We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

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Accepted versio

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.

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