Problems of Information Transmission, Vol. 40, No. 3, 2004, pp. 243–253. Translated from Problemy Peredachi Informatsii, No. 3, 2004, pp. 69–80.
Original Russian Text Copyright
2004 by Kuznetsov, Nazarov.
COMMUNICATION NETWORK THEORY
Analysis of a Communication Network Governed by
an Adaptive Random Multiple Access Protocol
in Critical Load
D. Yu. Kuznetsov
and A. A. Nazarov
Tomsk Polytechnic University
Tomsk State University
Received August 11, 2003; in ﬁnal form, April 13, 2004
Abstract—A mathematical model of an adaptive random multiple access communication net-
work is investigated. The value of the network critical load is found; in the critical load,
asymptotic probability distributions for states of the information transmission channel and for
the number of requests in the source of repeated calls are found. It is proved that distributions
of the normalized number of requests belong to the class of normal and exponential distribu-
tions, and it is shown how conditional normal distributions pass in the limit to the class of
To provide reliable data transmission in telecommunication networks, mathematical modeling
is required with the purpose of ﬁnding optimum characteristics of transmitting facilities and de-
signing algorithms guaranteeing message delivery in reasonable time. This problem is of most
importance for random access networks, which operate rather irregularly in high load of communi-
For stabilizing such communication networks, various modiﬁcations of random access protocols
are suggested [1–4]. Thus, in [1, 2], respectively, the asynchronous and synchronous models of a
dynamic random multiple access network are investigated. Dynamic protocols give eﬃcient results
in stabilizing a communication network but are not realizable technically since each user must
have information on the total number of retrial requests. In , a model of a synchronous random
multiple access system with the part-and-try protocol is considered. Another approach to stabi-
lization of random multiple access communication networks is realized by adaptive protocols [5–8].
Thus,  considers a model of a communication network similar to that described in the present
paper; however, in  only one network characteristic, the capacity, is considered.
An eﬃcient instrument of mathematical modeling for such communication networks is the queue-
ing theory apparatus. With the help of it, analytical models of data transmission networks are
constructed. Here, in contrast to classical queueing models, one has to consider retrial queueing
systems . Developing methods for analysis of retrial queues requires attention since such systems
are at present widely used in modeling.
One of such methods is the asymptotic analysis method . Using this method, in [6,7] two
models of adaptive random multiple access networks were analyzed: in overload ; when load ρ of
a communication channel is larger than some critical value S, for a network with a ﬁnite number, N,
of users; and in large load , when ρ tends to S from below, with inﬁnitely many users. Note that
in the latter case S has the sense of the capacity of the communication network.
2004 MAIK “Nauka/Interperiodica”