ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 4, pp. 385–394.
Pleiades Publishing, Inc., 2008.
Original Russian Text
G.Sh. Tsitsiashvili, M.A. Osipova, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 4, pp. 109–119.
COMMUNICATION NETWORK THEORY
Limiting Distributions in Queueing Networks
with Unreliable Elements
G. Sh. Tsitsiashvili and M. A. Osipova
Institute of Applied Mathematics, Far East Branch of the RAS, Vladivostok
Received April 4, 2006; in ﬁnal form, May 23, 2008
Abstract—We consider multichannel systems and open queueing networks with unreliable
elements: nodes, paths between nodes, and channels at nodes. Computation of limiting distri-
butions in a product form for these models is based on choosing recovery schemes for unreliable
elements (independent recovery, recovery at a single site, recovering network scheme), routing
algorithms, and service disciplines. Thus, by introducing a certain control, we constructively
relate queueing theory with reliability theory. Results of the paper can be transferred to closed
networks almost without changes.
We consider models of queueing networks with failure-resilient (recoverable) elements: nodes,
paths between nodes, and channels at nodes. The background for modeling queueing networks
with unreliable elements is a large body of the authors’ observations of traﬃc jam appearances,
cancellations in suburban train schedules, and ATM failures.
Queueing models with unreliable elements are of considerable interest in the analysis and design
of telecommunication networks [1–5]. The matter concerns service quality deterioration, length of
oﬀ periods (idle time), propagation of failure eﬀects. The above-mentioned papers mainly consider
systems but not queueing networks, which is, apparently, due to the complexity of computations.
In the present paper we construct models for failure and recovery of elements of queueing
networks that can be computed using product theorems, which considerably simplify computations.
To this end, we introduce special control in models of queueing networks with unreliable elements.
We consider various recovery schemes for unreliable elements (independent recovery, recovery at a
single site, recovering network scheme), various routing algorithms, and service disciplines.
2.1. Product Theorem
Let S and J denote ﬁnite or countable sets (they can be miltidimensional); deﬁne the set
X = S × J. Consider a Markov process x(t) with state set X and with transition rates (s, s
Λ((s, n), (s
)) = L
)I(s = s
)I(n = n
satisfying the following conditions (I(A) is the indicator function of an event A):
Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00063a, and the Far
East Branch of the Russian Academy of Sciences, project no. 06-III-A-01-016.