  We present a solution of a class of network utility maximization (NUM)
problems using minimal communication. The constraints of the problem are
inspired less by TCP-like congestion control but by problems in the area of
internet of things and related areas in which the need arises to bring the
behavior of a large group of agents to a social optimum. The approach uses only
intermittent feedback, no inter-agent communication, and no common clock.
  The proposed algorithm is a combination of the classical AIMD algorithm in
conjunction with a simple probabilistic rule for the agents to respond to a
capacity signal. This leads to a nonhomogeneous Markov chain and we show almost
sure convergence of this chain to the social optimum.
