Establishing Convergence of Infinite-Server Queues with Batch Arrivals to Shot-Noise Processes

Across domains as diverse as communication channels, computing systems, and public health management, a myriad of real-world queueing systems receive batch arrivals of jobs or customers. In this work, we show that under a natural scaling regime, both the queue-length process and the workload process associated with a properly scaled sequence of infinite-server queueing systems with batch arrivals converge almost surely, uniformly on compact sets, to shot-noise processes. Given the applicability of these models, our relatively direct and accessible methodology may also be of independent interest, where we invoke the Glivenko–Cantelli theorem when the Strong Law of Large Numbers fails to hold for the queue-length batch scaling yet then, exploit the continuity of stationary excess distributions and the classic strong law when the Glivenko–Cantelli theorem fails to hold in the workload batch scaling. These results strengthen a convergence result recently established in the work of de Graaf et al. [de Graaf WF, Scheinhardt WR, Boucherie RJ (2017) Shot-noise fluid queues and infinite-server systems with batch arrivals. Performance Evaluation 116:143–155] in multiple ways, and furthermore, they provide new insight into how the queue-length and workload limits differ from one another.