Optimization Analysis Of Multi-Server Repairable System

The repairable system with finite customers resource which may be called machine repairable system is an important direction of research, it has been widely used in communication systems, transport system and computer memory system and so on. It is significant to study the systems whether in queuing theory or reliability theory. It has broad application prospects.The M/M/R machine-repair system with balking, reneging and server breakdowns is studied in this paper. The systems that the service stations have the fixed service rate and service stations have the variable service rate are considered, respectively. They are expansions of other models in the previous documents. In the model, the number of repairmen is positive integer and may less than the number of the service stations. Therefore, the model of this paper reflects the actual system characteristics and is more general.First, the system that the service stations have the fixed service rate is studied. By the Markov process method, the steady-state probability equations are developed. By using the matrix geometric solution, we get the matrix form solution of the steady-state probabilities. In addition, we also obtain the explicit expression of system performance measures such as the average number of busy servers, the average number of idle servers, the average balking rate, the average reneging rate and the average rate of the customer loss. At the end of this part, we develop a cost model and analyze the influence of the parameters of the system to the optimal cost control and critical value.Second, the system that the service stations have the variable service rate is discussed. By the Markov process method, the steady-state probability equations are developed. By rewriting the transition rate matrix as a blocked one, the matrix form solution of the steady-state probability is obtained by using the inverse of the block matrix. In addition, we get the explicit expression of system performance measures and develop a cost model, the numerical example analysis is also presented.