A New Method for Computing the Expected Hitting Time Between Arbitrary Different Configurations of the Multiple–Urn Ehrenfest Model

Abstract

We study a multiple-urn version of the Ehrenfest model. In this setting, we denote the $n$ urns by Urn $1$ to Urn $n$, where $n\geq2$. Initially, $M$ balls are randomly placed in the $n$ urns. At each subsequent step, a ball is selected and put into the other $n-1$ urns with equal probability. The expected hitting time leading to a change of the $M$ balls' status is computed using the method of stopping times. As a corollary, we obtain the expected hitting time of moving all the $M$ balls from Urn $1$ to Urn $2$.