Extreme values for the waiting time in large fork-join queues

TL;DR

This paper analyzes the asymptotic behavior of maximum waiting times and queue lengths in large fork-join queues, showing they converge to a normal distribution and scaling logarithmically with the number of servers.

Contribution

It provides a rigorous proof of the normal convergence and logarithmic scaling of maximum waiting times in large fork-join queues, including extensions to multiple server classes.

Findings

Maximum steady-state waiting time scales as (1/γ) log N

Maximum queue length converges to a normal distribution

Results extend to multi-class server systems

Abstract

We prove that the scaled maximum steady-state waiting time and the scaled maximum steady-state queue length among $N$ $GI/GI/1$-queues in the $N$-server fork-join queue, converge to a normally distributed random variable as $N\to\infty$. The maximum steady-state waiting time in this queueing system scales around $\frac{1}{\gamma}\log N$, where $\gamma$ is determined by the cumulant generating function $\Lambda$ of the service distribution and solves the Cram\'er-Lundberg equation with stochastic service times and deterministic inter-arrival times. This value $\frac{1}{\gamma}\log N$ is reached at a certain hitting time. The number of arrivals until that hitting time satisfies the central limit theorem, with standard deviation $\frac{\sigma_A}{\sqrt{\Lambda'(\gamma)\gamma}}$. By using distributional Little's law, we can extend this result to the maximum queue length. Finally, we extend these results to a fork-join queue with different classes of servers.