Analysis of a repairable queuing system with server vacations and a spare server

In this paper, a repairable queuing system with server vacations and a spare server is considered, where one server goes on duty, the other one is kept on standby for a vacation. When the server has broken down, and if the repairman is idle, it can be repaired immediately; otherwise, it needs to wait for repairation. Customers have to wait for the end of server vacation or system restoration before the system continues to serve. Using the matrix-geometric solution's method, the existing condition of steady- state equilibrium and the steady-state probability vectors are discussed. Some performance measures of the system and numerical results are obtained.