Abstract
This paper considers a fork-join system (or: parallel queue), which is a two-queue network in which any arrival generates jobs at both queues and the jobs synchronize before they leave the system. The focus is on methods to quantify the mean value of the ‘system’s sojourn time’ S: with S i denoting a job’s sojourn time in queue i, S is defined as max{S 1, S 2}. Earlier work has revealed that this class of models is notoriously hard to analyze. In this paper, we focus on the homogeneous case, in which the jobs generated at both queues stem from the same distribution. We first evaluate various bounds developed in the literature, and observe that under fairly broad circumstances these can be rather inaccurate. We then present a number of approximations, that are extensively tested by simulation and turn out to perform remarkably well.
Highlights
Fork-join systems are service systems in which every arrival generates input in multiple queues
The BM bounds for the GI/G/1 parallel queue are ‘explicit’ in the sense that they reduce to standard formulas in terms of the distribution of the sojourn times of single GI/G/1 systems for the upper bound, and single D/G/1 systems for the lower bound
For i = 1, 2 we find, as before, the Laplace transforms of the sojourn times:
Summary
Fork-join systems (or: parallel queues) are service systems in which every arrival generates input in multiple queues. For the general M/G/1 fork-join system (and for the GI/G/1 variant), upper and lower bounds on the mean sojourn time were derived by Baccelli and Makowski (1985), relying on stochastic comparison techniques; see Baccelli et al (1989) These bounds are not always easy to compute, as they require the availability of explicit expressions or accurate approximations of the distribution function of the workload in related single-node M/G/1 and D/G/1 queues. An upper and lower bound for the general GI/G/1 case are presented by Baccelli and Makowski (1985), see Baccelli et al (1989); in the sequel we refer to these bounds as the BM bounds The idea behind these bounds is that the level of the variability of the fork-join system’s waiting time should be increasing in the level of variability of the stochastic arrival process of the system.
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