Markov-chain model and algorithmic procedure for the performance analysis of closed cyclic queues

This paper presents a model and an algorithmic procedure to analyze closed cyclic queues that are subject to blocking. We consider the first two moments of the processing time and present the fitting of phase-type distributions such that the number of phases and transitions is minimal. Using phase-type distributions, we enable the analysis of queueing systems with processing times with any coefficient of variation. We model the closed cyclic queues subject to blocking as continuous-time Markov chains. The implementation procedure covers the state-space generation and the determination of the infinitesimal generator matrix. Apart from rounding errors, we obtain exact results for the queueing model this way. The results are useful as reference values for the output of approximate approaches. Further, the algorithmic procedure enables a repeated analysis of different configuration alternatives as needed in optimization procedures. Though the method is very fast for small cyclic queues, it takes a long computation time for larger systems. Furthermore, the size of the queueing model to be analyzed is restricted due to the limited working memory with its present-day capacity. In a numerical study, the computation times for different configurations are investigated, limits in the size of the applicable queueing model are given, and numerical results of the performance measures are provided.