI level 1 and 2). In addition to demand-side benefits, such findings imply lower share of massively overcrowded bus trips and thus more efficient capacity utilization.
The above findings are also reflected in Fig. 8, depicting changes in passenger volumes subject to different crowding conditions along the line. The upper chart shows a higher number of passengers able to get a seated place, especially right after the high-demand, central line segment (from stops no. 117 and 118 onwards). Volume changes in the RTCI scenario range between extra 20 to 100 seated [pass.], i.e. up to 13% of total demand flows. The lower chart indicates a significant drop in passengers experiencing excessive on-board overcrowding in the RTCI scenario, which is evident in 2 parts of the network. Firstly, a decrease of 80-110 [pass.] is observable along the highest-demand line segment, as passenger loads are spread evenly across the buses thanks to WTW decisions at consecutive stops. Secondly, improved service regularity along downstream line segments essentially mitigates the on-board overcrowding risk, with up to 110 [pass.] able to travel in lesscrowded conditions.
To examine the spatial RTCI impact upon travel experience, in Fig. 9 we plot resultant changes in total travel disutility (i.e. PJT), distinguishing them for passengers boarding at various stops. The plot indicates that a vast majority of bus line users stand to gain from ubiquitous RTCI provision and utilization. Those boarding at initial, low-demand stops experience a 3-7% reduction in average PJT, compared to the no RTCI scenario. Here, the improved on-board (IVT) utility (experienced later en-route) outweighs the contractions in waiting time (WT) utility (endured at these stops). The only passengers who are likely to experience worse PJT in the RTCI scenario-albeit by a mere 3.5%-are those boarding at stops nos. 112 and 113. These stops are located along the peak-demand line segment and rising excess wait time becomes highly relevant for output travel experience at these locations. However, from this point onwards, average PJT savings exhibit a downward trend in the RTCI scenario-from 4% for users boarding at stops no. 115 to even 14-16% for those at stops no. 121-125. Passengers boarding downstream enjoy thus relatively the greatest improvements in journey experience and reductions in both IVT and WT disutility. This is, in turn, a favourable outcome of WTW decisions at upstream stops, as they ultimately prevent evolution of major bunching irregularities along the bus line, which are otherwise very prominent in the no RTCI scenario.

Impact of variable responsiveness to RTCI
The above presented no RTCI and 'full' RTCI scenarios assume that real-time crowding information is utilized in boarding decisions by 0% and 100% of passengers, respectively. More realistically, the RTCI utilization will vary depending on the information accessibility, propensity to observe and use the information etc. Therefore, to examine the RTCI effectiveness with variable penetration rate, we extend the above simulation set with intermediate scenarios. These are three additional scenarios, where 25%, 50% and 75% of (randomly selected) passengers obey the WTW choice algorithm in their boarding decisions, whereas the remainder follow the default 'take-the-first-available-bus' boarding principle. Likewise, outputs for each scenario are extracted from 30 simulation runs with randomized seed. Table 5 reports changes in experienced travel utility for different RTCI penetration rates, evaluated against the no RTCI scenario. It turns out that improvements are already noticeable for the 25% RTCI penetration rate, mainly in terms of waiting time (dis)utility and overall PJT savings equal to 2%. These increase to 4-5% for the RTCI penetration rates of 50% and 75%, when passengers stand to gain the most from reductions in denied-boarding disutility (ca. − 65%), on-board overcrowding (− 50%) excess wait time (− 9%). The overall PJT savings are preserved on the similar level as responsiveness to RTCI approaches the'full' 100% rate. These result then mostly from enhanced on-board (IVT) utility, as ubiquitous response to RTCI leads to limited improvements in waiting time disutility. Nevertheless, these results reveal positive influence of RTCI availability upon travel experience already at low RTCI utilization rates.
On the supply side, Fig. 10 depicts the impact of RTCI penetration rate upon service regularity. For the response rates of 25% and 50%, the equalization of bus passenger loads can be already observed, yet notwithstanding-bus bunching persists along the downstream line sections with significant headway variations. Once the RTCI penetration rate is prevalent, however, a much greater and positive impact upon headway regularity is noticeable. For the 75% RTCI penetration rate, maximum headway variation coefficient oscillates at ca. 0.5-0.6. Finally, with universal (100%) response rate, the evolution of bus bunching is effectively suppressed, and headway variations reflect a stable, downward trend. It is thus observable that higher RTCI penetration rates yield an increasingly advantageous impact upon headway regularity in our case-study PT network.

Discussion
The objective of this study is to investigate the potential application of real-time crowding information (RTCI) for counteracting the bus bunching problem. To this end, we describe the novel RTCI effects upon passengers' travel behavior using a PT simulation model and analyse their consequences for PT network performance. The contributions of this study are twofold. The first contribution pertains to the proposed methodology-i.e., an extended passenger path choice algorithm. Its purpose is to simulate how generating and providing RTCI concerning the two nearest vehicle arrivals induces the so-called willingness-to-wait (WTW) for a later bus (vehicle) departure to Fig. 7 Relative changes in passengers' on-board comfort experience (i.e. total in-vehicle travel time) in comparison with the no RTCI scenario. Distinguished for 4 crowding levels, corresponding to the RTCI categorization scale (Table 1). Total perceived IVT reduces by 7.5% in the RTCI scenario Fig. 8 Resultant changes in passenger volumes able to travel seated (top) and avoid excessive on-board overcrowding (bottom) in the RTCI scenario avoid (reduce) overcrowding among passengers waiting at urban PT stops. Model specification is based on own stated-preference survey findings of the WTW phenomenon among urban PT users (Drabicki et al. 2022). The WTW choice model is implemented within a dynamic agent-based PT assignment model and is thus applicable for analysing the network-wide implications of RTCI provision upon PT demand-supply interactions, travel experience and network performance.
The second contribution relates to its application onto a real-world bus corridor in the Warsaw. Findings reveal how the WTW behaviour incited by RTCI provision might suppress and even reverse the unravelling bunching process. RTCI access has been shown to substantially mitigate service irregularities-as witnessed by headway CV capped at the rates of 0.2 to 0.3-in service conditions which would otherwise spiral into a 'full-fledged bunching' state. The evolution of bus bunching was effectively mitigated both along the central, high-demand line segment, as well as in the downstream network dominated by alighting flows (which cannot be influenced by RTCI). Additionally, RTCI utilization resulted in a much more uniform passenger load distribution among consecutive bus trips, especially at high demand stops. This enabled a more efficient capacity utilization with fewer bus trips being alternately overcrowded or underutilized.  Network-wide availability of RTCI on upstream trip loads implied that even as many as 30-70% of passengers opted to wait further at the stop to board a less-crowded bus. The collective dynamics resultant from all the individual, RTCI-induced choices eventually led to a system-wide reduction in total travel disutility (i.e. perceived journey time) of 5%. This stemmed mostly from improved on-board journey comfort across the bus route, and importantly-despite RTCI encouraging the WTW behavior-total waiting disutility did not deteriorate in our case study. This was attributable to the counteracting effect of globally shorter waiting times associated with higher service regularity. Major benefits were especially reflected in lower prevalence of denied boardings and excessive on-board overcrowding-globally by as much as 40%, and locally these risks were eliminated altogether. Such benefits were also attainable for a major share of passengers in the central high-demand network segments, while those travelling in downstream, less-crowded network were additionally more likely to get a seated place on-board (up to extra 13% of total demand flows). Improvements in travel utility were in general experienced by passengers boarding at a vast majority stops in our case-study bus corridor, reaching on average 2% at central line segments, 6% at upstream locations, and even up to 12-16% in the downstream parts of the network. In monetary terms, passenger welfare benefits induced by RTCI provision along this bus corridor would be equivalent to approx. 0.22 m EUR annually (for the PM peak-hour period only). Furthermore, sensitivity analysis has shown that travel experience benefits are already achievable with a limited response rate to RTCI among passengers, with additional improvements in headway regularity once RTCI utilization becomes increasingly prevalent.
To summarise, our findings demonstrate that providing RTCI on upstream trip loads may stimulate a co-operative travel behaviour in form of WTW to reduce (over)crowding, helpful in mitigating the bus bunching problem. Favourably, ubiquitous response in individual boarding decisions across our case-study network was shown to act in a certain equilibrating manner, bringing network-wide benefits in terms of improved service regularity and journey experience. To the best of our knowledge, ours is the first study that investigates the impacts of RTCI provision on a real-world bus corridor model, in context of bus bunching and based on empirical (stated-preference) insights into WTW probability among urban PT users. Our results demonstrate the potential efficacy of WTW behaviour with RTCI to become a useful travel demand management strategy in Fig. 10 Impact of variable RTCI penetration rate upon coefficient of headway variation, evaluated for peak (intermediate) trips reducing PT disruptions such as bus bunching. Although bunching might not be fully avoidable, simulation results suggest that the behavioural response to RTCI can decelerate and under certain circumstances even halt the negative feedback loop underlying the bunching process. Hence, RTCI offers a soft bottom-up bunching mitigation measure which is consistent with and driven by passengers' aim to maximize travel utility, while simultaneously bringing system-wide performance benefits. Consequently, this can contribute to enhanced PT quality of service and travel experience, either as a sole measure or in conjunction with supply control strategies.
Findings reported in our work shed more light onto the hitherto uninvestigated WTW phenomenon, which might become a common travel behaviour pattern with the future emergence of RTCI solutions in urban PT networks. The development of adequate analytical tools is thus paramount to proper assessment of how advanced passenger information systems (such as the RTCI) can ensure better PT operational efficiency. This is particularly relevant given the increasing importance of passenger overcrowding in urban areas, fostered additionally by the (yet on-going) COVID-19 pandemic crisis (Gkiotsalitis 2021). Hence, RTCI can facilitate and stimulate the better distribution of demand flows across the network (in compliance with specific loading guidelines), possibly also in lower crowding conditions, and help passengers make more informed, crowding-aware travel choices (Adam et al. 2020).
The above mentioned RTCI benefits will be only achievable if information disseminated is deemed reliable and trustworthy for prospective passengers. This underlines the need for ensuring high-quality, accurate crowding information, possibly in conjunction with demand-anticipatory techniques (similarly to car traffic congestion information). The presented model specification can be extended in the future to account for different crowding prediction rules, and to evaluate their accuracy and network consequences.
Future research may investigate-either analytically or by means of simulation-how RTCI affects bus bunching under various conditions, depending on e.g. supply parameters (service headway, system capacity), dwell-time functions (and boarding regimes such as mingling etc.), network and demand configuration etc. Among these, studies may consider situations where several subsequent departures might be of interest, and possibly-include departures from multiple lines as well. The prevalence of advanced passenger information solutions may encourage the tendency to consider multiple travel options simultaneously, particularly in high-frequency urban PT, though travellers' bounded rationality and information acquisition and processing limitations need to be taken into account. Additionally, comparison and combination with other PT control strategies will be crucial to fully understand whether the WTW with RTCI can be synergic or interchangeable with strategies such as holding and expressing. Future work may also examine how pre-trip RTCI availability may affect the WTW and induce greater departure-time adjustments, as in our study we considered localized (stop-level) RTCI access only. Finally, real-world applications will allow for calibrating and validating the details of RTCI generation and its utilization by passengers. conceptualization; methodology; resources; software; validation; writing-original; writing-editing and review; supervision.

Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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The Application of The Arm's Length Buyer Rule in The Context of The Civil Code

: Based on the consideration of protecting the interests of transactions, article 404 of the civil code extends the normal business buyer rule to the chattel mortgage and stipulates the constituent elements of the rule. The business activities cannot exceed the scope of the business license, and the goods are for continuous sale; Actual payment is required, but not limited to monetary form; Ownership is required, but actual possession is not required. As for the excluded application, the continuous sale of production equipment is not included in the excluded application; The existence of control relationship does not necessarily exclude the application; Bona fide elements can be added to the identification of the cover clause. The application of this rule can block the recovery of chattel mortgage, but it is still bound by paragraph 2 of article 406.

Introduction
The establishment of the normal buyer rule is of great significance to protect the interests of market transactions, while its legal effect of blocking the retroactive effect of property rights makes the academic community pay more attention to the rule. The expansion of the scope of application of the normal buyer's rule from floating mortgage to general mortgage of movable property in the Civil Code has led the legislator to make corresponding provisions on the conditions of application of the rule in Article 404 of the Civil Code and Article 56 of the Judicial Interpretation on Security, many details of which still need further discussion. Such as whether the business activities can exceed the scope of business recorded in the business license; whether the acquisition of the subject matter requires actual possession; whether the subjective good faith of the buyer is required; and how to determine the quantity of goods purchased by the general buyer.

Consideration Based on Transaction Interests
According to Article 404 of the Civil Code, a buyer who acquires mortgaged property by paying a reasonable price in the course of normal business activities is subject to a mortgage on movable property. This article extends the scope of application of Article 189(2) of the Property Law to mortgages on movable property, which forms the basis for the rule of arm's-length buyer in the era of the Civil Code. Opponents argue that the establishment of the arm's-length buyer rule would shake the foundation of the mortgage security system because blocking the retroactive effect of a chattel mortgage would undermine the credit of a security right created for financing or other transactions prior to the sale of the chattel [1]. There is some truth to this argument, but the legislator chose the latter over the transactional interests represented by the arm's length buyer rule, making it an exception to the retroactive effect of a chattel mortgage. The interests of the transaction represented by the rule include the following two main points: First, the exemption of consumers from the obligation to inquire about transactions. According to Article 406, paragraph 1, of the Civil Code, movable property encumbered by a mortgage is transferable and the mortgage is not affected by the transfer. In this context, if the arm's-length buyer rule does not apply, a consumer who wishes to acquire a movable asset free of encumbrances must search the security register before the transaction to determine whether the movable asset is encumbered by an encumbrance, thus creating a duty of search for the consumer. The importance of the arm's length buyer rule is that it exempts consumers from the obligation to search prior to the transaction, allowing buyers who meet the conditions for application of the rule to acquire unencumbered title. Second, to maintain transaction security, reduce transaction costs and enhance transaction efficiency. According to the movable mortgage registration system, the movable mortgage can be recorded in the security register against third parties, which means that the consumer has to bear the risk of the mortgage of the goods. If the buyer's goods are encumbered by a chattel mortgage, the mortgagee may receive priority payment for the goods purchased by the buyer according to the retroactive effect of the mortgage, which will seriously affect the security of the transaction. Consumers have to search the security register before purchasing goods, which is obviously not in line with consumer habits and will bring heavy burden to consumers, and if the goods are subject to a mortgage, the mortgagee has to be notified, which will undoubtedly increase transaction costs and reduce transaction efficiency. Consumers are relieved of their obligation to search under the normal business buyer's rule, which reduces transaction costs and improves transaction efficiency; they obtain unencumbered ownership and improve transaction security.
It is important to emphasize that the object of protection of the normal business buyer rule is to a large extent the legitimate interests of consumers [2].But from the legal provisions of the understanding, the normal business buyer is not limited to consumers, all paid a reasonable price and has acquired the ownership of the movable property transactions of natural persons, legal persons or other organizations belong to the subject of the application of the rule. Such as the purchase of goods for profit merchants are also the subject of the application of the rules. The buyer in normal business activities is not limited to consumers, but also includes commercial subjects; the purpose of consumption is not limited to life, but also includes production, although there are differences in the subject but the needs in the interests of the transaction is the same. In the context of the increasing variety of movable property transactions and the increasing frequency of transactions, it is the reasonable choice of the legislator that the buyer of normal business activities acquires the ownership of movable property without the burden of rights by paying a reasonable price.

The Elements of The Application of
The Normal Business Buyer's Rule

The Applicable Circumstances for Normal Business Activities
The relevant provisions of the normal business buyer rule, in addition to Article 404 of the Civil Code, are supplemented by the Judicial Interpretation on Guarantees, which sets forth two requirements with respect to normal business activities: First, the seller's business activities belong to its business license clearly recorded in the scope of business. In line with the premise of the scope of the record is the seller has a business license, which can be reversed to the identity of the seller's operator, where the "operator" can refer to the application of the Civil Code, Article 312 of the good faith acquisition of lost property with business qualifications of the "operator". The requirement of having a business license to the applicable subject eliminates the possibility of applying the rule to natural persons without a business license, such as second-hand transactions between natural persons and natural persons are not applicable to the normal business buyer rule, through the "salted fish" and other second-hand trading platform completed by the individual to individual second-hand movable property transactions, although there is a network platform as The rules are not applicable even though there is a network platform as a medium.
On the judicial interpretation requires the seller's business activities in its business license clearly recorded within the scope, some scholars believe that the provisions are too strict. Because the practice of business behavior and business license business scope does not match the phenomenon of a few, the effectiveness of this legal act is not clearly denied by law; will be strictly limited to business activities in the business license business scope to a certain extent contrary to the rules to improve the efficiency of the original legislative intent. The view has a strong practical significance, engaged in business activities with the business license scope of different business activities in real life is very common, and even more, first arbitrarily registered business scope and then consider the type of sales of goods, if the strict implementation of the "clear" standard, the role of the rule will be greatly reduced.
However, although the law does not directly deny the effectiveness of operating beyond the scope of business license, but also did not recognize the legality of operating beyond the scope. Beyond the scope of business may produce a violation of the administrative license business behavior, or even illegal business behavior. For the violation of the administrative license, legal norms of business behavior to provide blocking the legal protection of the retroactive effect of the mortgage is obviously unreasonable. The rule requires that business activities must be within the scope of the business license also has the role of excluding the application of the rule for violations, illegal business behavior. In addition, in the judicial trial in strict compliance with the "clear" provisions have a strong convenience, usually against the business license is to identify the seller's business activities are normal the simplest way [3]. In judicial practice, based on whether the operator of the goods sold in the scope of the business license to determine whether the seller is engaged in business activities are normal, simple and clear with strong operability.
Second, the continued sale of similar goods. The goods should be in the scope of the business license records and for movable property. In judicial practice can refer to the "inventory" standard, to determine whether the seller of goods sold in line with the continuous sale of similar goods elements.
The seller of goods sold should have continuity, that is, in a period of continuous sales of certain types of goods, such as food sales sellers, should be in a period of continuous sales of food commodities, if there is an interval, or at a point in time interrupted the sale of food commodities may affect the "continuous" determination, as to how long the interval, how to determine the point in time. How long the interval, how to determine the time point, the judicial interpretation does not provide, which also leaves room for the discretion of the judge. In addition, the way in the transaction of goods, should be in line with their own or the industry's customary practices [4].

The Buyer Has Paid A Reasonable Price
The reasonable price paid by the buyer is the main means to balance the interests of the mortgagee, the buyer and the seller in the context of the rule. If the buyer and the seller have entered into a contract but the buyer has not paid a reasonable price, the normal business buyer rule cannot be applied. The judicial interpretation does not provide for "reasonable", and it is generally believed that the interpretation of reasonable should include: First, it can be interpreted in the same way as the "reasonable price" in Article 312 of the Civil Code, that is, the principle of equal price should be applied, and the price paid should be approximately equal to the price of the goods. Whether the price is reasonable, some scholars believe that it can be measured based on the normal market price, the parties' agreement, the delivery method, whether the movable property is unique and the transaction habits and other factors [5]. In addition, reference can be made to the relevant provisions that the reasonably fluctuating price of the transfer does not exceed thirty percent of the guide price of the place of transaction or the market transaction price, and not less than seventy percent.
Secondly, the price paid is not limited to the form of money, but other alternatives to money are also possible, and it is worth discussing whether in-kind is also included. It should be considered that the payment in kind can also be a way to pay the price, whether it is in money or in kind, it has property properties for the seller, and the bartering method is not much different from the payment in money.
Third, the payment of reasonable price does not require the payment of the full price, the buyer if the buyer has paid most of the price can also apply the normal business buyer rule, it is worth discussing whether the deposit has been paid can constitute the payment of reasonable price, the author believes that only the payment of the deposit does not meet the requirements of the payment of reasonable price. On the one hand, the deposit usually accounts for only a relatively small part of the reasonable price to be paid, which does not achieve the role of balancing interests, and although the deposit can prove the possibility of full payment by the buyer, it cannot guarantee that the buyer will definitely pay the remaining price in full; on the other hand, for the purpose of preventing malicious collusion between the seller and the buyer to the detriment of the mortgagee, if the deposit is paid, it is in line with the requirement of paying a reasonable price, which means that On the other hand, for the purpose of preventing malicious collusion between the seller and the buyer to the detriment of the mortgagee, if the payment of deposit meets the requirement of paying a reasonable price, it means that the buyer only needs to pay a small part of the actual price to acquire the ownership of the movable property free of the encumbrance.
Fourth, it is required that the price has actually been paid. If the defaulted payment has obtained possession of the movable property but still cannot apply the normal business buyer rule, for the sake of balancing interests and to prevent malicious collusion between the seller and the buyer, the price must have been actually paid, such as arrears, playing notes and other acts do not constitute actual payment.

The Subject Matter Is Movable Property and The Buyer to Obtain
The normal business buyer rule applies to all movable property, but does not include real estate, property rights. That is, the seller is required to sell goods should be movable property, housing, construction land use rights, sea use rights, etc. are not applicable to the normal business buyer rules.
The movable property sold should be acquired by the buyer, the judicial interpretation does not explain the "acquisition", but it is generally believed that the "acquisition" refers to the acquisition of ownership of movable property [6], that is, the buyer has completed the transfer of ownership based on the transfer of ownership rules, has acquired from the seller The buyer has acquired ownership of the movable property sold. If the buyer has not yet acquired the ownership of the movable property sold, that is, the ownership of the movable property has not been transferred to the seller, then the issue of the retroactive effect of the blocking mortgage will not arise, only when the buyer has acquired the ownership of the movable property sold, then the issue of the retroactive effect of the blocking mortgage attached to the movable property sold will arise. It can be seen that the acquisition should refer to the acquisition of the ownership of the movable property, and the mere possession of the movable property is not included in the semantics.
What needs to be discussed is whether direct possession is also required of a buyer who has acquired title to the movable property. The general view of the academic circle is that only the buyer in ordinary course of business is required to acquire the ownership of the movable property, and there is no requirement on whether to take possession of the movable property. On the one hand, when the buyer in normal operation has acquired the ownership of the movable property, the retroactive effect of the mortgage will be blocked, and it is no longer important whether to take possession or not. On the other hand, the requirement of taking possession of the movable property while acquiring ownership is too strict for the buyer in normal operation, and is also a restriction on the mode of delivery, when the seller and the buyer take possession of the movable property by way of redetermination and proposed delivery, and do not take direct possession of the movable property. Requiring the buyer to take direct possession of the movable property implies a restriction on the use of delivery methods such as possession modification and proposed delivery, which is obviously unreasonable.

Exclude the Application of Circumstances
Article 56 of the Judicial Interpretation on Warranties provides five circumstances that preclude the application of the arm's-length buyer rule.
First, the quantity of goods purchased significantly exceeds that of the general buyer. Here, the number of goods purchased by the general buyer is used as the judgment standard, and the number of goods purchased by them can be judged based on multiple factors such as industry norms, market environment, geographical characteristics, and seasonality. The judgment of the general buyer also differs depending on the identity of the buyer, for example, the quantity of goods purchased by enterprises and institutions is generally more than the quantity of goods purchased by ordinary consumers. The judgment of the number of goods purchased by the general buyer is complex, difficult to abstractly generalize, and difficult to have uniform regulations. In practice, the parties may combine a variety of factors mentioned above to prove the evidence.
Second, the purchase of the seller's production equipment. The production equipment in this subparagraph refers to equipment used by the seller to produce goods rather than equipment sold as goods. If the seller sells the production equipment as merchandise and continues to sell it, then this subparagraph cannot be used to exclude the application of the arm's length buyer rule. Article 404 of the Civil Code alone already excludes the application of the purchase of the seller's production equipment because the sale of production equipment by the seller is no longer a normal business activity. First, the scope of the business license would not normally include the seller's production equipment; second, the production equipment could not be sold continuously. Judicial interpretation will be listed in this case, one is because this situation in practice is very common, the second is generally higher production equipment security amount, the mortgagee's interests are more damaging.
Third, the purpose of the contract is to guarantee the performance of the debt by the seller or a third party. In this case, although it has the appearance of a sales contract, its essence should be a security contract, which is inconsistent with the purpose of the normal operating buyer's rules to protect the interests of the transaction, and should be excluded from the protection of the normal operating buyer's rules. Contracts entered into for the purpose of securing the performance of a debt shall be subject to the relevant provisions on the subordination of the validity of a mortgage by reference to Article 414 of the Civil Code. It is difficult to judge the true purpose of a sales contract, and in practice, it needs to be considered based on the content of the contract, the affiliation between the seller and the buyer, and other factors.
Fourth, the buyer has a direct or indirect control relationship with the seller. Control relationship refers to the dominant and decisive influence on personnel, financial, business and other important management matters of other enterprises. In most cases, the control relationship is reflected in the shareholding ratio, but judging whether there is a control relationship based on the shareholding ratio is only one of the ways, and because different companies have different equity settings, it is difficult to reach a fixed value of the shareholding ratio of the control relationship, in practice, should combined with the shareholding ratio, senior management, director seats, the existence of control agreements and other factors to determine the existence of the control relationship [7]. Practice should also pay attention to the difference between affiliated relationship and control relationship, affiliated enterprises do not mean that there is a control relationship. When the buyer and the seller have a controlling relationship, the two sides have the possibility and convenience of colluding in bad faith to damage the mortgage, but the controlling relationship between the buyer and the seller does not necessarily produce the effect of damaging the mortgage, if the buyer has paid a reasonable price and the transaction has the rightfulness, for the purpose of protecting the interests of the transaction, the normal operation of the buyer rule should not be excluded.
Fifth, the buyer should search the mortgage registration but did not search other circumstances. As the bottom clause of the exclusion of the normal business buyer rule, the provision leaves room for the legislator to supplement the exclusion of the circumstances, so as to facilitate the exclusion of the circumstances found in practice, to supplement the provisions. However, this provision also leaves a hidden danger for the judicial practice of the trial judge to exclude the application of the normal business buyer rule. Some scholars suggest that the standard of the buyer's good faith can be used as the standard, that is, the buyer knows that its purchase will infringe the mortgage right, the normal operation of the buyer rule can be excluded from the underwriting clause. [8] This view deserves recognition. The good faith criterion can be one of the considerations that should be inquired but not inquired, but the subjective good faith element imposed on the buyer can only be one of the reference factors for the application of the tout clause, because Article 404 of the Civil Code does not require the subjective good faith of the normal business buyer, and if subjective good faith is the general principle, it is easy to make people think that the buyer does not satisfy the subjective good faith to exclude the application of the rule. The bona fide element should not be the only factor to be considered for the underwriting clause, and the specific judgment should be made by considering the circumstances of the case and measuring the degree of damage to the mortgage.

The Legal Effect of The Normal
Operation Buyer's Rule

Exceptions constituting Article 403 and Article 406, paragraph 1 of the Civil Code
The important legal effect of the arm's length buyer is to create an exception for the arm's length buyer in the effect of the mortgage on movable property [9], so that the relevant effect of the mortgage does not affect the interests of the arm's length buyer and does not harm the interests of the arm's length buyer. The exception, in particular, is: First, the exception of the registration of the adversarialism of movable mortgage. Article 403 of the Civil Code provides for the adversarial system of registration of movable mortgage, and the movable mortgage can be registered against third parties. Although the Civil Code does not indicate which of Article 404 and Article 403 should take precedence, based on the legislative purpose, it is not difficult to infer that the normal operating buyer is not among the third parties against whom the registered movable mortgage can be registered, i.e., the movable mortgage can be registered against third parties in the security register, but according to the normal operating buyer rule, the "third parties " does not include the normal business buyer. Here, it should be clear that Article 404 of the Civil Code does not require the existence of subjective good faith of the buyer, and when the rule is applied, regardless of whether the chattel mortgage is registered or not [10], and regardless of whether the normal operating buyer is in good faith or not, any buyer who meets the requirements of the normal operating buyer rule can counter the retroactive effect of the chattel mortgage.
Second, the exception of "agreement of the parties". According to Article 406(1) of the Civil Code, the parties to a mortgage contract may agree on the transfer of the mortgaged property, such as agreeing that the mortgaged property is not transferable, or agreeing on the way to restrict the transfer of the mortgaged property. However, this agreement is only binding between the mortgagor and the mortgagee and does not have effect on third parties. The "third party" includes the normal business buyer, that is, the agreement of the parties on the transfer of the mortgaged property is not effective on the normal business buyer, and the normal business buyer is not bound by the retroactive effect of the mortgage, and the mortgaged property shall be movable property. If the parties to the security contract agree that the mortgaged property is not transferable and the mortgagor still transfers the mortgaged property, the mortgagee may pursue the mortgagor's liability for breach of contract according to the contract [11], but may not deny the effectiveness of the transfer contract according to the agreement, and the mortgagee may not exercise the retroactive effect of the mortgage when the object of the transfer is the buyer in normal operation. The mortgagee may not exercise the retroactive effect of the mortgage right, and may not discount, auction or sell the mortgaged property for priority payment.

The Mortgagee May Claim Subrogation in Rem of The Price
Article 404 of the Civil Code shall not be opposed to Article 406, paragraph 2, which states that the mortgagee shall not be entitled to priority in payment for the mortgaged property transferred by the arm's-length buyer, but may request the mortgagee to claim early settlement of the debt for the price paid by the buyer. The reasons are as follows: Firstly, from the perspective of the application of the law, the rule of the arm's length buyer can be regarded as "the transfer of the mortgaged property may impair the mortgage right", which meets the requirements of paragraph 406 (2). Secondly, although the Civil Code does not specify the order of application of Article 404 and Article 406, paragraph 2, nor does it use the proviso, from the perspective of protecting the legitimate rights and interests of the mortgagee and balancing the interests of the buyer, the mortgagee and the mortgagee, it should be considered that Article 404 cannot be opposed to Article 406, paragraph 2. Thirdly, when the mortgaged property is transformed into the price, i.e., it loses its specificity, the realization of the mortgage right loses its security. If the mortgagee is not granted the right to be paid in advance for the price paid by the buyer, the interests of the mortgagee will be extremely vulnerable to damage. Paragraph 2 of Article 406 involves the application of the normal operating buyer and the following issues should be noted: First, the burden of proof that the transfer of the mortgaged property may cause damage to the mortgage shall be borne by the mortgagee, but the burden of proof shall be lighter, when the retroactive effect of the mortgage is blocked by the normal operating buyer rule and the buyer acquires full ownership, the mortgagee shall be deemed to have completed the proof. Secondly, the paragraph does not presuppose that the mortgagee is notified in time, and if the mortgagee learns of the fact of transfer through other means, he may also request early liquidation and purification. The reason is that the paragraph requires the mortgagee to fulfill the obligation of notice in order to facilitate the mortgagee's timely claim for priority payment of the price paid by the buyer, so as to reduce the possibility of damage to the mortgage, rather than creating an obligation for the mortgagee. If the mortgagee learns of the timely request for early settlement through other means, it should be supported. Thirdly, the amount of the request shall be based on the mortgagee's claim, and if the price paid by the buyer is not enough to satisfy the debt, the mortgagee shall make up the amount, and the excess shall belong to the mortgagee.
The claim of subrogation in rem is the main means for the mortgagee to realize the satisfaction of the claim after the retroactive effect of the mortgage is blocked, and it is also an effective means to protect the interests of the mortgagee after the application of the normal operation buyer's rule. For the sake of protecting the interests of the mortgagee, the request of the mortgagee for subrogation in rem should be supported after the retroactive effect of the mortgage is blocked, so that the reasonable price paid by the buyer can actually satisfy the debt, so as to achieve the purpose of balancing the interests of the buyer, the mortgagee and the mortgagee in practice.