Nowadays, traffic congestion is a significant problem in the world. With the noticeable rise in vehicle usage in recent years and therefore congestion, there has been a wealth of study into possible ways that this congestion can be eased and the flow of traffic on the road improved. Controlling traffic congestion relies on good mathematical models of traffic systems. Creating accurate and reliable traffic control systems is one of the crucial steps for active congestion control. These traffic systems generally use algorithms that depend on mathematical models of traffic.
Day-to-day dynamic assignment models play a critical role in transport management and planning. These models can be either deterministic or stochastic and can be used to describe the day-to-day evolution of traffic flow across the network. This doctoral research is dedicated to understanding the difference between deterministic models and stochastic
models. Deterministic models have been studied well, but the properties of stochastic models are less well understood. We investigate how predictions of the long term properties of the system differ between deterministic models and stochastic models. We find that in contrast to systems with a unique equilibrium where the deterministic model can be a good approximation for the mean of the stochastic model, for a system
with multiple equilibria, the situation is more complicated. In such a case even when deterministic and stochastic models appear to have comparable properties over a significant
time frame, they may still behave very differently in the long-run. Markov models are popular for stochastic day-to-day assignment. Properties of such models are difficult to analyse theoretically, so there has been an interest in approximations which are more mathematically tractable. However, it is di cult to tell when approximation will work well, both in a stationary state and during transient periods following a network disruption.
The coefficient of reactivity introduced by Hazelton (2002) measures the degree to which a system reacts to a disruption. We propose that it can be used as a guide to when approximation models will work well. We study this issue through a raft of numerical experiments. We find that the value of the coefficient of reactivity is useful in predicting the accuracy of approximation models. However, the detailed interpretation of the coefficient of reactivity depends to a modest degree on properties of the network such as its size and number of routes.