Network control by Bayesian broadcast

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu</tex> and further simplifying. Each station keeps a copy of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu</tex> , transmits a packet with probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/\nu</tex> , and then updates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu</tex> in two steps: For collisions, increment <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu</tex> by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(e-2)^{-l}=1.39221 \cdots</tex> . For successes and holes, decrement <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu</tex> by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</tex> . Set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu</tex> to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\max (\nu + \hat{\lambda}, 1)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{\lambda}</tex> is an estimate of the arrival rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda &lt; e^{-1}</tex> are discussed.