A stochastic model for bus injection in an unscheduled public transport service

doi:10.1016/j.trc.2019.05.029

Transportation Research Part C: Emerging Technologies

Volume 113, April 2020, Pages 277-292

https://doi.org/10.1016/j.trc.2019.05.029 Get rights and content

Highlights

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An injection stochastic model based on the headway distribution second moment is proposed.
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An expression for the optimal headway threshold triggering the injection is proposed.
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A model to determine when a bus should be injected within the headway was developed.
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We identify at which moment within the headway a bus should be injected.
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Simulations with real data are used to test the proposed model, proving its accuracy.

Abstract

Randomness affecting the operation of public transport systems generates significant increments in waiting times. A strategy to deal with this randomness is bus injection, in which buses are kept in specific points along the route ready to be dispatched when an event such as an extremely long headway occurs. In this work, a stochastic model based on the second moment of the headways distribution is developed to determine if one or more buses are worth reserving for injection in a public transport service. A single stop approach is initially used to determine an expression for the optimal headway threshold triggering the injection. Then, a model for the complete service is developed and used to determine when the empty bus should be injected within the headway once the decision to inject it has been taken. We show that the bus should be injected approximately when 57% of the headway has passed. Simulations with real data are used to test the proposed model, proving its accuracy in terms of measuring the impact on waiting times. The results show that reserving a bus to be injected can be better than operating the entire fleet continuously.

Section snippets

Introduction and motivation

Randomness in passenger arrival times and vehicle travel times affects the operation of public transport systems significantly. If a vehicle gets delayed, the number of people waiting to ride increases, which reduces its speed even more. The vehicle that follows the delayed one will follow a short interval, so its speed will grow quite fast if no action is made to prevent it. This phenomena, known as bus bunching since it affects buses more than trains, damages waiting time reliability and

Model for injection considering a single stop

Consider a stop visited by a single bus service. The headways H between arriving buses follow a known distribution. Let W be the waiting time of a random passenger arriving to the stop who always boards the first arriving bus (so bus capacity never binds). Osuna and Newell show that if the arrival of passengers is independent to the arrival of the buses, then (Osuna and Newell, 1972) $E [W] = \frac{E [H^{2}]}{2 E [H]}$

We assume that a given fleet is available. If all the buses in the fleet are used for the

Injection model considering the impact at all stops

In order to correctly evaluate the impact of bus injection, the results from the previous section must be adapted to consider the effect in waiting times at all downstream stops along the service. To do this, we must evaluate the evolution of a headway sequence at downstream stops. The work by Marguier, later improved by Hickman (2001) is particularly useful as a starting point.

Conclusions

A model for analyzing the impact in waiting times of injecting a bus considering at a single stop is provided. The model recommends a threshold for the injection to be made based on the length of the headway preceding each bus. Then, a complete model predicting the trajectory of each bus and incorporating the stochasticity of travel times and passenger demand is provided. Examples and comparison with real-data and simulations show that this model, even with independence assumptions provides

Acknowledgements

This research was supported by the Centro de Desarrollo Urbano Sustentable (CEDEUS), CONICYT/FONDAP 15110020, the Bus Rapid Transit Centre of Excellence funded by Volvo Research and Educational Foundations (VREF) and FONDECYT1150657.

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^☆: This paper has been accepted for a poster presentation at the 23rd International Symposium on Transportation and Traffic Theory (ISTTT23) July 24–26, 2019 in Lausanne, CH.

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A stochastic model for bus injection in an unscheduled public transport service☆

Highlights

Abstract

Section snippets

Introduction and motivation

Model for injection considering a single stop

Injection model considering the impact at all stops

Conclusions

Acknowledgements

Transport. Res. Part A: Policy Pract.

Transport. Res. Part A: General

Transport. Res. Part A: Policy Pract.

Transport. Res. Part B: Methodol.

Transport. Res. Part B: Methodol.

Transport. Res. Part B: Methodol.

Transport. Res. Part B: Methodol.

Transport. Res. Part C: Emerg. Technol.

Transport. Res. Part B: Methodol.