Network control by Bayesian broadcast

A transmission control strategy is described for slotted-ALOHA-type broadcast channels with ternary feedback. At each time slot, each station estimates the probability that n stations are ready to transmit a packet for each
n
, using Bayes' rule and the observed history of collisions, successful transmissions, and holes (empty slots). A station transmits a packet in a probabilistic manner based on these estimates. This strategy is called Bayesian broadcast. An elegant and very practical strategy--pseudo-Bayesian broadcast--is then derived by approximating the probability estimates with a Poisson distribution with mean
\nu
and further simplifying. Each station keeps a copy of
\nu
, transmits a packet with probability
1/\nu
, and then updates
\nu
in two steps: For collisions, increment
\nu
by
(e-2)^{-l}=1.39221 \cdots
. For successes and holes, decrement
\nu
by
1
. Set
\nu
to
\max (\nu + \hat{\lambda}, 1)
, where
\hat{\lambda}
is an estimate of the arrival rate
\lambda
of new packets into the system. Simulation results are presented showing that pseudo-Bayesian broadcast performs well in practice, and methods that can be used to prove that certain versions of pseudo-Bayesian broadcast are stable for
\lambda < e^{-1}
are discussed.