On the stability of redundancy models

We investigate the stability condition of redundancy-$d$ multi-server systems. Each server has its own queue and implements popular scheduling disciplines such as First-Come-First-Serve (FCFS), Processor Sharing (PS), and Random Order of Service (ROS). New jobs arrive according to a Poisson process and copies of each job are sent to $d$ servers chosen uniformly at random. The service times of jobs are assumed to be exponentially distributed. A job departs as soon as one of its copies finishes service. Under the assumption that all $d$ copies are i.i.d., we show that for PS and ROS (for FCFS it is already known) sending redundant copies does not reduce the stability region. Under the assumption that the $d$ copies are identical, we show that (i) ROS does not reduce the stability region, (ii) FCFS reduces the stability region, which can be characterized through an associated saturated system, and (iii) PS severely reduces the stability region, which coincides with the system where all copies have to be \emph{fully} served. The proofs are based on careful characterizations of scaling limits of the underlying stochastic process. Through simulations we obtain interesting insights on the system's performance for non-exponential service time distributions and heterogeneous server speeds.