M^{[x]}/G/1 queue with two types of service subject to random breakdowns, multiple vacation and restricted admissibility

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 p1 and type 2 with probability p2 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