A Large Closed Queueing Network in Markov Environment and Its Application

A paper studies a closed queueing network containing a server station and k client stations. The server station is an infinite server queueing system, and client stations are single server queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. The total number of units in the network is N . The expected times between departures in client stations are (Nμj ) . After service completion in the server station a unit is transmitted to the jth client station with probability pj (j = 1, 2, . . . , k), and being processed in the jth client station the unit returns to server station. The network is assumed to be in Markov environment. The Markov environment is defined by initial state, and phase space of dimension d. Then the routing matrix pj as well as transmission rates (which are expressed via parameters of the network) depend on the Markov state of the environment. The paper studies the queue-length processes in client stations of this network, and is aimed to analysis of performance measures associated with this network. The questions risen in this paper have immediate relation to quality control of complex telecommunication networks.