The Maximum of an Asymmetric Simple Random Walk with Reflection

TL;DR

This paper analyzes the maximum value of a biased reflected Bernoulli random walk over finite time, providing asymptotic mean and variance results, and also addresses a related queueing theory problem.

Contribution

It offers new asymptotic results for the maximum of an asymmetric reflected random walk and connects it to a queueing theory problem.

Findings

Asymptotic mean and variance of the maximum are derived.

Results are specific to the biased case of the random walk.

An elementary queueing problem is also solved.

Abstract

Consider the extreme value of a Bernoulli random walk on the one-dimensional integer lattice, with reflection at 0, over a finite discrete time interval. Only the asymmetric (biased) case is discussed. Asymptotic mean/variance results are given as the time interval length approaches infinity. We similarly solve an elementary traffic light problem from queueing theory.