Dynamic Analysis of the M/G/1 Stochastic Clearing Queueing Model in a Three-Phase Environment

Abstract

In this paper, we consider the M/G/1 stochastic clearing queueing model in a three-phase environment, which is described by integro-partial differential equations (IPDEs). Our first result is semigroup well-posedness for the dynamic system. Utilizing a C0—semigroup theory, we prove that the system has a unique positive time-dependent solution (TDS) that satisfies the probability condition. As our second result, we prove that the TDS of the system strongly converges to its steady-state solution (SSS) if the service rates of the servers are constants. For this asymptotic behavior, we analyze the spectrum of the system operator associated with the system. Additionally, the stability of the semigroup generated by the system operator is also discussed.