M/G/1 Repairable Queueing System With Warm Standby Failure And Min (N,V)-Policy Based On Single Server Vacation

In this paper, we consider an M/G/1 repairable queueing system with single server vacation and Min(N,V)- policy, in which the service station may experience two types of breakdowns: One is that the service station may break during the "generalized busy period"of the servers, another is warm standby failure during the period of not providing service for customers. We assumed that the service station cannot be repaired at once under warm standby failure with the consideration of reality. Using the total probability decomposition technique,laplace transform and renewal process theory, we discuss the transient queue length distribution and the steady state queue length in detail, The recursive expression of the laplace transforma-tion for transient queue length distribution and the recursive expression for the steady-state queue length distribution are obtained. Then, the stochastic decomposition property of the steady state queue length distribution is proved. Meanwhile, some special cases are also dis-cussed. Furthermore, employing the standard results in renewal reward process, we get the optimal control policy N* by the numerical example under the constructed cost model. Finally,the reliability indexes of the service station under two types of failure are derived, and some numerical examples are presented.