Abstract
This paper investigates the performance of a queueing model with multiple finite queues and a single server. Departures from the queues are synchronised or coupled which means that a service completion leads to a departure in every queue and that service is temporarily interrupted whenever any of the queues is empty. We focus on the numerical analysis of this queueing model in a Markovian setting: the arrivals in the different queues constitute Poisson processes and the service times are exponentially distributed. Taking into account the state space explosion problem associated with multidimensional Markov processes, we calculate the terms in the series expansion in the service rate of the stationary distribution of the Markov chain as well as various performance measures when the system is (i) overloaded and (ii) under intermediate load. Our numerical results reveal that, by calculating the series expansions of performance measures around a few service rates, we get accurate estimates of various performance measures once the load is above 40% to 50%.
Highlights
Numerical methods for queueing systems involving multiple queues like queueing networks [1], polling systems [2], priority queues [3], and fork-join queues [4] often suffer from the state space explosion problem
State space explosion refers to the problem that multidimensionality of Markov processes leads to processes with a very large state space
In a coupled queueing system with nonequal arrival loads, the performance is mainly determined by the queues with the lowest loads, we first focus on a coupled queueing system with an equal arrival rate λ in all queues
Summary
Numerical methods for queueing systems involving multiple queues like queueing networks [1], polling systems [2], priority queues [3], and fork-join queues [4] often suffer from the state space explosion problem. Matrix-analytic methods for neither M/G/1-type nor G/M/1-type queueing systems apply, and there is no product form solution To overcome these challenges, literature proposes two alternative approaches, both focusing on approximations for various performance measures of the coupled queueing system. The system can be studied approximately by means of series expansion techniques if one limits the study to a subset of the parameter space This is the case in [9, 10] where the coupled queueing system was studied in overload. In these papers it was shown that the terms of the Maclaurin series expansion of the steady-state distribution in the service rate can be obtained at low computational cost.
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