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

Abstract

Stronger Connections and Uncovered Nuance for Queues with Batch Arrivals Many modern queueing systems face the challenge that arrivals occur in batches of customers or jobs. Such simultaneity in the arrival stream may place a particularly acute stress on the service system. Recently, researchers have identified that shot-noise processes, a family of stochastic models for which the dynamics are deterministic between arrival epochs, can be used to approximate batch-arrival queues as justified through batch scaling limits. In “Establishing Convergence of Infinite-Server Queues with Batch Arrivals to Shot-Noise Processes,” Daw, Fralix, and Pender both elevate and generalize these limits, moving from point-wise convergence in distribution for Markovian models to almost sure convergence on a process level (uniformly on compact sets) for general arrival and service processes, allowing for possible nonstationarity and dependence between arrivals. Additionally, the authors establish these batch-scaling limits for both the queue-length and workload processes, and this leads to an unexpected insight. By comparison with the Markovian case, in which the queue-length and workload limits are proportionally identical, for general service distributions, the batch-scaled queue-length and workload processes can differ significantly, revealing that the workload can persist even as the queue length dwindles.