Non-Stationary Queues with Batch Arrivals

Motivated by applications that involve setting proper staffing levels for multi-server queueing systems with batch arrivals, we present a thorough study of the queue-length process $\{Q(t); t \geq 0\}$, departure process $\{D(t); t \geq 0\}$, and the workload process $\{W(t); t \geq 0\}$ associated with the M$_{t}^{B_{t}}$/G$_{t}$/$\infty$ queueing system. With two fundamental assumptions of (non-stationary) Poisson arrivals and infinitely many servers, we otherwise maintain a highly general model, in which the service duration and batch size distributions may depend on time and, moreover, where the service durations within a batch may be arbitrarily dependent. Nevertheless, we find that the Poisson and infinite server assumptions are enough to show that for each $t > 0$, the law of $Q(t)$ is that of a weighted sum of mutually independent Poisson random variables. We further invoke this type of decomposition to derive various joint Laplace-Stieltjes transforms associated with the queue-length and departure processes. Next, we study the time-dependent behavior of the workload process, and we conclude by establishing almost sure convergence of the queue-length and workload processes (when properly scaled) to two different shot-noise processes, elevating the weak convergence results shown previously.