# Distributed Integer Balancing under Weight Constraints in the Presence of Transmission Delays and Packet Drops

## TL;DR

This paper presents a distributed algorithm for integer weight balancing in directed networks that accounts for unreliable communication with delays and packet drops, ensuring convergence under certain conditions.

## Contribution

It introduces a novel distributed iterative method that guarantees convergence to a balanced weight configuration despite communication delays and packet drops.

## Key findings

- Algorithm converges with probability one under bounded delays and packet drops.
- Convergence occurs after a finite number of iterations.
- The method works as long as circulation conditions on weight limits are satisfied.

## Abstract

We consider the distributed weight balancing problem in networks of nodes that are interconnected via directed edges, each of which is able to admit a positive integer weight within a certain interval, captured by individual lower and upper limits. A digraph with positive integer weights on its (directed) 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 work, 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 communication between neighboring nodes is bidirectional, but unreliable since it may be affected from bounded or unbounded delays (packet drops), independently between different links and link directions. We show that, even when communication links are affected from bounded delays or occasional packet drops (but not permanent communication link failures), the proposed distributed algorithm allows the nodes to converge to a set of weight values that solves the integer weight balancing problem, after a finite number of iterations with probability one, as long as the necessary and sufficient circulation conditions on the lower and upper edge weight limits are satisfied. Finally, we provide examples to illustrate the operation and performance of the proposed algorithms.

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Source: https://tomesphere.com/paper/1907.04062