Problems of Information Transmission, Vol. 37, No. 3, 2001, pp. 236–247. Translated from Problemy Peredachi Informatsii, No. 3, 2001, pp. 55–66.
Original Russian Text Copyright
2001 by Evdokimovich, Malinkovskii.
COMMUNICATION NETWORK THEORY
Queueing Networks with Dynamic Routing and
Dynamic Stochastic Bypass of Nodes
V. E. Evdokimovich and Yu. V. Malinkovskii
Received November 22, 2000; in ﬁnal form, March 2, 2001
Abstract—We consider open exponential networks with routing matrices that depend on a
network state. A customer entering a node is either independently of other customers queued
with probability that depends on the network state or instantly bypasses the node with com-
plementary probability. After bypassing a node, customers are routed according to a stochastic
matrix that depends on the network state and may diﬀer from the routing matrix. Under cer-
tain restrictions on parameters of the model, we establish a suﬃcient ergodicity condition and
ﬁnd the ﬁnal stationary distribution.
Models of exponential networks with dynamic service and routing parameters were considered
in [1–4]. In , a network model was introduced where a customer arriving at a node (from
outside or from another node) either joins the queue with probability that depends on the state
of this node or, with complementary probability, instantly bypasses the node and then moves
according to the routing matrix, i.e., in the same way as a customer served at this node. With
appropriate modiﬁcations, this model allows for constraints on the number of customers at a node
or on expected idle times of customers entering a node. It was proved in  that the network
output ﬂows of customers are independent and Poissonic. Various generalizations of this model
were considered in [6–11]. All of these papers were restricted to the following essential assumptions.
First, routing matrix does not depend on the network state. Second, for a customer that is routed
to a node, the probability to join the queue at this node depends on the state of the node only but
does not depend on the state of a network as a whole. Third, the matrix that governs the routing of
customers after bypassing a node coincides with the routing matrix. However, in practice, behavior
of customers that are denied servicing at the moment when they are directed to a node, generally
speaking, may drastically diﬀer from the behavior of customers served at this node and may depend
on the number of customers at the other nodes, i.e., a network state. As a consequence of these
restrictions, stationary distributions in [6–11] have a multiplicative form, where factors depend
on states of particular nodes. Moreover, it was shown in [9, 10] that for some service disciplines
stationary probabilities are not sensitive to distributions of service times with ﬁxed initial moments.
In the present paper, we lift the above-mentioned restrictions. Because of this, we have to
impose some conditions on parameters of the model in order that the stationary distribution has
the assumed form. Formally, this distribution is also a product but factors depend not only on
states of the nodes but on the whole network state.
Note that results of the paper remain valid for a particular case of no instantaneous bypass.
2. SETTING OF THE PROBLEM
The state of a queueing network consisting of N single-server nodes at time t ≥ 0isgivenbya
stochastic vector n(t)=(n
(t)), where n
(t) is the number of customers at the ith
node at time t. For the convenience of description, we denote a network state and its ith coordinate
2001 MAIK “Nauka/Interperiodica”