ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 2, pp. 184–200.
Pleiades Publishing, Inc., 2010.
Original Russian Text
A.M. Turlikov, S.G. Foss, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 2, pp. 91–109.
COMMUNICATION NETWORK THEORY
On Ergodic Algorithms in Random Multiple Access
Systems with “Success-Failure” Feedback
A. M. Turlikov
and S. G. Foss
St. Petersburg State University of Aerospace Instrumentation
Sobolev Institute of Mathematics, Siberian Branch of the RAS, Novosibirsk
Heriot-Watt University, Edinburgh
Received August 27, 2009; in ﬁnal form, February 15, 2010
Abstract—We consider a decentralized multiple access system with a binary “success-failure”
feedback. We introduce a family of algorithms (protocols) called “algorithms with delayed
intervals” and study stability conditions of one of them. Then we discuss some numerical
results and a number of related and interesting problems and hypotheses.
In the late 1970s, Tsybakov and Mikhailov  and Capetanakis  considered a model with
inﬁnite number of users and a single transmission channel which is available to all users and
transmits messages between them. The authors proposed an algorithm that allows to transmit
messages with a ﬁnite mean delay given that the input intensity is below a certain threshold.
The algorithm is based on the use of the so-called ternary feedback. This means that the users
can observe the channel output and distinguish three possible situations, either no transmissions
(“Empty”), or transmission from a single user (“Success”), or a collision of messages from two
or more users (“Conﬂict”). Soon afterwards, following [1, 2], algorithms with binary feedback,
“Empty–Nonempty” and “Conﬂict–Nonconﬂict”, were introduced and studied.
Performance of the model with “Success–Nonsuccess” (S–NS) feedback is less deﬁnite. Here a
user cannot distinguish collisions and empty slots. Several algorithms that were proposed in [3–5]
could guarantee a stable behaviour of the system only given that certain model extensions are made,
like introducing a special testing ﬁle, etc. In  the authors proposed an idea of a new algorithm
that may provide stable performance of the system with S–NS feedback and without a model
extension, but they did not give a precise description. Then a description of one such algorithm
was given in , where the authors presented balance equations for the stationary distribution of a
corresponding Markov chain and numerically found the capacity of this algorithm, i.e., a number λ
such that the balance equations have a solution if and only if the input rate λ is smaller than λ
We are unaware of any further studies on algorithms for systems with the “S–NS” feedback.
Modern communication systems deal with a variety of multiple access algorithms including ran-
dom multiple access; see, for example, systems based on standards IEEE 802.11 and IEEE 802.16.
Moreover, one may say that these standards deal with the “S–NS” feedback. In particular, in stan-
dard IEEE 802.16, the base station does not distinguish collisions from empty slots. It is known
that the algorithms used in practice do not provide stable dynamics if the number of users is inﬁnite
Supported in part by the Russian Foundation for Basic Research, project no. 07-01-00077.