Queue with Two Types of Service Subject to Random Breakdowns, Multiple Vacation and Restricted Admissibility

the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We consider an M [x] /G/1 queue with two types of service subject to random breakdowns having multiple server vacation, where the customers arrive to the system in batches of variable size. We assume that a single server provides two types of service, type 1 with probability p 1 and type 2 with probability p 2 with the service times following general distribution and each arriving customer may choose either type of service. The server takes vacation each time the system becomes empty and the vacation period is assumed to be general. On returning from vacation if the server finds no customer waiting in the system, then the server again goes for vacation until he finds at least one customer in the system. The system may breakdown at random and repair time follow exponential distribution. In addition we assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times. The probability generating function for the number of customers in the queue is found using the supplementary variable technique. The mean number of customers in the queue and the system are also found.