An efficient algorithm is developed to calculate the periodic steady-state distribution and moments of the remaining workload W y at time yc within a cycle of length c, 0 ≤ y < 1, in a single-server queue with a periodic arrival-rate function. The algorithm applies exactly to the GI t /GI/1 model, where the arrival process is a time-transformation of a renewal process. A new representation of W y makes it possible to apply a modification of the classic rare-event simulation for the stationary GI/GI/1 model exploiting importance sampling using an exponential change of measure. We establish bounds between the periodic workload and the stationary workload with the average arrival rate that enable us to prove that the relative error in estimates of P(W y > b) is uniformly bounded in b. With the aid of a recent heavy-traffic limit theorem, the algorithm also applies to compute the periodic steady-state distribution of (i) reflected periodic Brownian motion (RPBM) by considering appropriately scaled GI t /GI/1 models and (ii) a large class of general G t /G/1 queues by approximating by GI t /GI/1 models with the same heavy-traffic limit. Simulation examples demonstrate the accuracy and efficiency of the algorithm for both GI t /GI/1 queues and RPBM.

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We consider a single-server queue with unlimited waiting space, the first-come, first-served discipline, a periodic arrival-rate function, and independent and identically distributed service requirements, where the service-rate function is subject to control. We previously showed that a rate-matching control, whereby the service rate is made proportional to the arrival rate, stabilizes the queue-length process but not the (virtual) waitingtime process. To minimize the maximum expected waiting time (and stabilize the expected waiting time), we now consider a modification of the service-rate control involving two parameters: a time lag and a damping factor. We develop an efficient simulation search algorithm to find the best time lag and damping factor. That simulation algorithm is an extension of our recent rare-event simulation algorithm for the GI t /GI/1 queue to the GI t /GI t /1 queue, allowing the time-varying service rate. To gain insight into these controls, we establish a heavy-traffic limit with periodicity in the fluid scale. This produces a diffusion control problem for the stabilization, which we solve numerically by the simulation search in the scaled family of systems with ρ ↑ 1. The state space collapse in that theorem shows that there is a time-varying Little's law in heavy traffic, implying that the queue length and waiting time cannot be simultaneously stabilized in this limit. We conduct simulation experiments showing that the new control is effective for stabilizing the expected waiting time for a wide range of model parameters, but we also show that it cannot stabilize the expected waiting time perfectly. History: Former designation of this paper was SSy-2018-012.R2.

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