Iryo, T. orcid.org/0000-0003-0839-0835, Watling, D. orcid.org/0000-0002-6193-9121 and Hazelton, M. orcid.org/0000-0001-7831-725X (2024) Estimating Markov Chain Mixing Times: Convergence Rate Towards Equilibrium of a Stochastic Process Traffic Assignment Model. Transportation Science. ISSN 0041-1655
Abstract
Network equilibrium models have been extensively used for decades. The rationale for using equilibrium as a predictor is essentially that (i) a unique and globally stable equilibrium point is guaranteed to exist and (ii) the transient period over which a system adapts to a change is sufficiently short in time that it can be neglected. However, we find transport problems without a unique and stable equilibrium in the literature. Even if it exists, it is not certain how long it takes for the system to reach an equilibrium point after an external shock onto the transport system, such as infrastructure improvement and damage by a disaster. The day-to-day adjustment process must be analysed to answer these questions. Among several models, the Markov chain approach has been claimed to be the most general and flexible. It is also advantageous as a unique stationary distribution is guaranteed in mild conditions, even when a unique and stable equilibrium does not exist. In the present paper, we first aim to develop a methodology for estimating the Markov chain mixing time (MCMT), a worst-case assessment of the convergence time of a Markov chain to its stationary distribution. The main tools are coupling and aggregation, which enable us to analyse MCMTs in large-scale transport systems. Our second aim is to conduct a preliminary examination of the relationships between MCMTs and critical properties of the system, such as travellers’ sensitivity to differences in travel cost and the frequency of travellers’ revisions of their choices. Through analytical and numerical analyses, we found key relationships in a few transport problems, including those without a unique and stable equilibrium. We also showed that the proposed method, combined with coupling and aggregation, can be applied to larger transport models.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | This item is protected by copyright. This is an author produced version of an article published in Transportation Science. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Markov chain mixing time, day-to-day dynamics, stochastic traffic assignment process |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Environment (Leeds) > Institute for Transport Studies (Leeds) > ITS: Spatial Modelling and Dynamics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 15 May 2024 09:22 |
Last Modified: | 15 May 2024 09:22 |
Published Version: | https://pubsonline.informs.org/doi/10.1287/trsc.20... |
Status: | Published online |
Publisher: | Institute for Operations Research and the Management Sciences |
Identification Number: | 10.1287/trsc.2024.0523 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:212527 |