Problems of Information Transmission, Vol. 39, No. 3, 2003, pp. 299–308. Translated from Problemy Peredachi Informatsii, No. 3, 2003, pp. 77–86.
Original Russian Text Copyright
2003 by Kotsyuruba, Nazarov.
COMMUNICATION NETWORK THEORY
Analysis of Asymptotic Average Characteristics
for Non-Markovian Models of
Unstable Random Access Networks
P. I. Kotsyuruba and A. A. Nazarov
Tomsk State University
Received November 12, 2002; in ﬁnal form, March 24, 2003
Abstract—A mathematical non-Markovian model of a random access communication network
is investigated asymptotically under the condition of a large delay. An ordinary diﬀerential
equation is obtained, which determines the average number of calls in the source of repeat calls.
Various expressions for it are found in the form of implicit functions, depending on relations
between network parameters.
Random access networks constitute a vast majority among local computation networks due to a
large number of their advantages: high transmission rate, good range, error resistance, possibility
to transmit diﬀerent types of information, low relative cost, etc. However, these networks have a
number of drawbacks, the main of them being network operation instability, inherent in all types
of random access networks.
One of the most important characteristics of data transmission networks is the delay required
for transmitting a message from a source to destination, which, in random access networks, is not
regular and is therefore studied by methods of probability theory and theory of random processes.
The main methodological basis for the analysis of random access networks is queueing the-
ory . However, as is mentioned in this monograph, application of queueing theory often requires
simpliﬁed assumptions since, unfortunately, more realistic assumptions make an appropriate anal-
ysis extremely complicated. For this reason, precise quantitative estimation of the delay based on
queueing theory models is impossible.
Here, we have the following situation. Classical queueing models, for which analytical Erlang
and Pollaczek–Khinchin formulas are obtained, priority systems are far from real-world random
access networks. More adequate models are queueing systems (QS) with a source of repeat calls
(SRC) but, as was already mentioned, analysis of such models is extremely complicated and is
therefore implemented by either graphical methods on the level of average characteristics or im-
itation simulation, which gives numerical results only and does not reﬂect a qualitative picture.
At the same time, SRC systems have very strange properties and a large number of nodes. These
properties become rather clear in the analytical examination of QSs considered.
In the present paper, a non-Markovian model is constructed, in the form of a QS with a source
of repeat calls, of a random access network with channel reservation and collision warning that
corresponds to the original version of the 802.3 IEEE standard for media access control, which
is called the carrier sense multiple access with collision detection (CSMA/CD) and, in essence,
coincides with the access method in the Ethernet network.
2003 MAIK “Nauka/Interperiodica”