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.