NON-STATIONARY DISTRIBUTION OF THE PROBABILITY STATES OF THE MARKOV NETWORK WITH INFINITE-SERVER QUEUING SYSTEMS OPERATING AT HIGH LOAD

Abstract

The object of research is the Markov queuing network with infinite-server queues. The disciplines of the customer’s service in queuing systems (QS) are FIFO (first come – first served), service rates of customers are distributed exponentially with their own rates for each QS in each line of QS. The purpose of the research is to obtain sufficient conditions for representability of non-stationary state probabilities of such a network operating within the heavy-traffic regime in the multiplicative form. In the introduction, the field of applications of Markov networks with infinite-server queues has been described; the relevance of this work has also been indicated; a brief overview of the previous results on this subject has been given. In the main part, the network has been shown; the system of Kolmogorov’s difference-differential equations for the state probabilities of the network conditions has been derived. The main result of this article is as follows, i.e. the multiplicative form of the non-stationary state probabilities of the above-mentioned Markov network operating within the heavy-traffic regime is formulated and proved as a theorem. The obtained results can be used for modeling the behavior of information and computer systems and networks, transportation systems, insurance companies, banking networks and other facilities, the stochastic models which are the queuing networks.