Parallel server systems with cancel-on-completion redundancy

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.