Stability of Redundancy Systems with Processor Sharing

We investigate the stability condition for redundancy-d systems where each of the servers follows a processor-sharing (PS) discipline. We allow for generally distributed job sizes, with possible dependence among the d replica sizes being governed by an arbitrary joint distribution. We establish that the stability condition is characterized by the expectation of the minimum of d replica sizes being less than the mean interarrival time per server. In the special case of identical replicas, the stability condition is insensitive to the job size distribution given its mean, and the stability condition is inversely proportional to the number of replicas. In the special case of i.i.d. replicas, the stability threshold decreases (increases) in the number of replicas for job size distributions that are NBU (NWU). We also discuss extensions to scenarios with heterogeneous servers.