Asymptotic stability of the M/G/1 queueing system with optional second service

An M/G/1 queueing system with second optional service is considered in this paper. We are devoted to studying the asymptotic stability of this kind of system by using C0C0-semigroup theory. By analyzing the spectral distribution of the system operator, we derive that 0 is an eigenvalue and is the only spectral point on the imaginary axis. It shows that the time-dependent solution of the system converges to the steady-state solution as time approaches infinity. Using the steady-state solution, we obtain the mean queue length.