Problems of Information Transmission, Vol. 38, No. 3, 2002, pp. 227–236. Translated from Problemy Peredachi Informatsii, No. 3, 2002, pp. 72–82.
Original Russian Text Copyright
2002 by Tyurikov.
COMMUNICATION NETWORK THEORY
Multiplicativity of Markov Chains
with Multiaddress Routing
M. Yu. Tyurikov
Received October 27, 2000; in ﬁnal form, January 17, 2002
Abstract—A broad class of network Markov processes (including open queueing networks)
with multiaddress routing and one type of calls is considered. Under such routing, the same
call can simultaneously arrive at several nodes. For these processes, we found necessary and
suﬃcient conditions of multiplicativity, that is, conditions of representability of a stationary
distribution as a product of factors characterizing separate nodes.
A multiplicativity criterion of network Markov processes with one type of calls was obtained
in  (see also [2, 3]). The present paper extends the criterion of  to multiaddress networks.
Although we use the terminology of queueing network theory, the result obtained can be applied
to multidimensional Markov processes. The presented way of network description allows us to
interpret a call as an arbitrary action that a node applies to other nodes.
We use a general description of network nodes (see [1–4]) in terms of transition intensities,
which are divided into three components that correspond to node events. According to concepts of
Markov processes, there are three types of events: “arrival,” “departure,” and “internal transition.”
In the case of traditional routing, a call either passes from one node to another (“departure”
from a node results in an “arrival” at another node) or leaves the network (“departure” from a
node has no inﬂuence on other nodes). In multiaddress routing, a departing call is addressed to
all nodes of a certain set (maybe, empty); that is, “departure” from a node results in “arrivals”
at all nodes belonging to the set. As an example of multiaddress routing, one may consider e-mail
service, where the same message can simultaneously be directed to several recipients.
As in [1–3], we investigate the interrelation between a stationary distribution of a network and
those of isolated nodes. In addition, we analyze the relation between partial balances of isolated
nodes and the multiplicativity property.
In multiaddress routing, conditions of node quasi-reversibility [4–15] and reversibility of random
processes [16, 17] are not so important as in traditional networks.
2. DEFINITION OF A NETWORK WITH MULTIADDRESS ROUTING
We consider open stochastic networks with a ﬁnite set of nodes M.Anexternalsourceis
presented by node 0, 0 ∈ M .Theith node is given by a state x
from a countable state space X
The network is described by a continuous-time Markov chain with state space X =
Strictly speaking, instead of open networks, we investigate networks whose stationary distribu-
tion is positive at each point of the space X.
Anodei is deﬁned by the set
2002 MAIK “Nauka/Interperiodica”