On a unifying product form framework for redundancy models

Abstract : In this paper, we present a unifying analysis for redundancy systems with cancel-on-start ($c.o.s.$) and cancel-on-complete ($c.o.c.$) with exponentially distributed service requirements. With $c.o.s.$ ($c.o.c.$) all redundant copies are removed as soon as one of the copies starts (completes) service. As a consequence, $c.o.s.$ does not waste any computing resources, as opposed to $c.o.c.$. We show that the $c.o.s.$ model is equivalent to a queueing system with multi-type jobs and servers, which was analyzed in \cite{Visschers12}, and show that $c.o.c.$ (under the assumption of i.i.d. copies) can be analyzed by a generalization of \cite{Visschers12} where state-dependent departure rates are permitted. This allows us to show that the stationary distribution for both the $c.o.c.$ and $c.o.s.$ models have a product form. We give a detailed first-time analysis for $c.o.s$ and derive a closed form expression for important metrics like mean number of jobs in the system, and probability of waiting. We also note that the $c.o.s.$ model is equivalent to Join-Shortest-Work queue with redundancy (JSW($d$)). In the latter, an incoming job is dispatched to the server with smallest workload among $d$ randomly chosen ones. Thus, all our results apply mutatis-mutandis to JSW($d$). {Comparing the performance of $c.o.s.$ with that of $c.o.c.$ with i.i.d copies gives the unexpected conclusion (since $c.o.s.$ does not waste any resources) that $c.o.s.$ is worse in terms of mean number of jobs. As part of ancillary results, we illustrate that this is primarily due to the assumption of i.i.d copies in case of $c.o.c.$ (together with exponentially distributed requirements) and that such assumptions might lead to conclusions that are qualitatively different from that observed in practice.
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Urtzi Ayesta, Tejas Bodas, Ina Verloop. On a unifying product form framework for redundancy models. Performance Evaluation, Elsevier, 2018, 127-128, pp.93-119. ⟨10.1016/j.peva.2018.09.008⟩. ⟨hal-01713937v2⟩

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