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 [1], and show that c.o.c. (under the assumption of i.i.d. copies) can be analyzed by a generalization of [1] 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.

Computation of the normalising constant for product-form models of distributed systems with synchronisation

“…Product-form SPNs can model instantaneous signal propagation as shown in [33], fork-join constructs [34] (including RAID systems, MapReduce), database conflicts [35]. Similar results are being studied for the transmission redundancy, showing that product-form solution makes the analysis of complex systems feasible [36,37,38] Testing the sufficient conditions for product-forms in SPNs is efficient, and can be done in polynomial time in the number of places and transitions. Similarly, the computation of the functions g k of Equation (1) is very efficient, while obtaining the expression for G is, in general, quite difficult.…”

Pass-and-swap queues

“…4, we turn to the analysis of closed networks of P&S queues in Sect. 5. We demonstrate the applicability of these models to the modeling of resource-management protocols in Sect.…”

“…We shall also see that, in addition to their theoretical appeal, P&S queues can be used to describe the dynamics of load-distribution and scheduling protocols in a cluster of machines in which jobs have assignment constraints. This cluster model, which can represent various computer clusters or manufacturing systems in which not every machine is able to fulfill the service requirement of any job, has played a central role in several studies of product-form queueing models over the past decade; see, for example, [2][3][4][5]9,[12][13][14][15]. Roughly speaking, this cluster model can be interpreted as a P&S queue where customers represent jobs and servers represent machines.…”

“…In a similar vein, P&S queues can be used to model redundancy scheduling [4,5,15] in clusters of machines, where replicas of a job may be routed to (the buffers of) multiple machines, and redundant replicas are canceled whenever a replica completes service at a machine (cancel-on-completion) or enters service at a machine (cancelon-start). One can verify that the dynamics of the cancel-on-completion protocol are tantamount to the first aforementioned protocol, and, as such, can be modeled by an OI queue (and thus also by a P&S queue).…”

“…Several other studies pointed out connections between the above-mentioned applications and placed them in a more general context of product-form queueing models; cf. [1,5].…”