Reliability Analysis of Repairable MX/G/1 Queue with Multiphase Optional Services and General Setup Time

Many queueing systems are characterized by the feature that all arrivals demand the first essential service, whereas only some of them demand second optional service which is provided by the same server. In this work, we study a single non reliable server Mx/G/1 queue with multiphase optional services. The customers arrive in batches according to a Poisson process. Two types of services are provided to the customers, the first “essential” service and second multiphase “optional” service. After the completion of the essential service, the customer either leaves the system with probability (1-r1) or join the first optional service with probability r1; again after completing the first phase optional service, either he leaves or joins second phase of optional service with probability r2 and similarly in continuation at the end of (k-1)th phase optional service, he may opt kth phase of optional service with probability rk or may leave the system with probability (1-rk). Both essential and optional services are provided by same single server. While the server is working, he is subject to breakdown according to Poisson process. When the server breaks down, he requires repair at repair facility where a repairman renders repair of failed server according to general distribution. By introducing supplementary variable technique and generating function method, some queueing and reliability characteristics of the system are derived. We facilitate numerical results to illustrate the effect of different parameters on several performance indices. Key-words: Batch arrivals, Unreliable server, Setup time, Multiphase optional service, Supplementary variable, Generating function, Queue size, Reliability.