ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 1, pp. 86–102.
Pleiades Publishing, Inc., 2010.
Original Russian Text
A.A. Nazarov, E.A. Sudyko, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 1, pp. 94–111.
COMMUNICATION NETWORK THEORY
Method of Asymptotic Semiinvariants for Studying
a Mathematical Model of a Random Access Network
A. A. Nazarov and E. A. Sudyko
Tomsk State University
Received June 25, 2009; in ﬁnal form, December 7, 2009
Abstract—To study a mathematical model of a random access network with a ﬁnite number
of sources, retrials, and a conﬂict warning stage, we propose a method of asymptotic semiin-
variants under a growing number of sources, which allows us to ﬁnd the asymptotic probability
distribution of the number of requests in a retrial pool. We present results of numerical imple-
mentation of a prelimit distribution of the number of requests in the retrial pool. We compare
the prelimit and asymptotic semiinvariants.
There are many papers devoted to the study of random access computer communication net-
works. Thus, papers [1–8] are devoted to the analysis of communication networks and random
However, despite a large number of papers devoted to the study of such models, many problems
remain unsolved and interesting to investigate. Thus, one of the most important characteristics of
a data transmission network is the number of requests in a retrial pool, which determines the delay
required for delivery of a message from its source to destination; this characteristic is irregular in
random access networks.
In real-world systems, there often occurs the eﬀect of retrial attempts to get service, and con-
ﬂicts require consideration of models that are beyond the framework of classical queueing systems.
Therefore, there is growing interest to the analysis of such real-world systems. In this connection,
there appeared a large number of papers devoted to the study of retrial systems (see, e.g., [9–13]).
Also, many papers are devoted to the analysis of random access networks with queues and to
the problem of ﬁnding a suitable request in a queue. For a description of this research direction,
see [14–16], where numerical analysis methods are developed.
An alternative approach is applying the asymptotic analysis method to such systems, which
allows one to ﬁnd the asymptotic probability distribution of the number of requests in a retrial
By the asymptotic analysis method in queueing theory, we call solving equations that describe
certain characteristics of a system under a certain limiting condition; the form of such a condition
should be speciﬁed for particular models and investigation problems.
Supported in part by the Federal Agency of Education of the Russian Federation Analytical Departmental
Target Program “Development of the Scientiﬁc Potential of the Higher School (2009–2010),” project
“Designing Analysis Methods for Non-Markovian Queueing Systems and Their Application to Complex
Economic Systems and Computer Communication Networks.”