Parallel server systems with cancel-on-completion redundancy

TL;DR

This paper analyzes a complex parallel server system with cancel-on-completion redundancy, establishing stability, workload tightness, and conditions for asymptotic independence of server workloads in large systems.

Contribution

It provides the first analysis of stability and workload distribution properties for systems with cancel-on-completion redundancy, including conditions for steady-state independence.

Findings

System stability and workload tightness are established.

Conditions for steady-state asymptotic independence are derived.

SSAI-FRL holds when job components are i.i.d. with increasing-hazard-rate distribution.

Abstract

We consider a parallel server system with so-called cancel-on-completion redundancy. There are $n$ servers and multiple job classes $j$. An arriving class $j$ job consists of $d_j$ components, placed on a randomly selected subset of servers; the job service is complete as soon as $k_j$ components out of $d_j$ (with $k_j \le d_j$) complete their service, at which point the unfinished service of all remaining $d_j-k_j$ components is canceled. The system is in general non-work-conserving, in the sense that the average amount of new workload added to the system by an arriving class $j$ job is not defined a priori -- it depends on the system state at the time of arrival. This poses the main challenge for the system analysis. For the system with a fixed number of servers $n$ our main results include: the stability properties; the property that the stationary distributions of the relative server workloads remain tight, uniformly in the system load. We also consider the mean-field asymptotic regime when $n\to\infty$ while each job class arrival rate per server remains constant. The main question we address here is: under which conditions the steady-state asymptotic independence (SSAI) of server workloads holds, and in particular when the SSAI for the full range of loads (SSAI-FRL) holds. (Informally, SSAI-FRL means that SSAI holds for any system load less than $1$.) We obtain sufficient conditions for SSAI and SSAI-FRL. In particular, we prove that SSAI-FRL holds in the important special case when job components of each class $j$ are i.i.d. with an increasing-hazard-rate distribution.