Patch-based Modelling of City-centre Bus Movement with Phase-type Distributions

Abstract

We propose a methodology for constructing a stochastic performance model of a public transportation network using real-world data. Our main data source consists of Automatic Vehicle Location (AVL) measurements of buses in the Edinburgh region. Although the data has a relatively low frequency, we can use it to parameterise a model in which a bus moves between predefined patches in the city. We fit the probability distributions of the sojourn times in the patches to phase-type distributions using the tool HyperStar. We then translate the output from HyperStar to a model of a complete part of a bus route expressed in the reactive modules language of the PRISM model checker. Finally, we demonstrate how we can use the numerical techniques implemented in PRISM to answer meaningful questions about the performance of the bus network in the context of a case study involving the addition of trams to a busy section of Edinburgh's city centre.

Finding Optimal Timetables for Edinburgh Bus Routes

“…This tension tends to separate them sufficiently that the overall journey time tends to become more predictable. In other work [6], HyperStar has returned hyper-Erlang distributions. The HyperStar fitting procedure produces the following results for the three stages of the journey.…”

Probabilistic Modelling of the Impact on Bus Punctuality of a Speed Limit Proposal in Edinburgh

“…As we demonstrate in this paper, the accuracy of these predictions -particularly for timetable adherence probabilities -can be affected by the modelling choice. In previous work [16], the hyper-Erlang distribution was chosen to model the time spent in patches for several reasons, including its general applicability and the fact that the resulting models are Continuous-Time Markov Chains, for which many efficient analysis techniques exist. However, as we discuss later on, an important alternative, namely the probability distribution that is recommended for traffic engineers, may not yield a Markov chain, but the resulting model can still be expressed using the framework of Probabilistic Timed Automata, allowing us to use the tool UPPAAL [4] and its powerful stochastic simulation engine.…”

“…This paper broadly follows the same approach as in [16], in which a patch-based analysis was carried out to evaluate the impact of the introduction of trams to Edinburgh's city centre, and [12], in which Erlang distributions were used to model the time spent by buses between major stops, and to suggest improvements to bus timetables. The contributions of this paper are the use of a probability distribution recommended in the traffic engineering literature [13], the use of the statistical model checking engine of UP-PAAL [10] and the application to a new case study that has received considerable media attention in Edinburgh.…”

“…In this paper we perform a case study on two bus routes in Edinburgh and provide a quantitative answer to the question of how the proposed speed limit impacts bus punctuality. Our methodology for developing bus models involves an expanded and modified implementation of the work done in [16] and [12]. Using this approach, we build a model that represents the continuous-time movement of buses as they move along predefined segments of a route.…”

Towards the Automated Verification of Weibull Distributions for System Failure Rates

“…In both cases, results are contrasted with those obtained via simulation. In [10], a stochastic performance model is constructed and the hyper Erlang distribution of real-world data used in PRISM to analyse a public bus transportation network in Edinburgh. In [11], phase-type distributions are used to analyse a collaborative editing system in PRISM.…”