ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 2, pp. 156–165.
Pleiades Publishing, Inc., 2016.
Original Russian Text
S.G. Foss, B. Hajek, A.M. Turlikov, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 2, pp. 61–71.
COMMUNICATION NETWORK THEORY
Doubly Randomized Protocols for a Random Multiple
Access Channel with “Success–Nonsuccess” Feedback
S. G. Foss
, and A. M. Turlikov
Sobolev Institute of Mathematics, Siberian Branch
of the Russian Academy of Sciences, Novosibirsk, Russia
Novosibirsk State University, Novosibirsk, Russia
Heriot-Watt University, Edinburgh, UK
University of Illinois at Urbana-Champaign, USA
St. Petersburg State University of Aerospace Instrumentation, St. Petersburg, Russia
Received January 8, 2015; in ﬁnal form, December 28, 2015
Abstract—We consider a model of a decentralized multiple access system with a nonstandard
binary feedback where the empty and collision situations cannot be distinguished. We show
that, like in the case of a ternary feedback, for any input rate λ<e
there exists a “doubly
randomized” adaptive transmission protocol which stabilizes the behavior of the system. We
discuss also a number of related problems and formulate some hypotheses.
We consider a decentralized multiple access system model with an inﬁnite number of users,
a single transmission channel, and an adaptive transmission protocol; we consider a class of pro-
tocols where the user cannot observe the individual history of messages and the total number of
messages. With any such a protocol, all users transmit their messages in time slot [n, n +1)with
equal probabilities p
that depend on the history of feedback from the transmission channel.
Algorithms with ternary feedback “Empty–Success–Collision” were introduced in [1, 2]. It is
assumed that the users can observe the channel output and distinguish among three possible sit-
uations: either no transmission (“Empty”) or transmission from a single server (“Success”) or a
collision of messages from two or more users (“Conﬂict”).
It is known since 1980s (see, e.g., [3,4]) that if the feedback is ternary, then the channel capacity
: if the input rate is below e
, then there is a stable transmission protocol, and if the input
rate is above e
, then any transmission protocol is unstable. A stable protocol may be constructed
recursively as follows: given probability p
in time slot [n, n + 1) and a feedback at time n +1,
is greater than p
if the slot [n, n +1) was empty, p
if there was a
successful transmission, and p
is less than p
if there is a conﬂict.
In  there was considered a multiplicative increase/decrease and assumed that the random
number, ξ, of arrivals per typical slot has a ﬁnite exponential moment, E e
In  there was considered an additive increase/decrease assuming the second moment E ξ
be ﬁnite. Later  it was shown that, without further assumptions on the input, the condition
Supported in part by the Ministry of Education and Science of the Republic of Kazakhstan, grant