We analyze the performance of redundancy in a multi-type job and multi-type server system. We assume the job dispatcher is unaware of the servers capacities, and we set out to study under which circumstances redundancy improves the performance. With redundancy an arriving job dispatches redundant copies to all its compatible servers, and departs as soon as one of its copies completes service. As a benchmark comparison, we take the non-redundant system in which a job arrival is routed to only one randomly selected compatible server. Service times are generally distributed and all copies of a job are identical, i.e., have the same service requirement. In our first main result, we characterize the sufficient and necessary stability conditions of the redundancy system. This condition coincides with that of a system where each job type only dispatches copies into its least-loaded servers, and those copies need to be fully served. In our second result, we compare the stability regions of the system under redundancy to that of no redundancy. We show that if the servers capacities are sufficiently heterogeneous, the stability region under redundancy can be much larger than that without redundancy. We apply the general solution to particular classes of systems, including redundancy-d and nested models, to derive simple conditions on the degree of heterogeneity required for redundancy to improve the stability. As such, our result is the first in showing that redundancy can improve the stability and hence performance of a system when copies are non-i.i.d..
Download