A study on \(M^x/G(a,b)/1\) queue with server breakdown without interruption and controllable arrivals during multiple adaptive vacations (Q2205080)

Summary: A single server model, after completion of a bulk service, if there is no breakdown with probability \((1-\psi\)) and queue length (queue) \(\geq a\), then the bulk service continues, otherwise, the server performs closedown work is analysed. At the end of bulk service, if there is a breakdown occurs with probability (\(\psi)\), then the server performs renovation process. After that, if the queue is \(\geq a\), then he performs bulk service otherwise the server perform closedown work follows a vacation. After that, if the queue is less than \(a\), then he takes at most \(M\) vacations successively. After \(M\) vacations, if the queue is still less than \(a\), then he remains in the system. However, the customers enter the service station with probability \(p\) (\(0 \leq p \leq 1\)) during multiple adaptive vacations. The probability generating function (PGF) of queue size and performance measures are obtained and cost model is developed.