Analysis Of Operators Of Queueing System

Queueing systems are the most important systems discussed in the operations research, and also the primary subject studied in applied mathematics. This paper mainly investigates the exponential stability of the M/G/1queueing system with additional optional service and no waiting capacity, which are formulated by supplementary variable technique. It is significant for the analysis of the systems. Many scholars have done a lot of work about the steady state solution and asymptotic stability of the systems. However, the exponential stability of systems is not well solved.In this paper, first of all, under some reasonable assumptions, the system can be written as Abstract Cauchy Problem in Banach space.Then we discuss the properties of the main operator of the system. And estimate the upper spectral bound of the main operator. We show that the main operator of the system generates a Co-semigroup S(t) and proof that the growth bound of the semigroup is equal to upper spectral bound of the main operator using the concept of cofinal and relative theory. Then we prove the existence and uniqueness of non-negative time-dependent solution of the system.At last, we discuss the spectral distribution of the system operator. Along with the growth bound of the two semigroup of operators and perturbation theory for operators, we prove the solution of the system is exponentially stable.