Dynamic bus holding strategies for schedule reliability: Optimal linear control and performance analysis

As is well known, bus systems are naturally unstable. Without control, buses on a single line tend to bunch, reducing their punctuality in meeting a schedule. Although conventional schedule-based strategies that hold buses at control points can alleviate this problem these methods require too much slack, which slows buses. This delays on-board passengers and increases operating costs.It is shown that dynamic holding strategies based on headways alone cannot help buses adhere to a schedule. Therefore, a family of dynamic holding strategies that use bus arrival deviations from a virtual schedule at the control points is proposed. The virtual schedule is introduced whether the system is run with a published schedule or not. It is shown that with this approach, buses can both closely adhere to a published schedule and maintain regular headways without too much slack.A one-parameter version of the method can be optimized in closed form. This simple method is shown to be near-optimal. To put it in practice, the only data needed in real time are the arrival times of the current bus and the preceding bus at the control point relative to the virtual schedule. The simple method was found to require about 40% less slack than the conventional schedule-based method. When used only to regulate headways it outperforms headway-based methods.     

Real-time control of buses for schedule coordination at a terminal

Recently, bus transit operators have begun to adopt technologies that enable bus locations to be tracked from a central location in real-time. Combined with other technologies, such as automated passenger counting and wireless communication, it is now feasible for these operators to execute a variety of real-time strategies for coordinating the movement of buses along their routes. This paper compares control strategies that depend on technologies for communication, tracking and passenger counting, to those that depend solely on local information (e.g., time that a bus arrived at a stop, and whether other connecting buses have also arrived). We also develop methods to forecast bus arrival times, which are most accurate for lines with long headways, as is usually the case in timed transfer systems. These methods are tested in simulations, which demonstrate that technology is most advantageous when the schedule slack is close to zero, when the headway is large, and when there are many connecting buses.     

Reducing bunching with bus-to-bus cooperation

Schedule-based or headway-based control schemes to reduce bus bunching are not resilient because they cannot prevent buses from losing ground to the buses they follow when disruptions increase the gaps separating them beyond a critical value. (Following buses are then overwhelmed with passengers and cannot process their work quick enough to catch up.) This critical gap problem can be avoided, however, if buses at the leading end of such gaps are given information to cooperate with the ones behind by slowing down.This paper builds on this idea. It proposes an adaptive control scheme that adjusts a bus cruising speed in real-time based on both, its front and rear spacings much as if successive bus pairs were connected by springs. The scheme is shown to yield regular headways with faster bus travel than existing control methods. Its simple and decentralized logic automatically compensates for traffic disruptions and inaccurate bus driver actions. Its hardware and data requirements are minimal.     

Comparing static and dynamic threshold based control strategies

This research extends a static threshold based control strategy used to control headway variation to a dynamic threshold based control strategy. In the static strategy, buses are controlled by setting a threshold value that holds buses at a control point for a certain amount of time before allowing the bus to continue along the route. The threshold remains constant each time the bus stops at the control point. The dynamic strategy involves the same principle of holding buses at a bus stop; however, a different threshold value is chosen each time the bus holds at a control point. The results indicate that in cases where the static threshold is set equal to the scheduled headway, very low headway variation and passenger system times result; however, passengers on board the bus are penalized by extra delay on the bus while waiting at the control point. The dynamic strategy reduces the penalty to passengers delayed on-board the bus at a control point at the expense of a slight increase in overall passenger system time.The results indicate that in most cases, the tradeoff of the slight increase in waiting time for the significant decrease in on-board delay penalty makes the dynamic strategy an acceptable choice.     

What is behind fare evasion in urban bus systems? An econometric approach

Fare evasion is a problem in many public transport systems around the world and policies to reduce it are generally aimed at improving control and increasing fines. We use an econometric approach to attempt explaining the high levels of evasion in Santiago, Chile, and guide public policy formulation to reduce this problem. In particular, a negative binomial count regression model allowed us to find that fare evasion rates on buses increase as: (i) more people board (or alight) at a given bus door, (ii) more passengers board by a rear door, (iii) buses have higher occupancy levels (and more doors) and (iv) passengers experience longer headways. By controlling these variables (ceteris paribus), results indicate that evasion is greater during the afternoon and evening, but it is not clear that it is higher during peak hours. Regarding socioeconomic variables, we found that fare evasion at bus stops located in higher income areas (municipalities) is significantly lower than in more deprived areas. Finally, based on our results we identified five main methods to address evasion as alternatives to more dedicated fine enforcement or increased inspection; (i) increasing the bus fleet, (ii) improving the bus headway regularity, (iii) implementing off-board payment stations, (iv) changing the payment system on board and (v) changing the bus design (number of doors or capacity). Our model provides a powerful tool to predict the reduction of fare evasion due to the implementation of some of these five operational strategies, and can be applied to other bus public transport systems.     

Development of a probabilistic model to optimize disseminated real‐time bus arrival information for pre‐trip passengers

In the advent of Advanced Traveler Information Systems (ATIS), the total wait time of passengers for buses may be reduced by disseminating real‐time bus arrival times for the next or series of buses to pre‐trip passengers through various media (e.g., internet, mobile phones, and personal digital assistants). A probabilistic model is desirable and developed in this study, while realistic distributions of bus and passenger arrivals are considered. The disseminated bus arrival time is optimized by minimizing the total wait time incurred by pre‐trip passengers, and its impact to the total wait time under both late and early bus arrival conditions is studied. Relations between the optimal disseminated bus arrival time and major model parameters, such as the mean and standard deviation of arrival times for buses and pre‐trip passengers, are investigated. Analytical results are presented based on Normal and Lognormal distributions of bus arrivals and Gumbel distribution of pre‐trip passenger arrivals at a designated stop. The developed methodology can be practically applied to any arrival distributions of buses and passengers.     

A model of bus bunching under reliability-based passenger arrival patterns

If bus service departure times are not completely unknown to the passengers, non-uniform passenger arrival patterns can be expected. We propose that passengers decide their arrival time at stops based on a continuous logit model that considers the risk of missing services. Expected passenger waiting times are derived in a bus system that allows also for overtaking between bus services. We then propose an algorithm to derive the dwell time of subsequent buses serving a stop in order to illustrate when bus bunching might occur. We show that non-uniform arrival patterns can significantly influence the bus bunching process. With case studies we find that, even without exogenous delay, bunching can arise when the boarding rate is insufficient given the level of overall demand. Further, in case of exogenous delay, non-uniform arrivals can either worsen or improve the bunching conditions, depending on the level of delay. We conclude that therefore such effects should be considered when service control measures are discussed.     

Bus control strategies in corridors with signalized intersections

This paper proposes a new dynamic bus control strategy aimed at reducing the negative effects of time-headway variations on route performance, based on real-time bus tracking data at stops. In routes with high demand, any delay of a single vehicle ends up causing an unstable motion of buses and producing the bus bunching phenomena. This strategy controls the cruising speed of buses and considers the extension of the green phase of traffic lights at intersections, when a bus is significantly delayed. The performance of this strategy will be compared to the current static operation technique based on the provision of slack times at holding points. An operational model is presented in order to estimate the effects of each controlling strategy, taking into account the vehicle capacity constraint. Control strategies are assessed in terms of passenger total travel time, operating cost as well as on the coefficient of headway variation. The effects of controlling strategies are tested in an idealized bus route under different operational settings and in the bus route of highest demand in Barcelona by simulation. The results show that the proposed dynamic controlling strategy reduces total system cost (user and agency) by 15–40% as well as the coefficient of headway variation 53–78% regarding the uncontrolled case, providing a bus performance similar to the expected when time disturbance is not presented.     

Analysis of real-time control strategies in a corridor with multiple bus services

Control strategies have been widely used in the literature to counteract the effects of bus bunching in passenger‘s waiting times and its variability. These strategies have only been studied for the case of a single bus line in a corridor. However, in many real cases this assumption does not hold. Indeed, there are many transit corridors with multiple bus lines interacting, and this interaction affects the efficiency of the implemented control mechanism.This work develops an optimization model capable of executing a control scheme based on holding strategy for a corridor with multiple bus lines.We analyzed the benefits in the level of service of the public transport system when considering a central operator who wants to maximize the level of service for users of all the bus lines, versus scenarios where each bus line operates independently. A simulation was carried out considering two medium frequency bus lines that serve a set of stops and where these two bus lines coexist in a given subset of stops. In the simulation we compared the existence of a central operator, using the optimization model we developed, against the independent operation of each line.In the simulations the central operator showed a greater reduction in the overall waiting time of the passengers of 55% compared to a no control scenario. It also provided a balanced load of the buses along the corridor, and a lower variability of the bus headways in the subset of stops where the lines coexist, thus obtaining better reliability for all types of passengers present in the public transport system.     

A stochastic optimization model for transit network timetable design to mitigate the randomness of traveling time by adding slack time

Transit network timetabling aims at determining the departure time of each trip of all lines in order to facilitate passengers transferring either to or from a bus. In this paper, we consider a bus timetabling problem with stochastic travel times (BTP-STT). Slack time is added into timetable to mitigate the randomness in bus travel times. We then develop a stochastic integer programming model for the BTP-STT to minimize the total waiting time cost for three types of passengers (i.e., transferring passengers, boarding passengers and through passengers). The mathematical properties of the model are characterized. Due to its computational complexity, a genetic algorithm with local search (GALS) is designed to solve our proposed model (OPM). The numerical results based on a small bus network show that the timetable obtained from OPM reduces the total waiting time cost by an average of 9.5%, when it is tested in different scenarios. OPM is relatively effective if the ratio of the number of through passengers to the number of transferring passengers is not larger than a threshold (e.g., 10 in our case). In addition, we test different scale instances randomly generated in a practical setting to further verify the effectiveness of OPM and GALS. We also find that adding slack time into timetable greatly benefits transferring passengers by reducing the rate of transferring failure.     

Bus frequency determination using passenger count data

The importance of ridership information has led transit properties to increase the amount of manually collected data or alternatively to introduce automated surveillance techniques. Naturally, the bus operators are expected to gain useful information for operations planning by obtaining more accurate passenger counts. This paper describes and analyzes several appropriate data collection approaches for the bus operator in order to set the bus frequencies/headways efficiently. Four different methods are presented to derive the bus frequency: two are based on point check (maximum load) data and two propose the use of ride check (load profile) data. A ride check provides more complete information than a point check, but at a greater cost, and there is a question as to whether the additional information gained justifies the expense. Based on available old profiles, the four methods provide the bus scheduler with adequate guidance in selecting the type of data collection procedure. In addition, the scheduler can evaluate the minimum expected bus runs when the load standard is released and avoid overcrowding (in an average sense) at the same time. Alternative timetables are also investigated in conjunction with minimizing the required bus runs and number of buses for a single route. In this way, the derived headways can be analyzed within an acceptable range while considering the possible changes incurred indirectly to the fleet size. The integration between resource saving and frequency determination procedures allows the scheduler's performance to be improved.     

Mitigation of disruptions in public transit by Bee Colony Optimization

Dispatchers in many public transit companies face the daily problem of assigning available buses to bus routes under conditions of bus shortages. In addition to this, weather conditions, crew absenteeism, traffic accidents, traffic congestion and other factors lead to disturbances of the planned schedule. We propose the Bee Colony Optimization (BCO) algorithm for mitigation of bus schedule disturbances. The developed model takes care of interests of the transit operator and passengers. The model reassigns available buses to bus routes and, if it is allowed, the model simultaneously changes the transportation network topology (it shortens some of the planned bus routes) and reassigns available buses to a new set of bus routes. The model is tested on the network of Rivera (Uruguay). Results obtained show that the proposed algorithm can significantly mitigate disruptions.     

Multi-objective re-synchronizing of bus timetable: Model,complexity and solution

This work is originally motived by the re-planning of a bus network timetable. The existing timetable with even headways for the network is generated using line by line timetabling approach without considering the interactions between lines. Decision-makers (i.e., schedulers) intend to synchronize vehicle timetable of lines at transfer nodes to facilitate passenger transfers while being concerned with the impacts of re-designed timetable on the regularity of existing timetable and the accustomed trip plans of passengers. Regarding this situation, we investigate a multi-objective re-synchronizing of bus timetable (MSBT) problem, which is characterized by headway-sensitive passenger demand, uneven headways, service regularity, flexible synchronization and involvement of existing bus timetable. A multi-objective optimization model for the MSBT is proposed to make a trade-off between the total number of passengers benefited by smooth transfers and the maximal deviation from the departure times of the existing timetable. By clarifying the mathematical properties and solution space of the model, we prove that the MSBT problem is NP-hard, and its Pareto-optimal front is non-convex. Therefore, we design a non-dominated sorting genetic (NSGA-II) based algorithm to solve this problem. Numerical experiments show that the designed algorithm, compared with enumeration method, can generate high-quality Pareto solutions within reasonable times. We also find that the timetable allowing larger flexibility of headways can obtain more and better Pareto-optimal solutions, which can provide decision-makers more choice.     

Plan-based flexible bus bridging operation strategy

Bus bridging has been widely used to connect stations affected by metro disruptions such that stranded passengers could resume their journeys. Previous studies generally assumed that a bus operates exclusively on one bridging route with given frequency, which limits the service flexibility and reduce the operational efficiency. We propose a strategy to instruct buses to operate on predefined bridging routes once they are dispatched from depots. Buses are allowed to flexibly serve different bridging routes. Each bus operates based on a bridging plan that lists the stations to serve in sequence instead of route frequencies. A two-stage model is developed to optimize the bridging plans and their assignments to buses with the objectives that balance the operational priorities between minimizing bus bridging time and reducing passenger delay. A Weight Shortest Processing Time first (WSPT) rule based heuristic algorithm is developed to solve the proposed model. The developed model is further incorporated in a rolling horizon framework to handle dynamic passenger arrivals during the disruption period. The effectiveness of the proposed strategy is demonstrated in comparison with alternative strategies in real-world case studies.     

Integrated optimization of bus priority operations in connected vehicle environment

Most previous works associated with transit signal priority merely focus on the optimization of signal timings, ignoring both bus speed and dwell time at bus stops. This paper presents a novel approach to optimize the holding time at bus stops, signal timings, and bus speed to provide priority to buses at isolated intersections. The objective of the proposed model is to minimize the weighted average vehicle delays of the intersection, which includes both bus delay and impact on nearby intersection traffic, ensuring that buses clear these intersections without being stopped by a red light. A set of formulations are developed to explicitly capture the interaction between bus speed, bus holding time, and transit priority signal timings. Experimental analysis is used to show that the proposed model has minimal negative impacts on general traffic and outperforms the no priority, signal priority only, and signal priority with holding control strategies (no bus speed adjustment) in terms of reducing average bus delays and stops. A sensitivity analysis further demonstrates the potential of the proposed approach to be applied to bus priority control systems in real‐time under different traffic demands, bus stop locations, and maximum speed limits. Copyright © 2016 John Wiley & Sons, Ltd.     

Isolating high-priority metro and feeder bus transfers using smart card data

Fixed-rail metro (or ‘subway’) infrastructure is generally unable to provide access to all parts of the city grid. Consequently, feeder bus lines are an integral component of urban mass transit systems. While passengers prefer a seamless transfer between these two distinct transportation services, each service’s operations are subject to a different set of factors that contribute to metro-bus transfer delay. Previous attempts to understand transfer delay were limited by the availability of tools to measure the time and cost associated with passengers’ transfer experience. This paper uses data from smart card systems, an emerging technology that automatically collects passenger trip data, to understand transfer delay. The primary objective of this study is to use smart card data to derive a reproducible methodology that isolates high priority transfer points between the metro system and its feeder-bus systems. The paper outlines a methodology to identify transfer transactions in the smart card dataset, estimate bus headways without the aid of geographic location information, estimate three components of the total transfer time (walking time, waiting time, and delay time), and isolate high-priority transfer pairs. The paper uses smart card data from Nanjing, China as a case study. The results isolate eight high priority metro-bus transfer pairs in the Nanjing metro system and finally, offers several targeted measures to improve transfer efficiency.     

Optimal slack time for timed transfers at a transit terminal

At transit terminals where two routes interchange passengers, total system costs may be reduced by allowing some “slack” time in the vehicle schedules to decrease the probability of missed connections. Transfer cost functions are formulated and used to determine optimal slack time for simple systems with transfers between one bus route and one rail line. Some analytic results are derived for empirical discrete and Gumbel distributions of bus arrival times. Relations between the optimal slack times and headways, transfer volumes, passenger time values, bus operating costs, and standard deviations of bus and train arrivals are also developed numerically using normally distributed arrivals. However, the proposed numerical approach can optimize slack times for any observed arrival distributions. The results provide some guidelines on desirable slack times and show that schedule coordination between the two routes is not worth attempting when standard deviations of arrivals exceed certain levels. Possible extensions of this work are suggested in the last section.     

Optimizing bus-size and headway in transit networks

Optimization models for calculating the best size for passenger carrying vehicles in urban areas were popular during the 1980s. These studies were abandoned in the ‘90s concluding that it was more efficient to use smaller buses at higher frequencies. This article returns to this controversial question, starting from the point of view that any calculation of bus size can only be made after considering the demand for each of the routes on the system. Therefore, an optimization model for sizing the buses and setting frequencies on each route in the system is proposed in accordance with the premises detailed below. The proposed model is a bi-level optimization model with constraints on bus capacity. The model allows buses of different sizes to be assigned to public transport routes optimizing the headways on each route in accordance with observed levels of demand. At the upper level the model considers the optimization of the system’s social and operating costs, these are understood to be the sum of the user’s and operator’s costs. At the lower level there is an assignment model for public transport with constraints on vehicle capacity which balances the flows for bus sizes and headways at each iteration. By graphically representing the results of the model applied to a real case, a series of useful conclusions are reached for the management and planning of a fleet of public transport vehicles.     

Investigating the irregularity of bus routes: highlighting how underlying assumptions of bus models impact the regularity results

A bus route is inherently unstable: when the system is uncontrolled, buses fail to maintain their time‐headways and tend to bunch. Several mathematical bus motion models were proposed to reproduce the bus behavior and assess management strategies. However, no work has established how the choice of a model impacts the irregularity of modeled bus systems, that is, the non‐respect of scheduled headways. Because of this gap, a large body of existing works assumes that the ability of these models to reproduce instability comes only from stochasticity, although the link between stochastic inputs and the level of irregularity remains unknown. Moreover, some recognized phenomena such as a change of travel conditions during a day or delays at signalized intersections are ignored. To address these shortcomings, this paper provides an overview of existing dynamic bus‐focused models and proposes a simple way to classify them. Commonly used deterministic and stochastic models are compared, which allows quantifying the relative influence of stochasticity of each model component on outputs. Moreover, we show that a change in the system equilibrium in a full deterministic system can lead to irregularity. Finally, this paper proposes a refinement of travel time models to account for non‐dynamic signals. In presence of traffic signals, we show that a bus system can be self‐regulated. Especially, these insights could help to calibrate bus model inputs to better reproduce real data. Copyright © 2014 John Wiley & Sons, Ltd.     

Rethinking bus punctuality by integrating Automatic Vehicle Location data and passenger patterns

This paper investigates punctuality at bus stops. Although it is typically evaluated from the point of view of bus operators, it must also account for users, as required in recent service quality norms. Therefore, evaluating punctuality at bus stops is highly important, but may also be a complex task, because data on both bus arrivals (or departures) and users must be taken into account and processed. Data on buses can be collected by Automatic Vehicle Location (AVL) systems, but several challenges must be addressed in order to use them effectively. Passengers data at bus stops cannot be derived from AVL, but they can be used to derive passenger patterns and need to be integrated into processed AVL data. This paper proposes a new punctuality measure defined as the fraction of passengers who will be served within an acceptably short interval after they arrive. A method is proposed to determine this measure: it provides (i) several rules to handle AVL collected data, (ii) a procedure integrating processed AVL data and potential passengers’ patterns and (iii) a hierarchical process to perform the punctuality measure on each bus route direction of a transit network, as well as for every bus stop and time period. The paper illustrates the experimentation of this method on more than 4,000,000 data of a real bus operator and represents outcomes by easy-to-read control dashboards.