This paper introduces a deterministic fluid model that approximates the many-server G t /GI/s t + GI queueing model, and determines the time-dependent performance functions. The fluid model has time-varying arrival rate and service capacity, abandonment from queue, and non-exponential service and patience distributions. Two key assumptions are that: (i) the system alternates between overloaded and underloaded intervals, and (ii) the functions specifying the fluid model are suitably smooth. An algorithm is developed to calculate all performance functions. It involves the iterative solution of a fixed-point equation for the time-varying rate that fluid enters service and the solution of an ordinary differential equation for the time-varying head-of-line waiting time, during each overloaded interval.Simulations are conducted to confirm that the algorithm and the approximation are effective.

An algorithm is developed to determine time-dependent staffing levels to stabilize the time-dependent abandonment probabilities and expected delays at positive target values in the M t /GI/s t + GI many-server queueing model, which has a nonhomogeneous Poisson arrival process (the M t ), has general service times (the first GI), and allows customer abandonment according to a general patience distribution (the +GI). New offered-load and modified-offered-load approximations involving infinite-server models are developed for that purpose. Simulations show that the approximations are effective. A many-server heavy-traffic limit in the efficiency-driven regime shows that (i) the proposed approximations achieve the goal asymptotically as the scale increases, and (ii) it is not possible to simultaneously stabilize the mean queue length in the same asymptotic regime.

A many-server heavy-traffic FCLT is proved for the $G_t/M/s_t+\mathit {GI}$ queueing model, having time-varying arrival rate and staffing, a general arrival process satisfying a FCLT, exponential service times and customer abandonment according to a general probability distribution. The FCLT provides theoretical support for the approximating deterministic fluid model the authors analyzed in a previous paper and a refined Gaussian process approximation, using variance formulas given here. The model is assumed to alternate between underloaded and overloaded intervals, with critical loading only at the isolated switching points. The proof is based on a recursive analysis of the system over these successive intervals, drawing heavily on previous results for infinite-server models. The FCLT requires careful treatment of the initial conditions for each interval.Comment: Published in at http://dx.doi.org/10.1214/13-AAP927 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

To describe the congestion in large-scale service systems, we introduce and analyze a non-Markovian open network of many-server fluid queues with customer abandonment, proportional routing, and time-varying model elements. Proportions of the fluid completing service from each queue are immediately routed to the other queues, with the fluid not routed to one of the queues being immediately routed out of the network. The fluid queue network serves as an approximation for the corresponding non-Markovian open network of many-server queues with Markovian routing, where all model elements may be time varying. We establish the existence of a unique vector of (net) arrival rate functions at each queue and the associated time-varying performance. In doing so, we provide the basis for an efficient algorithm, even for networks with many queues.

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