Nonhomogeneous Place-Dependent Markov Chains, Unsynchronised AIMD, and   Network Utility Maximization
  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. 
Fabian Wirth and Sonja Stuedli and Jia Yuan Yu and Martin Corless and   Robert Shorten
math.OC math.PR