Probabilistic decomposition method on the serverindices of an \(\mathrm{M}^\xi/\mathrm{G}/1\) vacation queue (Q1714536)

Summary: This paper develops a probabilistic decomposition method for an \(\mathrm{M}^\xi/\mathrm{G}/1\) repairable queueing system with multiple vacations, in which the customers who arrive during server vacations enter the system with probability \(p\). Such a novel method is used to analyze the main performance indices of the server, such as the unavailability and the mean failure number during \((0,t]\). It is derived that the structures of server indices are two convolution equations. Further, comparisons with existing methods indicate that our method is effective and applicable for studying server performances in single-server \(\mathrm{M}^\xi/\mathrm{G}/1\) vacation queues and their complex variants. Finally, a stochastic order and production system with a multipurpose production facility is numerically presented for illustrative purpose.