ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 1, pp. 46–56.
Pleiades Publishing, Inc., 2011.
Original Russian Text
A.N. Starovoitov, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 1, pp. 54–65.
COMMUNICATION NETWORK THEORY
On Conditions for a Product-Form Stationary
Probability Distribution of States
of a Multimode Service Network
A. N. Starovoitov
Belarusian State University of Transport, Gomel, Belarus
Received November 10, 2009; in ﬁnal form, November 2, 2010
Abstract—We consider open and closed queueing networks with Markovian routing and sym-
metric service policies. Single-server nodes may operate in several modes. In each node, the
amount of work required for servicing an arrival or for switching the mode is distributed arbi-
trarily. The performance rate for each of these operations depends on the residual amount of
work. For open networks, the arrival ﬂow is Poissonian. We establish conditions for a product-
form stationary distribution of states of the piecewise continuous process that describes the
One of methods for analyzing non-Markovian queueing networks is the phase space extension
method (extra variables method). According to this method, the process describing the network
behavior is augmented by extra components such that the resulting process is Markovian. In the
case where the amount of work required to service a customer has an arbitrary distribution, residual
amounts of work are taken as such extra variables. Depending on the form of these extra variables,
piecewise linear and piecewise continuous processes  are distinguished. Namely, a process is said
to be piecewise linear if the extra variables decay linearly and is said to be piecewise continuous
Multimode service networks were introduced in [2,3]. In [4–6], multimode networks with various
symmetric service policies were studied, where the network operation is described by a piecewise
linear process. In the present paper, we consider multimode networks with arbitrarily distributed
amount of work required for servicing a customer and for switching server modes. Rates of these
operations depend on the residual workload of servers. This dependence is described by a continuous
function. Thus, the process describing the network operation is piecewise continuous.
For this class of networks, we ﬁnd a stationary probability distribution of states of this piecewise
continuous process. As a consequence, we ﬁnd a stationary distribution of the “main” (non-Marko-
vian) process and prove its invariance.
Results of this paper were announced in .
2. PROBLEM SETTING
We consider a queueing network consisting of N single-server nodes. In the case of an open
network, the arrival ﬂow is Poissonian with intensity λ, and each arrival independently of the
others is sent to the th node with probability π
= 1). In the case of a closed