Problems of Information Transmission, Vol. 37, No. 2, 2001, pp. 165–171. Translated from Problemy Peredachi Informatsii, No. 2, 2001, pp. 88–95.
Original Russian Text Copyright
2001 by Nazarov, Shokhor.
COMMUNICATION NETWORK THEORY
Ergodicity of Band Graph Markov Chains and Their
Application to Problems of the Analysis of Existence
of a Stationary Regime in a Dynamic Random
Multiple Access Communication Network
A. A. Nazarov and S. L. Shokhor
Received March 16, 2000; in ﬁnal form, October 24, 2000
Abstract—A concept of the class of band graph Markov chains is introduced. Simple er-
godicity criteria for this class of stochastic processes are found. We show how these criteria
can be applied to prove the existence of a stationary operation regime in a dynamic access
communication network with conﬂict warning.
The problem of existence of a stationary regime in a communication network is one of the
main problems in network analysis since ergodicity conditions make it possible to ﬁnd the network
As mathematical models of computer communication networks, usually [1–3] Markov and, in
the more general case, semi-Markov queueing systems are considered.
The main goal of this paper is the analysis of ergodicity of random processes which describe
operation of such systems.
As follows from the Foster theorem , ergodicity of a continuous-time Markov chain is deter-
mined by ergodicity of its embedded chain. A similar statement  holds for semi-Markov processes
as well. Ergodicity properties of a semi-Markov process are completely determined by properties
of its embedded Markov chain provided that mean delays for the states of the semi-Markov process
According to the main limit theorem , for an irreducible aperiodic Markov chain, its ergodicity
follows from recurrence and positivity of states.
As a rule, it is easy to check that a Markov chain is irreducible and aperiodic, whereas verifying
the recurrence and positivity is not so easy. Therefore, other versions of suﬃcient conditions for
ergodicity are proposed, for example, the Foster condition , Mustafa condition , etc. However,
verifying these conditions still makes certain diﬃculties. Therefore, ﬁnding classes for which these
conditions can be reduced to simpler constructive criteria is of interest.
As is shown below, Markov chains with homogeneous band graphs form such a class.
2. MARKOV CHAINS WITH A HOMOGENEOUS BAND GRAPH
Consider a discrete-time Markov chain whose states are given by a two-dimensional vector
(k, i) and the transition probabilities, q
(k, j), from a state (n, i)toastate(k, j), where k takes
values from a ﬁnite set and j is denumerable. For deﬁniteness, we assume k =0, 1,...,K and
j =0, 1, 2,.... We call such a Markov process a band graph Markov chain. For instance, in a
queueing system, the component k may describe a server state and i the number of calls in a buﬀer
or in a source of repeat calls (SRC).
2001 MAIK “Nauka/Interperiodica”