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Stochastic processes and queueing theory
Stability of infinite closed Jackson networks
Dmitri Khmelev
Moscow State University, Moscow, Russia
Eugene Spodarev
Friedrich Schiller University, Jena, Germany
Consider the following particle system: a countable number of parti­
cles circulates in the network \Lambda of nodes being controlled by a routing
matrix P = fp xy g x;y2\Lambda : if there are some particles in the node x 2 \Lambda
then one of them (particles do not have their individuality) instantly
moves to a node y 2 \Lambda with probability p xy at the moment of time
distributed exponentially with parameter fl(x); fl =
P
x2\Lambda
fl(x) Ÿ 1.
There is a finite number of particles in each node at the initial moment
t = 0. The system informally described above is a generalisation of
some particle systems introduced by Spitzer and then considered e.g.
by Liggett et al. It can be also viewed as a closed Jackson­type network
with infinitely many customers inside that connects this problem to the
queueing theory. Under certain conditions on P and fl(x) there exists
the strongly continious Markov process
\Phi
X(t)
\Psi
t–0
=
\Phi
(X y (t)) y2\Lambda
\Psi
t–0
on the state space
\Gamma
Z \Lambda
+ ; B \Lambda
\Delta
which describes the system: for any y 2 \Lambda
X y (t) is a number of particles in y at time t (Z + being the set of all
n – 0, n 2 Z). One can say that the system is non­ergodic if there
does not exist such unique stationary measure ¯ that for any distribu­
tion of X(0) the measures correspondent to X(t) weakly converge to
¯ as t ! 1. In this report the long­time behaviour of the system is
studied: cases of non­ergodicity are explored, the class of stationary
measures is described. Some further developments are made in the

case fl ! 1: then one can consider an embedded Markov chain that
facilitates the proofs. Here the results about the stochastical bound­
edness of X y (t) for a concrete y 2 \Lambda and the existence of a functional
of X(t) that appears to be a supermartingale are obtained.
[ Dmitri Khmelev, Moscow State University, Faculty of Mathematics
and Mechanics, Chair of Probability Theory, 117234 Moscow, Russia;
dima@vvv.srcc.msu.su]
[ Eugene Spodarev, Friedrich­Schiller Universit¨at, Institut f¨ur Stochastik,
Ernst Abbe Platz 1­4, Jena 07743, Deutschland; seu@minet.uni­jena.de]
http://www.minet.uni­jena.de/~seu/