Heavy traffic limits for queues with non-stationary path-dependent arrival processes

Abstract

In this paper, we develop a diffusion approximation for the transient distribution of the workload process in a standard single-server queue with a non-stationary Polya arrival process, which is a path-dependent Markov point process. The path-dependent arrival process model is useful because it has the arrival rate depending on the history of the arrival process, thus capturing a self-reinforcing property that one might expect in some applications. The workload approximation is based on heavy-traffic limits for (i) a sequence of Polya processes, in which the limit is a Gaussian–Markov process, and (ii) a sequence of P/GI/1 queues in which the arrival rate function approaches a constant service rate uniformly over compact intervals.

Heavy-traffic limits for parallel single-server queues with randomly split Hawkes arrival processes

“…They can be used to model, for example, internet/social media traffic flows, patient flows during a pandemic, neuron interaction processes, and high-frequency transaction processes in limit order books. Queues with Hawkes input are studied in [8], [11], [16], [22], and [26], and single-server queues with Pólya arrival processes are considered in [14] and [15]. On the other hand, many systems have parallel servers, such as data centers and internet processors, and various randomized routeing schemes have been developed in the literature (see e.g.…”