A new class of doubly stochastic day-to-day dynamic traffic assignment models

  • Katharina Parry
  • David P. Watling
  • Martin L. Hazelton
Research Paper


Real-life systems are known to exhibit considerable day-to-day variability. A better understanding of such variability has increasing policy-relevance in the context of network reliability assessment and the design of intelligent transport systems. Conventional equilibrium models are ill-suited, because deterministic models such as these do not account for any kind of variability. At best, these types of models are restricted to finding a steady state of the mean flow patterns, they cannot capture the variance in flows as well. A more suitable alternative are stochastic day-to-day dynamic models studied by Cascetta in Trans Res 23:1–17, (1989). These types of traffic assignment models represent the traffic flows via a Markov process, where the current route flows are modelled as a function of previous traffic conditions. Day-to-day dynamic models differ from equilibrium models in that day-to-day changes in the system are modelled dependent on the time and thus allow for a far wider representation of traveller behaviour. However, to some degree they still suffer from some of the limitations of equilibrium analyses, in that while they permit variation they are still wedded to the concept of ‘stationarity’. In this paper, we show how these Markovian day-to-day dynamic traffic assignment models can be extended by replacing a subset of the fixed parameters in the Markov model with random processes. The resulting models are analogous to Cox process models. They are conditionally non-stationary given any realization of the parameter processes. We present numerical examples that demonstrate that this new class of doubly stochastic day-to-day traffic assignment models can indeed reproduce features such as the heteroscedasticity of traffic flows observed in real-life settings.


Markov Transportation Network Doubly stochastic Heteroscedasticity Day-to-day 



The authors acknowledge the helpful comments of three referees, which led to improvements in the paper.


  1. Bollerslev T (1986) Generalised autoregressive conditional heteroskedasticity. J Econ 31:307–327CrossRefGoogle Scholar
  2. Brix A, Diggle PJ (2001) Spatiotemporal prediction for log-Gaussian Cox processes. J R Stat Soc Ser B (Statistical Methodology) 63(4):823–841CrossRefGoogle Scholar
  3. Cantarella GE, Cascetta E (1995) Dynamic processes and equilibrium in transportation networks: towards a unifying theory. Transp Sci 29:305–329CrossRefGoogle Scholar
  4. Cascetta E (1989) A stochastic process approach to the analysis of temporal dynamics in transportation networks. Trans Res Part B 23:1–17CrossRefGoogle Scholar
  5. Cascetta EA, Nuzzolo F, Russo A, Vitetta A (1996) A modified logit route choice model overcoming path overlapping problems: specification and some calibration results for interurban networks. In: Lesort B (ed) Transportation and Traffic Theory. Proceedings from the Thirteenth International Symposium on Transportation and Traffic Theory, Lyon, France, Pergamon, pp 697–711Google Scholar
  6. Daganzo CF, Sheffi Y (1977) On stochastic models of traffic assignment. Trans Sci 11:253–274CrossRefGoogle Scholar
  7. Davis GA, Nihan NL (1993) Large population approximations of a general stochastic traffic assignment model. Oper Res 41(1):169–178CrossRefGoogle Scholar
  8. Hazelton ML, Watling DP (2004) Computation of equilibrium distributions of Markov traffic assignment models. Trans Sci 38:331–342CrossRefGoogle Scholar
  9. Horowitz JL (1984) The stability of stochastic equilibrium in a two-link transportation network. Trans Res Part B 18(1):13–28CrossRefGoogle Scholar
  10. Møller J, Waagepetersen RP (2007) Modern statistics for spatial point processes. Scand J Stat 34:643–684Google Scholar
  11. R Core Team (2013) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0
  12. Thavaneswaran A, Appadoo SS, Bector CR (2006) Recent developments in volatility modeling and applications. J Appl Math Decis Sci 2006:1–23CrossRefGoogle Scholar
  13. Wardrop JG (1952) Some theorectical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, Part II(1):325–378Google Scholar
  14. Watling DP, Hazelton ML (2003) The dynamics and equilibria of day-to-day assignment models. Netw Spatial Econ 3:349–370CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and EURO - The Association of European Operational Research Societies 2013

Authors and Affiliations

  1. 1.Institute of Fundamental SciencesMassey UniversityPalmerston NorthNew Zealand
  2. 2.Institute for Transport Studies University of LeedsLeedsUK
  3. 3.School of Computing and Mathematical Sciences Auckland University of TechnologyAucklandNew Zealand

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