Stability and tail behavior 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 for homogeneous servers the stability condition for the associated fluid-limit model 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 threshold 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 the extension to heterogeneous servers. For heavy-tailed job sizes we characterize the tail behavior of the response time distribution. In particular, for regularly varying job sizes with tail index −ν it is shown that the tail index of the response time equals −ν and −dν for identical and i.i.d. replicas, respectively.