Batch queues, reversibility and first-passage percolation

TL;DR

This paper extends discrete-time queue models with batch arrivals and services using Bernoulli and geometric distributions, proving reversibility properties and applying these results to solve new first-passage percolation problems.

Contribution

It introduces a generalized queue model with Bernoulli-geometric batches and proves Burke's theorem variants using reversibility, also applying these to first-passage percolation.

Findings

Reversibility properties hold for the generalized queue models.

Exact solutions for new first-passage percolation problems are derived.

Extensions to continuous-time models are discussed.

Abstract

We consider a model of queues in discrete time, with batch services and arrivals. The case where arrival and service batches both have Bernoulli distributions corresponds to a discrete-time M/M/1 queue, and the case where both have geometric distributions has also been previously studied. We describe a common extension to a more general class where the batches are the product of a Bernoulli and a geometric, and use reversibility arguments to prove versions of Burke's theorem for these models. Extensions to models with continuous time or continuous workload are also described. As an application, we show how these results can be combined with methods of Seppalainen and O'Connell to provide exact solutions for a new class of first-passage percolation problems.