Access control on networks with unique origin–destination paths

This paper presents improved time-dependent control strategies for small freeway networks with bottlenecks and unique origin–destination (O–D) paths. It is assumed that there are no spill-overs from any of the freeway exits so that freeway queues and delays can be completely avoided by regulating access to the system so as to maintain bottleneck flows strictly below capacity. It is also assumed that the time-dependent origin–destination table and the time-dependent bottleneck capacities are known, although not always a priori. The proposed control strategies attempt to minimize the total delay (including both system delay and access delay) while avoiding queues inside the system. The problem is formulated as a constrained calculus of variations exercise that can be cast in the conventional form of optimal control theory and can also be discretized as a mathematical program. Although the first-in-first-out requirement for the access queues introduces undesirable non-linearities, exact solutions for four important special cases can be obtained easily. More specifically, for networks with (1) a single origin or (2) a single bottleneck, a myopic strategy which requires the solution of a sequence of simple linear programs is optimal. For networks with (3) a single destination the non-linearities disappear and the problem becomes a large-scale linear program. This is also true for general networks if (4) the fractional distribution of flow across destinations for every origin is independent of time. A greedy heuristic algorithm is proposed for the general case. It has been programmed for a personal computer running Windows. The algorithm is non-anticipative in that it regulates access at the current time without using future information. As a result, it is computationally efficient and can be bolstered with dynamically-updated information. Globally optimal for cases (1) and (2), the heuristic has been developed with slow-varying O–D tables in mind. Significant improvements will likely require anticipatory information. An illustrative example is given.