ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 4, pp. 371–378.
Pleiades Publishing, Inc., 2006.
Original Russian Text
A.N. Starovoitov, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 4, pp. 121–128.
COMMUNICATION NETWORK THEORY
Invariance of the Stationary Distribution of States
of Multimode Service Policy Networks
A. N. Starovoitov
Francysk Skaryna Gomel State University
Received July 26, 2005; in ﬁnal form, April 3, 2006
Abstract—We consider open and closed preemptive-resume queueing systems with absolute
priority of incoming customers. Single-server nodes have several service modes (regimes); the
time of switching between the modes is exponential. Switching can be made to adjacent modes
only. The amount of work required for servicing an incoming customer (workload) is a random
variable with an arbitrary distribution function. For an open network, the input ﬂow is Poisso-
nian. We prove that the stationary distribution of the network states is invariant with respect
to a functional form of workload distributions if the ﬁrst moments are ﬁxed.
In the study of queueing networks, an important role is played by the problem of invariance of
the stationary probability distribution of states with respect to a functional form of distributions
of workload required for servicing customers at nodes. This is due to the fact that in real-world
networks this distribution is most often not exponential. In , ﬁrst signiﬁcant results in this direc-
tion were obtained; namely, it was proved for an M|G|m|0 system that the stationary probability
distribution of states is independent of the service length distribution function.
Appropriateness of studying networks with unreliable servers was many times pointed out in the
literature; however, it is rather complicated to ﬁnd an invariant measure for such networks. For
instance, in [2, pp. 288–294], Markov networks with unreliable service lines were considered, and
main asymptotic terms of the stationary distribution of states for such networks were found.
In [3–5], networks were studied where servers may partially fail and pass to operation in a
partial load mode. However, in these papers it was assumed that service lengths are exponentially
distributed. In the present paper we consider similar networks where service lengths have an
arbitrary distribution law, and service policies at the nodes have the following property. A customer
arriving at a node has absolute priority among all other customers at the node. A displaced
customer is wedged in the head of the queue, moving all customers in the queue; when a customer
accesses the server again, it gets resumed service in the same mode in which the server was at that
moment. We prove that the stationary distribution of states of such networks does not depend on
distributions of workload required for servicing customers at the nodes if the ﬁrst moments of these
distributions are ﬁxed.
2. PROBLEM SETTING
We consider a queueing network consisting of N single-server nodes. In the case of an open
network, input ﬂow is Poissonian with rate λ, and each incoming customer independently of others
goes with probability π
to the th node ( = 1,N;
= 1). In the case of a closed network,