Uunreliable multi-server machine repairable system with variable breakdown rates

We consider the machine repair system in which K identical unreliable machines are maintained by N servers. Each server is subject to breakdown even if no failed machines are in the system. There are c reliable repairmen in the system to repair the breakdown servers. Failure and service times of the machines, and breakdown and repair times of the servers, are assumed to follow a negative exponential distribution. Each unreliable server has different breakdown rates in its busy times and idle times. The system performance measures are obtained by a matrix geometric method. Some important relationships between the system performance measures are given. An optimization model is developed for the optimum numbers of servers and repairmen to minimize the expected total cost of the system. A numerical example is given for illustration.