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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%.

Numerical methods for queueing systems involving multiple queues like queueing networks [

This is the case for the queueing system investigated in this paper. We consider a queueing system with

Service-coupled queueing system.

We study the service-coupled queueing system under Markovian assumptions. That is, we assume independent Poisson arrivals to all queues with arrival rates

To overcome these challenges, literature proposes two alternative approaches, both focusing on approximations for various performance measures of the coupled queueing system. The first approach aims at decomposing the queueing system into a number of independent queueing systems which can be analysed in isolation [

Series expansion techniques for Markov chains go by different names in literature, including perturbation techniques, the power series algorithm, and light-traffic approximations. While the naming is not absolute, perturbation methods are mainly motivated by sensitivity analysis of the results with respect to some system parameter. In particular singular perturbations where the perturbation does not preserve the class-structure of the nonperturbed chain have received considerable attention in literature [

The present contribution builds on the results of [

Balancing computational cost and accuracy, we investigate the use of Taylor series expansions to calculate the performance measures for a wider range of the service rate. In contrast to the Maclaurin series expansions in [

For any iterative method, a good initial guess of the solution can reduce the number of required iterations considerably. In the present setting, such an initial guess is available if one considers a sequence of Taylor series expansions around increasing values of the service rate starting at

The remainder of this paper is organised as follows. The model at hand and the numerical evaluation method are described in the next section. We then illustrate our approach by numerical examples in Section

We consider a queueing system with

In view of the Markovian assumptions on both arrival and service processes, the state (in the Markovian sense) of the queueing system is completely described by the numbers of customers in the different queues. That is, the state of the system is described by a vector

Arrival in queue

Departure: when all queues are nonempty (

Given the summary of the possible transitions above, the balance equations of the Markov process are readily retrieved. For

As the system of (

First, when

While the terms in the Maclaurin series expansion can be calculated efficiently, the resulting expansion only converges to the exact solution in a neighbourhood of

Plugging the series expansion (

In contrast to the Maclaurin expansion above, the system of equations (

This iterative approach is computationally feasible as the number of possible transitions from a state is far less than the number of states (the generator matrix is sparse). More precisely, the number of transitions is related to the number of queues such that the numerical complexity of a single iteration for finding the

If

Once the terms in the series expansion are found, we can find approximations for various performance measures. For instance, the

Analogously, let the system content

The effective load is defined as the fraction of time that the server is serving. As the server is serving whenever all queues are nonempty, we find the following

Finally, let the blocking probability be the fraction of customers that cannot be accepted upon arrival in the queueing system. The effective load allows for calculating the blocking probability

We now evaluate our numerical approximation approach by some numerical examples. We focus on the mean and standard deviation of the queue content as well as the blocking probability. Noting that, in a coupled queueing system with nonequal arrival loads, the performance is mainly determined by the queues with the lowest loads (the queues with higher load can be neglected when studying the overall performance), we first focus on a coupled queueing system with an equal arrival rate

Figures

For the coupled queueing system under study with

The effect of increasing

Next, we study an example with nonequal arrival rates at the different queues. In particular, we consider a system with

Figures

As a final example, we assess the impact of the number of queues involved. To this end, we compare the performance of the queueing system with

In this paper we presented a numerical approach for the performance evaluation of coupled queueing systems. The study was motivated by an assembly-like system, where inventory replenishments can be modelled by Poisson processes. The presented method focuses on coupled queueing systems working under intermediate load and builds on a previously designed method for such systems in overload. We showed that the region where an accurate estimation is obtained can be extended to lower loads by iteratively calculating the terms of the Taylor series expansion of the steady-state probability vector.

An important contribution of the study is that the problem is tackled numerically, while existing analysis methods for large-scale queueing systems mainly rely on simulation. We showed that our analysis method allows for performance evaluation under intermediate load, although the specific region of accuracy may vary depending on the system size and structure.

The authors declare that they have no conflicts of interest.