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Markov Processes with Restart

Published online by Cambridge University Press:  30 January 2018

Konstantin Avrachenkov*
Affiliation:
Inria Sophia Antipolis
Alexey Piunovskiy*
Affiliation:
The University of Liverpool
Yi Zhang*
Affiliation:
The University of Liverpool
*
Postal address: Inria Sophia Antipolis, 2004 Route des Lucioles, Sophia Antipolis, 06902, France. Email address: [email protected]
∗∗ Postal address: Department of Mathematical Sciences, University of Liverpool, M&O Building, L69 7ZL, UK.
∗∗ Postal address: Department of Mathematical Sciences, University of Liverpool, M&O Building, L69 7ZL, UK.
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Abstract

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We consider a general homogeneous continuous-time Markov process with restarts. The process is forced to restart from a given distribution at time moments generated by an independent Poisson process. The motivation to study such processes comes from modeling human and animal mobility patterns, restart processes in communication protocols, and from application of restarting random walks in information retrieval. We provide a connection between the transition probability functions of the original Markov process and the modified process with restarts. We give closed-form expressions for the invariant probability measure of the modified process. When the process evolves on the Euclidean space, there is also a closed-form expression for the moments of the modified process. We show that the modified process is always positive Harris recurrent and exponentially ergodic with the index equal to (or greater than) the rate of restarts. Finally, we illustrate the general results by the standard and geometric Brownian motions.

Type
Research Article
Copyright
© Applied Probability Trust 

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