Queueing system with general operation and vacation time

An M/M/1 queueing system with general operation and vacation time is considered. The server is nonexhaustive, that is to say, server may be in vacation when the system is not empty. The probability generating function of queue length under stability is obtained by supplementing variables, where p⁰(x) may be found numerically by solving first kind Fredholm integral equation. At last, the expected value of queue length is obtained.