Maximum waiting time in heavy-tailed fork-join queues

TL;DR

This paper analyzes the asymptotic behavior of the maximum waiting time in large heavy-tailed fork-join queues, revealing convergence to an extremal process after specific scaling, with implications for understanding large system delays.

Contribution

It establishes the convergence of the maximum waiting time process in heavy-tailed fork-join queues to an extremal process, including steady-state results, under new scaling regimes.

Findings

Maximum waiting time converges to an extremal process after scaling.

Derived specific temporal and spatial scaling laws for convergence.

Proved steady-state convergence of the maximum waiting time.

Abstract

In this paper, we study the maximum waiting time $\max_{i\leq N}W_i(\cdot)$ in an $N$-server fork-join queue with heavy-tailed services as $N\to\infty$. The service times are the product of two random variables. One random variable has a regularly varying tail probability and is the same among all $N$ servers, and one random variable is Weibull distributed and is independent and identically distributed among all servers. This setup has the physical interpretation that if a job has a large size, then all the subtasks have large sizes, with some variability described by the Weibull-distributed part. We prove that after a temporal and spatial scaling, the maximum waiting time process converges in $D[0,T]$ to the supremum of an extremal process with negative drift. The temporal and spatial scaling are of order $\tilde{L}(b_N)b_N^{\frac{\beta}{(\beta-1)}}$, where $\beta$ is the shape parameter in the regularly varying distribution, $\tilde{L}(x)$ is a slowly varying function, and $(b_N,N\geq 1)$ is a sequence for which holds that $\max_{i\leq N}A_i/b_N\overset{\mathbb{P}}{\longrightarrow}1$, as $N\to\infty$, where $A_i$ are i.i.d.\ Weibull-distributed random variables. Finally, we prove steady-state convergence.