Computing hitting times via fluid approximation: application to the coupon collector problem

TL;DR

This paper introduces a stochastic approximation method using fluid models to efficiently estimate hitting times in Markov processes, applied specifically to a generalized coupon collector problem involving multiple chains.

Contribution

It develops a novel fluid approximation approach to compute hitting times in complex stochastic systems, extending analysis to multiple interacting Markov chains.

Findings

Hitting time bounded by N log N + b N log log N + O(N)

Constants depend on eigenvalues of the Markov chain

Method applicable to generalized coupon collector scenarios

Abstract

In this paper, we show how to use stochastic approximation to compute hitting time of a stochastic process, based on the study of the time for a fluid approximation of this process to be at distance 1/N of its fixed point. This approach is developed to study a generalized version of the coupon collector problem. The system is composed by N independent identical Markov chains. At each time step, one Markov chain is picked at random and performs one transition. We show that the time at which all chains have hit the same state is bounded by a N log N + b N log log N + O(N) where a and b are two constants depending on eigenvalues of the Markov chain.