Day-to-day variation in Markovian traffic assignment models

Markovian traffic assignment processes form an intuitively attractive and flexible class of models. However, the intrinsic complexity of these models (especially when applied to large road networks) makes their theoretical analysis very difficult. Simulation of the process in question can be used to learn something of the characteristics of the assignment model, but such simulation is typically computationally expensive. Furthermore, interpretation of simulation results is often far from straightforward. For example, it is difficult to discover to what extent day-to-day variability in traffic flows can be attributed to travellers' reactions to changing mean travel costs, and to what extent these variations are simply explained by entirely haphazard fluctuations in individual travellers' preferences. Nonetheless, this is an important issue, not least because it is crucial in understanding how a Markovian assignment process will react to various types of one-off event (such as the temporary closure of a road link). In this paper a new analytical tool called the coefficient of reactivity is introduced. This coefficient summarises the extent to which traveller learning mechanisms provoke day-to-day volatility in Markovian assignment models. Properties of the coefficient of reactivity, and its more easily calculated asymptotic form, are discussed. Uses and interpretations of these new coefficients are illustrated through a number of examples.