Convergence analysis of asynchronous iterations with stochastic delays

The authors present a general linear model of asynchronous iterations, the communication delays of which are stochastic with a Markovian character. This model allows static or dynamic allocation of the iterate vector components to processors. It also permits simultaneous updating of the same vector component by multiple processors. Sufficient conditions under which the model of asynchronous iterations converges in the second moment (and in the mean) to the sought resolution are provided. For the specialization of the Markov case when the communication delays are independent and identically distributed, the authors provide sufficient conditions for convergence in the second moment and necessary and sufficient conditions for convergence in the mean