The RESET and MARC Techniques, with Application to Multiserver-Job Analysis

TL;DR

This paper introduces the RESET and MARC techniques to analytically approximate mean response time in multiserver-job FCFS systems with phase-type job durations, advancing understanding beyond stability analysis.

Contribution

It provides the first explicit approximation of mean response time in MSJ FCFS systems using novel RESET and MARC techniques, reducing complex models to more tractable forms.

Findings

Derived explicit mean response time formula up to negligible additive error.

Reduced MSJ FCFS analysis to an M/M/1 with Markovian service rate.

Introduced the concept of relative completions for analytical tractability.

Abstract

Multiserver-job (MSJ) systems, where jobs need to run concurrently across many servers, are increasingly common in practice. The default service ordering in many settings is First-Come First-Served (FCFS) service. Virtually all theoretical work on MSJ FCFS models focuses on characterizing the stability region, with almost nothing known about mean response time. We derive the first explicit characterization of mean response time in the MSJ FCFS system. Our formula characterizes mean response time up to an additive constant, which becomes negligible as arrival rate approaches throughput, and allows for general phase-type job durations. We derive our result by utilizing two key techniques: REduction to Saturated for Expected Time (RESET) and MArkovian Relative Completions (MARC). Using our novel RESET technique, we reduce the problem of characterizing mean response time in the MSJ FCFS system to an M/M/1 with Markovian service rate (MMSR). The Markov chain controlling the service rate is based on the saturated system, a simpler closed system which is far more analytically tractable. Unfortunately, the MMSR has no explicit characterization of mean response time. We therefore use our novel MARC technique to give the first explicit characterization of mean response time in the MMSR, again up to constant additive error. We specifically introduce the concept of "relative completions," which is the cornerstone of our MARC technique.