  Volume 55, Issue 3
A New Method for Computing the Expected Hitting Time between Arbitrary Different Configurations of the Multiple–Urn Ehrenfest Model

J. Math. Study, 55 (2022), pp. 254-270.

Published online: 2022-09

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• 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 $2$.

• Keywords

Ehrenfest urn model, Markov chain, random walk, hitting time.

60C05, 60J10

abc_sophia123@126.com (Sai Song)

qyao@sfs.ecnu.edu.cn (Qiang Yao)

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@Article{JMS-55-254, author = {Song , Sai and Yao , Qiang}, title = {A New Method for Computing the Expected Hitting Time between Arbitrary Different Configurations of the Multiple–Urn Ehrenfest Model}, journal = {Journal of Mathematical Study}, year = {2022}, volume = {55}, number = {3}, pages = {254--270}, 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 $2$.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v55n3.22.03}, url = {http://global-sci.org/intro/article_detail/jms/20975.html} }
TY - JOUR T1 - A New Method for Computing the Expected Hitting Time between Arbitrary Different Configurations of the Multiple–Urn Ehrenfest Model AU - Song , Sai AU - Yao , Qiang JO - Journal of Mathematical Study VL - 3 SP - 254 EP - 270 PY - 2022 DA - 2022/09 SN - 55 DO - http://doi.org/10.4208/jms.v55n3.22.03 UR - https://global-sci.org/intro/article_detail/jms/20975.html KW - Ehrenfest urn model, Markov chain, random walk, hitting time. AB -

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 $2$.

Sai Song & Qiang Yao. (2022). A New Method for Computing the Expected Hitting Time between Arbitrary Different Configurations of the Multiple–Urn Ehrenfest Model. Journal of Mathematical Study. 55 (3). 254-270. doi:10.4208/jms.v55n3.22.03
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