Approximations for the Queue Length Distributions of Time-Varying Many-Server Queues

This paper presents a novel and computationally efficient methodology for approximating the queue length (the number of customers in the system) distributions of time-varying non-Markovian many-server queues (e.g., Gt/Gt/nt queues), where the number of servers (nt) is large. Our methodology consists of two steps. The first step uses phase-type distributions to approximate the general interarrival and service times, thus generating an approximating Pht/Pht/nt queue. The second step develops strong approximation theory to approximate the Pht/Pht/nt queue with fluid and diffusion limits whose mean and variance can be computed using ordinary differential equations. However, by naively representing the Pht/Pht/nt queue as a Markov process by expanding the state space, we encounter the lingering phenomenoneven when the queue is overloaded. Lingering typically occurs when the mean queue length is equal or near the number of servers, however, in this case it also happens when the queue is overloaded and this time...

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