  The belief propagation (BP) algorithm is an efficient way to solve
"inference" problems in graphical models, such as Bayesian networks and Markov
random fields. The system-state probability distribution of CSMA wireless
networks is a Markov random field. An interesting question is how BP can help
the analysis and design of CSMA wireless networks. This paper explores three
such applications. First, we show how BP can be used to compute the throughputs
of different links in the network given their access intensities, defined as
the mean packet transmission time divided by the mean backoff countdown time.
Second, we propose an inverse-BP algorithm to solve the reverse problem: how to
set the access intensities of different links to meet their target throughputs?
Third, we introduce a BP-adaptive CSMA algorithm to find the link access
intensities that can achieve optimal system utility. BP solves the three
problems with exact results in tree networks. It may, however, lose accuracy in
networks with a loopy contention graph. We show how a generalized version of
BP, GBP, can be designed to solve the three problems with high accuracy for
networks with a loopy contention graph. Importantly, we show how the BP and GBP
algorithms in this paper can be implemented in a distributed manner, making
them useful in practical CSMA network opera-tion.
