Multiserver-job Response Time under Multilevel Scaling

TL;DR

This paper analyzes the response time of multiserver jobs in a multilevel scaling limit, revealing how queue lengths behave under different load regimes and showing a peak near balanced load.

Contribution

It characterizes the asymptotic stability boundary and mean queue length in a multilevel scaling setting for multiserver jobs, a novel analysis in this domain.

Findings

Mean queue length peaks near balanced load.

Asymptotic growth rate of the stability boundary varies with load regime.

Theoretical, numerical, and simulation results confirm the analysis.

Abstract

We study the multiserver-job setting in the load-focused multilevel scaling limit, where system load approaches capacity much faster than the growth of the number of servers $n$. We consider the ``1 and $n$'' system, where each job requires either one server or all $n$. Within the multilevel scaling limit, we examine three regimes: load dominated by $n$-server jobs, 1-server jobs, or balanced. In each regime, we characterize the asymptotic growth rate of the boundary of the stability region and the scaled mean queue length. We demonstrate that mean queue length peaks near balanced load via theory, numerics, and simulation.