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As job redundancy has been recognized as an effective means to improveperformance of large-scale computer systems, queueing systems with redundancyhave been studied by various authors. Existing results include methods tocompute the queue length distribution and response time but only when theservice discipline is First-Come-First-Served (FCFS). For other servicedisciplines, such as Processor Sharing (PS), or Last-Come-First-Served (LCFS),only the stability conditions are known. In this paper we develop the firstmethods to approximate the queue length distribution in a queueing system withredundancy under various service disciplines. We focus on a system withexponential job sizes, i.i.d. copies, and a large number of servers. We firstderive a mean field approximation that is independent of the scheduling policy.In order to study the impact of service discipline, we then derive refinementsof this approximation to specific scheduling policies. In the case of ProcessorSharing, we provide a pair and a triplet approximation. The pair approximationcan be regarded as a refinement of the classic mean field approximation andtakes the service discipline into account, while the triplet approximationfurther refines the pair approximation. We also develop a pair approximationfor three other service disciplines: First-Come-First-Served, Limited ProcessorSharing and Last-Come-First-Served. We present numerical evidence that showsthat all the approximations presented in the paper are highly accurate, butthat none of them are asymptotically exact (as the number of servers goes toinfinity). This makes these approximations suitable to study the impact of theservice discipline on the queue length distribution. Our results show that FCFSyields the shortest queue length, and that the differences are more substantialat higher loads.

Approximations to Study the Impact of the Service Discipline in Systems with Redundancy