Large fork-join queues with nearly deterministic arrival and service times

TL;DR

This paper analyzes large-scale fork-join queues with nearly deterministic arrivals and services, deriving a fluid limit for maximum queue length as the number of servers grows, using extreme value theory and diffusion approximations.

Contribution

It introduces a fluid limit for maximum queue length in large fork-join queues with deterministic-like times, incorporating initial conditions and advanced probabilistic methods.

Findings

Fluid limit for maximum queue length as N→∞

Dependence of the limit on initial number of tasks

Development of extreme value theory and diffusion approximations

Abstract

In this paper, we study an $N$ server fork-join queue with nearly deterministic arrival and service times. Specifically, we present a fluid limit for the maximum queue length as $N\to\infty$. This fluid limit depends on the initial number of tasks. In order to prove these results, we develop extreme value theory and diffusion approximations for the queue lengths.