Multi server machine repair problem with unreliable server and two types of spares under asynchronous vacation policy

In many industrial situations, machines are subjected to random breakdowns. In this paper, we consider the asynchronous vacation policy for the multi-server machine repair problem with server breakdown and two types of spares. According to this policy, when a certain number of servers become idle at a service completion instant, the servers will take an asynchronous vacation of random length together. If the servers find no waiting failed unit in the system and the number of servers on vacation is less than the total number of servers, then these servers can take vacation individually and continue taking vacation if the vacation condition is satisfied. Each server is subject to breakdowns in working state only and can be repaired according to exponential service time distribution. Matrix geometric method is used to obtain the queue size distribution. Various performance measures viz. the expected number of the idle servers, busy servers and vacationing servers, etc. are determined. The sensitivity analysis is made to examine the effect of parameters on various performance measures.