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Model Representation & Decision-Making in an Ever-Changing World: The Role of Stochastic Process Models of Transportation Systems

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Abstract

We review and advance the state-of-the-art in the modelling of transportation systems as a stochastic process. The conceptual and theoretical basis of the approach is explained in detail. A variety of examples are given to motivate its use in the field. While the examples cover a wide range of modelling philosophies, in order to provide focus they are restricted to modelling a special class of problems involving driver route choice in networks. Our overall objective is to establish the applicability of this approach as a ‘unifying framework’ for modelling approaches involving dynamic and stochastic elements, developing further the ideas put forward in Cantarella & Cascetta (Transportation Science 29, 305–329, 1995). Directions for further development and research are identified.

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Notes

  1. Note that we could combine these cases as a single case by use of a Riemann-Stieltjes integral.

  2. If our interest had only been in models with a discrete, finite state space, then a simpler, standard specification would be as q (t) = Pq (t–1), with the states in \( \mathcal{S} \) labelled 1, 2, …, |\( \mathcal{S} \)|, the transition probabilities in a |\( \mathcal{S} \)| × |\( \mathcal{S} \)| matrix P, and the time-dependent state probabilities in a |\( \mathcal{S} \)| × 1 column vector q (t) .

  3. At this stage this is not a necessary assumption, and in fact we may wish to consider models in which endogenous factors vary over the time of the process, e.g. economic factors, seasonal changes in demand. The theoretical properties described later (section 4.2) are established under the assumption of a constant parameter vector (leading to a time-homogenous process with time-independent transition probabilities), but it is certainly possible to consider and model time-inhomogeneous processes. This is an interesting possibility left for future research to consider.

  4. In fact this is trivially generalised to non-separable cost functions if desired.

  5. In practice, we may wish to simplify the specification by using a model of the form given in Example 1 to generate the flow on day 1, given the flow on day 0, and then apply the model given here starting from day 2, given knowledge of the probabilities of the pair of states on days 0 and 1.

  6. For general Markov processes or Markov chains/processes, the notion of what we mean by convergence is a study in itself, since we may define a variety of norms over which convergence may be studied. The results established here concern what is termed strong convergence (see Stokey and Lucas 1989, pp 338–344).

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Acknowledgments

We would like to thank the anonymous reviewers, as well as colleagues at DTA2012, for constructive comments that helped us to improve an earlier version of this paper. This work was partially supported by UNISA local grant ORSA091208 (financial year 2009) and ORSA118135 (financial year 2011), and by UK EPSRC grant refs. EP/I00212X/1 (2011–12) and EP/I00212X/2 (2012–16). The financial assistance of Prof Terry Friesz is also gratefully acknowledged, in supporting the visit of the first-named author to deliver the keynote paper at the DTA 2012 International Symposium on Dynamic Traffic Assignment (Martha’s Vineyard, USA), upon which the present paper is based.

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Authors and Affiliations

  1. Institute for Transport Studies, University of Leeds, Leeds, UK

    David P. Watling

  2. Department of Civil Engineering, University of Salerno, Salerno, Italy

    Giulio E. Cantarella

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  1. David P. Watling
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  2. Giulio E. Cantarella
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Correspondence to David P. Watling.

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Watling, D.P., Cantarella, G.E. Model Representation & Decision-Making in an Ever-Changing World: The Role of Stochastic Process Models of Transportation Systems. Netw Spat Econ 15, 843–882 (2015). https://doi.org/10.1007/s11067-013-9198-2

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