ISSN 0032-9460, Problems of Information Transmission, 2013, Vol. 49, No. 2, pp. 167–178.
Pleiades Publishing, Inc., 2013.
Original Russian Text
A.A. Nazarov, A.N. Moiseev, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 2, pp. 78–91.
COMMUNICATION NETWORK THEORY
Analysis of an Open Non-Markovian GI − (GI |∞)
Queueing Network with High-Rate
Renewal Arrival Process
A. A. Nazarov and A. N. Moiseev
Tomsk State University
Received August 10, 2012; in ﬁnal form, January 10, 2013
Abstract—We analyze an open non-Markovian queueing network with high-rate renewal ar-
rival process, Markovian routing, arbitrary service policy, and unlimited number of servers at
nodes. We obtain mean values for the number of busy servers at nodes of the queueing network
in question. We show that, under an inﬁnitely increasing arrival rate, the multivariate distri-
bution of the number of busy servers at network nodes can be approximated by a multivariate
normal distribution; we ﬁnd parameters of this distribution.
There are a lot of papers devoted to analysis of queueing networks (see, e.g., survey ). Network
models that were originally formulated for solving telecommunication problems  rather quickly
became popular in analyzing and designing computers and computer networks . This is due to the
fact that a mathematical model in the form of queueing network very well represents information
transmission processes both inside computers and in computer networks. Presently, queueing theory
ﬁnds its application in various areas, both classical for it (such as communication systems ,
information processes inside computers , data transmission in computer networks ) and new:
industrial problems , economical problems , transportation networks  (see also [10, Appendix
by A.A. Zamyatin and V.A. Malyshev]).
A rather complete material on investigations accumulated up to now in queueing theory is
presented in . However, most studies in this area are devoted to networks with Poisson arrivals.
In the present paper, we analyze a queueing network with renewal arrival process under high
arrival rate, recurrent customer servicing, and unlimited number of servers at nodes. Such models
are reasonable to use in analyzing processes in telecommunication networks whose nodes are a
collection of a large number of computing (processing) devices. For example, such nodes can be
computer clusters, multicore supercomputers, or specialized routing nodes with a large number of
2. MATHEMATICAL MODEL OF A QUEUEING NETWORK
Consider an open  queueing network with a high-rate renewal arrival process . The
network has K nodes; each of them is a queueing system with an unlimited number of servers
and an arbitrary service time distribution B
(x) for customers at the kth node, the same for all
customers of this node (k =