Distributed Integer Balancing Under Weight Constraints in the Presence of Transmission Delays

Abstract

We consider the distributed weight balancing problem in networks of nodes that are interconnected via directed edges, each of which admits a positive integer weight. A digraph with positive integer weights on its edges is weight balanced if, for each node, the sum of the weights of the incoming edges equals the sum of the weights of the outgoing edges. In this article, we develop a distributed iterative algorithm, which solves the integer weight balancing problem in the presence of arbitrary (time-varying and inhomogeneous) time delays that might affect transmissions at particular links. We assume that each positive weight is constrained to lie within a certain interval, captured by individual lower and upper limits and that communication between neighboring nodes is bidirectional. We show that even when different transmissions on communication links are affected from bounded delays, the proposed distributed algorithm allows nodes to obtain a set of weights that solves the integer weight balancing problem, after a finite number of iterations, as long as a feasible solution exists. Finally, we provide examples to illustrate the operation and performance of the proposed algorithm.