  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.
