Asymptotic behavior of the solution of a stochastic clearing queueing system

In this paper, we investigate the asymptotic behavior of the time-dependent solution for the M/G/1 stochastic clearing queueing system operating in a three-phase environment. The mathematical model of this system is characterized by an infinite set of integro-partial differential equations, with boundary conditions that incorporate integral equations. Initially, we employ probability generating functions to demonstrate that 0 is an eigenvalue of the system operator, possessing a geometric multiplicity of one. Subsequently, by invoking Greiner's boundary perturbation method, we establish that all points on the imaginary axis, with the exception of 0, reside within the resolvent set of the system operator. Furthermore, we highlight that 0 also serves as an eigenvalue of the adjoint operator of the system operator, with a geometric multiplicity of unity. As a result, we conclude that the time-dependent solution of the system converges strongly to its steady-state solution.