Stability analysis to an M/G/1 queueing system with additional optional services and no waiting capacity

Asymptotic behavior of an M/G/1 queueing system with additional optional service and no waiting capacity is studied. By normalizing the system which is described by differential equations, we transform the system to an abstract Cauchy problem in a Banach space. Then, by the theory of bounded linear operator semigroup, we show that the unique and nonnegative stability solution of system is the eigenvector of system operator corresponding to eigenvalue 0. By spectral analysis of the system, we find that 0 is the unique spectral point of system on the imaginary axis. As a result, the asymptotic stability of the system is obtained.