Three m-failure group maintenance models for M/M/N unreliable queuing service systems

This paper considers group maintenance problems for an unreliable service system with N independent operating servers and a Markovian queue. A specific class of group maintenance policies is developed where the repair is started as soon as the number of failed servers reaches a predetermined threshold. This is actually a Quasi Birth-and-Death Process with two dimensions, the level for the arrival/service process and the phase for the failure/repair process. Two models with positive repair time and another with instantaneous repair are considered. The matrix geometric approach is applied to calculate the steady state distribution and the expected average cost for all three models. For the theoretical analysis, this paper proves that there exists an optimal group maintenance parameter m*, which can find the minimal average cost for all three models. Additionally, some mathematical properties and sensitivity analyses are numerically demonstrated based on various parameters. Finally, the comparisons of these three proposed models in many aspects are also discussed.

► Three group maintenance models are developed for M/M/N unreliable service systems.
► The expected number of customers is obtained by applying matrix geometric method.
► Three m-failure group maintenance policies are proved by mathematical analysis.
► Numerical results also demonstrate the properties of the above theoretical results.