# Randomized Load Balancing on Networks with Stochastic Inputs

## TL;DR

This paper analyzes the average-case performance of load balancing algorithms on various network topologies with stochastic inputs, providing bounds on discrepancy that highlight differences from worst-case scenarios.

## Contribution

It introduces new bounds on load discrepancy for multiple network types under stochastic inputs, extending previous worst-case analyses to average-case scenarios.

## Key findings

- Bounds on discrepancy for cycles, tori, hypercubes, and expanders.
- Significant difference between worst-case and average-case convergence.
- Applicable to various probability distributions including unbounded ones.

## Abstract

Iterative load balancing algorithms for indivisible tokens have been studied intensively in the past. Complementing previous worst-case analyses, we study an average-case scenario where the load inputs are drawn from a fixed probability distribution. For cycles, tori, hypercubes and expanders, we obtain almost matching upper and lower bounds on the discrepancy, the difference between the maximum and the minimum load. Our bounds hold for a variety of probability distributions including the uniform and binomial distribution but also distributions with unbounded range such as the Poisson and geometric distribution. For graphs with slow convergence like cycles and tori, our results demonstrate a substantial difference between the convergence in the worst- and average-case. An important ingredient in our analysis is new upper bound on the t-step transition probability of a general Markov chain, which is derived by invoking the evolving set process.