ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 4, pp. 400–405.
Pleiades Publishing, Inc., 2009.
Original Russian Text
G.Sh. Tsitsiashvili, M.A. Osipova, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 4, pp. 115–120.
COMMUNICATION NETWORK THEORY
Parameter Estimation for Product-Form
Distributions of Queueing Networks
G. Sh. Tsitsiashvili and M. A. Osipova
Institute of Applied Mathematics, Far East Branch of the RAS, Vladivostok
Received March 17, 2009; in ﬁnal form, August 4, 2009
Abstract—Basic parameters of a queueing network are its routing matrix, arrival ﬂow rate,
and service rates at network nodes. To estimate these parameters, one has to solve a system
of balance equations. In turn, a product-form limiting distribution of the number of customers
at the network nodes is deﬁned through loading factors. Therefore, in the paper we propose to
estimate loading factors through estimates of the limiting distribution based on observations
of the number of customers at the nodes. This makes it possible to avoid solving a system
of balance equations. This algorithm is realized for Jackson networks: classical, in a random
environment, with blocked transitions.
Queueing networks are widely used in modeling communication and data transmission systems,
data processing systems, etc. [1–4]. The main mathematical tool for modeling such systems is
product-form theorems based on solving a system of balance equations. At the same time, there is
a large number of papers devoted to estimation of parameters of particular queueing systems [5–8].
However, it is quite diﬃcult to transfer these methods to estimation of parameters of queueing
networks whose modeling is based on product-form theorems.
The problem of estimating network parameters—arrival ﬂow rate, service rates at network nodes,
and routing matrix—is in a sense inverse to the modeling problem. At ﬁrst sight, it looks more
complicated than the direct problem and should include a solution of balance equations with an
indeterminate routing matrix.
Product-form limiting distribution of the number of customers at network nodes is directly
expressed through loading factors of the nodes (and some normalizing factor) but not through the
routing matrix, arrival ﬂow rate, and service rates at the nodes. Therefore, ﬁrst, it is reasonable to
concentrate upon estimating the parameters that are directly involved in the product-form limiting
distribution. Second, one has to select formulas that express these parameters through the limiting
distribution at some points. In turn, the limiting distribution at the selected points should be
approximately estimated via observations of the number of customers at the network nodes during
a large enough time interval. The quality of such approximation is characterized by well-known
ergodic theorems and the law of large numbers for Markov processes.
Thus, the proposed estimation procedure for network parameters does not involve solution of a
system of balance equations, which in a number of cases signiﬁcantly complicates computations.
In the present paper, this algorithm is realized for open and closed Jackson networks, networks in
a random environment, and networks with blocked transitions.
Supported in part by the Far East Branch of the Russian Academy of Sciences, project nos. 09-1-P2-07