On Dropping Needles and WiFi Link Crossing

TL;DR

This paper analyzes the probability of a random walk crossing a WiFi link, deriving its asymptotic behavior using advanced mathematical tools and connecting it to classical probability problems from history.

Contribution

It introduces a novel application of Perron-Frobenius theory to WiFi link crossing probabilities and links the problem to Buffon's needle, providing new insights into the crossing behavior.

Findings

Crossing probability is proportional to step size and inversely proportional to room width.

Asymptotic crossing probability derived using Perron-Frobenius theory.

Connection established between WiFi link crossing and Buffon's needle problem.

Abstract

In a general simulation of random walking (with the angle of motion picked uniformly), it can be seen that the probability of crossing a WiFi TX-RX link is directly proportional to the per-step distance and inversely proportional to the lateral dimension of the room. The asymptotic value of the said crossing probability is derived using Perron-Frobenius theory to determine the limit distribution of the said Markov model. Surprisingly, we can establish a bijection to a scenario explored nearly 300 years ago by Georges-Louis Leclerc, Comte de Buffon to get the result. Furthermore we can use the generalizations of the latter problem to ascertain some interesting observations about the original one.