Multi-Server M/M/c Queue and Multiple Working Vacation under Phase Repair

This investigation aims to analysis multi-server queuing system (MSQS) with server breakdown wherein two phase of essential repair is provided to failed items. Moreover, the servers may choose multiple working vacations (MWV) in case there are no customers available in the system. After coming back from their first vacation, if the servers find that the customer is waiting in the system, they immediately start providing services to the waiting customers. Otherwise, they can continue their vacation. The customer joins the system according to the Poisson distribution. The life-time, service time and vacations time follow an exponential distribution. Further, Runge-Kutta (RK) method has been applied for solving the time-dependent probability. In addition, numerical results are also provided. At last, the conclusion is given.