# Heat-Diffusion: Pareto Optimal Dynamic Routing for Time-Varying Wireless Networks

## TL;DR

The paper introduces Heat-Diffusion, a dynamic routing policy for wireless networks that optimally balances throughput, routing cost, and delay using only current network conditions, and links it to heat equation mathematics.

## Contribution

It develops a novel, throughput and delay optimal routing policy that operates with current network state information and establishes a connection to heat equation analysis.

## Key findings

- HD achieves Pareto optimal tradeoff between routing cost and delay.
- HD is throughput optimal and minimizes average delay.
- Fluid limit of HD follows the graph heat equation.

## Abstract

A dynamic routing policy, referred to as Heat-Diffusion (HD), is developed for multihop uniclass wireless networks subject to random traffic, time-varying topology and inter-channel interference.The policy uses only current condition of queue occupancies and channel states, with requiring no knowledge of traffic and topology.Besides throughput optimality, HD minimizes an average quadratic routing cost defined by endowing each channel with a time-varying cost factor. Further, HD minimizes average network delay in the class of routing policies that base decisions only on current condition of traffic congestion and channel states. Further, in this class of routing policies, HD provides a Pareto optimal tradeoff between average routing cost and average network delay, meaning that no policy can improve either one without detriment to the other. Finally, HD fluid limit follows graph combinatorial heat equation, which can open a new way to study wireless networks using heat calculus, a very active area of pure mathematics.