Single-Server Queues with State-Dependent Hawkes Arrivals

Abstract

We study single-server queues with Hawkes arrivals whose intensity process depends on the queue length through the self-exciting function, and with independent and identically distributed general service times, under the first-come first-served discipline. We prove the functional law of large numbers and functional central limit theorems (FCLT) for the joint processes of the arrivals, queue-length, and workload processes, in the heavy traffic regime. The fluid limit is given by a set of nonlinear integral equations such that the fluid queue or workload has a reflection at zero. We analyze the transient and equilibrium behaviors of the fluid limit, in particular, identifying the equilibrium points for the queue or workload fluid model. We assume that the fluid limit is at an equilibrium point in order to establish the FCLT for the joint diffusion-scaled arrival, queue-length, and workload processes in the critically loaded regime. When the equilibrium point of the queue is at zero, the limit for the joint processes satisfies a stochastic differential equation such that the queue-length or workload limit has a reflection at zero. In particular, the queue-length or workload process is equivalent in distribution to a generalized (possibly nonlinear drift) Ornstein-Uhlenbeck (OU) diffusion with reflection at zero. When the equilibrium point is positive, the limit for the joint processes is a diffusion process without reflection; in particular, the queue-length or workload process is a generalized OU diffusion. Because of the interacting effects between the Hawkes arrivals and queue-length or workload processes, the standard approaches to prove functional limit theorems for Hawkes processes and for single-server queues cannot be applied directly. We develop a new method to prove the functional limit theorems for the joint Hawkes and queueing dynamics, by using a localization argument and exploiting convergence of martingales and stochastic integrals. Funding: G. Pang was partly supported by the National Science Foundation [Grants DMS 2216765 and CMMI 2452829]. Supplemental Material: The online companion is available at https://doi.org/10.1287/moor.2024.0855 .