Time-varying single-server queues with state-dependent Hawkes arrivals

We study a non-stationary state-dependent Hawkes single-server queueing model with both time-varying arrival and service rates, under the first-come first-served discipline. The arrival process is a Hawkes process with a time-varying baseline rate function, and a self-exciting function that depends on the state of the queue at the arrival times. The service requirements for the jobs are i.i.d. and are met with a common time-varying rate function, so that the realized service times are non-stationary. We establish both the functional law of large numbers (FLLN) and the functional central limit theorem (FCLT) for the joint dynamics of the Hawkes arrival, queueing and workload processes. The limit for the Hawkes arrival and queueing processes in FLLN is given by a set of nonlinear (state-dependent) differential equations with the queueing component having a reflection at zero, and the fluid limit for the workload process is given by a nonlinear functional of the fluid queueing limit involving a time-shifted cumulative service rate function. Since the traffic intensity is both time and state dependent, we analyze the fluid dynamics in the properly defined underloaded and overloaded intervals. The FCLT is proved for the CLT-scaled processes that are centered around the corresponding transient fluid limits, under the underloaded and overloaded intervals. The diffusion limits for the joint Hawkes arrival and queueing processes are given by a set of nonlinear stochastic differential equations with time-varying coefficients; in particular, the queueing limit is a reflected diffusion at zero over the underloaded intervals. The limit for the CLT-scaled workload process takes a complicated form in the overloaded intervals, involving the limits of the time-shifted cumulative service rate function in both fluid and diffusion scales.