Controlling Variability in Split-Merge Systems

Abstract

Abstract. We consider split-merge systems with heterogeneous subtask service times and limited output buffer space in which to hold completed but as yet unmerged subtasks. An important practical problem in such systems is to limit utilisation of the output buffer. This can be achieved by judiciously delaying the processing of subtasks in order to cluster subtask completion times. In this paper we present a methodology to find those deterministic subtask processing delays which minimise any given percentile of the difference in times of appearance of the first and the last subtasks in the output buffer. Technically this is achieved in three main steps: firstly, we define an expression for the distribution of the range of samples drawn from n independent heterogeneous service time distributions. This is a generalisation of the well-known order statistic result for the distribution of the range of n samples taken from the same distribution. Secondly, we extend our model to incorporate deterministic delays applied to the processing of subtasks. Finally, we present an optimisation scheme to find that vector of delays which minimises a given percentile of the range of arrival times of subtasks in the output buffer. A case study illustrates the applicability of our proposed approach.

Trading Off Subtask Dispersion and Response Time in Split-Merge Systems

“…Relaxing the assumption that the X i should be identically distributed leads to the theory of heterogeneous order statistics [4,20]. Definition 2.…”

“…There is a conflicting tension between these two metrics. Indeed, in previous research we have considered how to minimise mean subtask dispersion [12,19] and percentiles of subtask dispersion [20] by delaying the processing of the subtasks in such a way as to cluster subtask completion times. Whilst we apply a constraint to ensure that no delay is added to the bottleneck server, the introduction of subtask delays does naturally have an adverse impact on mean task response time (as quantified in [12]), with a corresponding reduction in maximum sustainable system throughput.…”

“…Consider a split-merge system with 3 parallel servers having heterogeneous service time density functions (as considered in [20]): Using our previously developed optimisation technique designed to minimise mean subtask dispersion without regard to impact on response time [19] we obtain the vector of optimal subtask delays: Fig. 6.…”

“…the time between the arrival of the first and last subtasks originating from a given task in the output buffer). The former class has been the subject of extensive research [2, 3, 6, 7, 9-11, 13, 14, 16, 18, 21, 22], while the latter class has received less attention [19,20], although reducing subtask dispersion can be of critical importance in certain real-life contexts.…”

“…Here we focus on an analytically tractable subclass of such systems, namely split-merge systems. In our previous research [19,20] we showed that reduction of subtask dispersion in such systems can be achieved through the application of judiciously chosen delays to subtask processing. This naturally has an adverse impact on task response time (as characterised in [12]), and therefore on maximum sustainable system throughput.…”