On the Distributions of Infinite Server Queues with Batch Arrivals

TL;DR

This paper analyzes infinite server queues with batch arrivals, deriving transient and steady-state distributions, and exploring their limiting behaviors, with applications to various stochastic distributions and queueing models.

Contribution

It provides a comprehensive analysis of infinite server queues with batch arrivals, including transient, steady-state, and limit behaviors, extending classical models with new distributional insights.

Findings

Steady-state distribution is a sum of scaled Poisson variables.

Transient mean, variance, and MGFs are derived for time-varying rates.

Connections to harmonic numbers, Hermite distributions, and polylogarithms are established.

Abstract

Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as "batch" (or, in some cases, "bulk") arrival queueing systems. In this work we study the affect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the order statistics of its service distribution. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.