Asymptotic Analysis of a Closed Queueing Network in a Markov Random Environment

The primary purpose of this paper is an asymptotic analysis of a closed homogeneous exponential queueing network in a Markov random environment. The zero node acts as a generator of finite arrivals to the network and is loaded with departures from the network nodes. The arrival rate depends on the state of the environment. The Markov random environment is defined by an infinitesimal generator matrix on a finite set of states. The asymptotic analysis is performed under the critical assumption that a large number of customers circulating in the network. The mathematical approach used is based on the approximation of a discrete random process describing the state of the environment and the network by a mixed discrete-continuous process. The theorem is formulated, and it is proved that the conditional probability density function of a mixed process satisfies the generalized Fokker–Planck–Kolmogorov equation; the precision of the asymptotic approximation is indicated. An analytical form of its drift and diffusion coefficients is found. The ordinary differential equations are given for the first- and second-order moments of the number of customers at network nodes. The presented asymptotic method makes it possible to calculate the main statistical characteristics of queue lengths in the network both in steady state and, importantly, in transient state. The study of a closed queueing network in a Markov random environment significantly expands the scope of practical application of network queueing models.