Asymmetrical resource networks. I. Stabilization processes for low resources

We consider dynamic resource allocation processes in two-sided asymmetrical resource networks with loops and study their stabilization conditions. We show that for a unit total resource, the resource reallocation process defines a regular Markov chain, and the limit state vector corresponds to the vector of limit probabilities and is an eigenvector of the stochastic matrix corresponding to the capacity matrix. We show that (a) for a resource not exceeding some threshold value, the limit state vector also exists, is unique, and is proportional to the limit probability vector; (b) for any connected network, the threshold value of total resource exists and is unique.