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.

Abstract. In many real-world systems incoming tasks split into subtasks which are processed by a set of parallel servers. In such systems two metrics are of potential interest: response time and subtask dispersion. Previous research has been focused on the minimisation of one, but not both, of these metrics. In particular, in our previous work, we showed how the processing of selected subtasks can be delayed in order to minimise expected subtask dispersion and percentiles of subtask dispersion in the context of split-merge systems. However, the introduction of subtask delays obviously impacts adversely on task response time and maximum sustainable system throughput. In the present work, we describe a methodology for managing the trade off between subtask dispersion and task response time. The objective function of the minimisation is based on the product of expected subtask dispersion and expected task response time. Compared with our previous methodology, we show how our new technique can achieve comparable subtask dispersion with substantial improvements in expected task response time.

Abstract. Fork-join and split-merge queueing systems are well-known abstractions of parallel systems in which each incoming task splits into subtasks that are processed by a set of parallel servers. A task exits the system when all of its subtasks have completed service. Two key metrics of interest in such systems are task response time and subtask dispersion. This paper presents a technique applicable to a class of fork-join systems with heterogeneous exponentially distributed service times that is able to reduce subtask dispersion with only a marginal increase in task response time. Achieving this is challenging since the unsynchronised operation of fork-join systems naturally militates against low subtask dispersion. Our approach builds on our earlier research examining subtask dispersion and response time in split-merge systems, and involves the frequent application and updating of delays to the subtasks at the head of the parallel service queues. Numerical results show the ability to reduce dispersion in fork-join systems to levels comparable with or below that observed in all varieties of split-merge systems while retaining the response time and throughput benefits of a fork-join system.

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