Morning commute problem with queue-length-dependent bottleneck capacity

When traffic control is demand-responsive, a bottleneck can have a queue-length-dependent capacity. Motivated by this fact, we studied the morning commute problem in which the bottleneck capacity increases from s1 to s2 as the queue length exceeds a threshold D1 and decreases back to s1 as the queue length reduces and becomes smaller than another threshold D2 (D2 ≤ D1). It has been found that multiple equilibria exist when D2 < D1. Their stability has been preliminarily studied via the day-to-day dynamics, and we found it is likely that only the equilibrium state with the lowest cost is stable against large disturbances. In the case where D2 = D1, there is no multiple user equilibrium state, and the capacity does not change more than twice. Moreover, there exists a parameter range in which there is no solution unless the bottleneck capacity can switch back and forth instantaneously. The policy of opening the shoulder lane on a 2-lane highway is discussed as an application example.