The joint distribution of value and local time of simple random walk and reflected simple random walk

TL;DR

This paper derives explicit formulas for the joint distribution of value and local time for simple and reflected random walks with asymmetric ±1 steps, extending known results from Brownian motion and recurrent random walks.

Contribution

It provides new closed-form formulas for the joint distribution of value and local time in asymmetric simple and reflected random walks, addressing a practical queue control problem.

Findings

Closed-form formulas for asymmetric simple random walk

Joint distribution formulas for reflected random walk

Application to queue size control during pandemics

Abstract

The joint distribution of value and local time for Brownian Motion has been reported by Borodin and Salminen. Its asymptotic behavior for recurrent random walk has been presented by Jain and Pruitt. Motivated by the need for queue size control during a pandemic (Hassin, Meilijson and Perlman), the current study presents closed form formulas for random walk and reflected random walk with $\pm 1$ increments, not necessarily fair.