Traffic Control and Route Choice; Capacity Maximization and Stability

This paper presents idealised natural general and special dynamical models of day-to-day re-routeing and of day to day green-time response. Both green-time response models are based on the responsive control policy P₀ introduced in Smith (1979a, b, c 1987). Several results are proved. For example, it is shown that, for any steady feasible demand within a flow model, if the general day to day re-routeing model is combined with the general day to day green-time response model then under natural conditions any (flow, green-time) solution trajectory cannot leave the region of supply-feasible (flow, green-time) pairs and costs are bounded. Throughput is maximised in the following sense. Given any constant feasible demand; this demand is met as any routeing / green-time trajectory evolves (following either the general or the special dynamical model). The paper then considers simple "pressure driven" responsive control policies, with explicit signal cycles of fixed positive duration. A possible approach to dynamic traffic control allowing for variable route choices is outlined. It is finally shown that modified Varaiya (2013) and Le at al (2013) pressure-driven responsive controls may not maximise network capacity, by considering a very simple one junction network. It is shown that (with each of these two modified policies) there is a steady demand within the capacity of the network for which there is no Wardrop equilibrium consistent with the policy. In contrast, responsive P₀ on this simple network does maximise throughput at a quasi-dynamic user equilibrium consistent with P₀; queues and delays remain bounded in natural dynamical evolutions in this case. It is to be expected that this P₀ result may be extended to allow for certain time-varying demands on a much wider variety of networks; to show that this is indeed the case is a challenge for the future.