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Computing absorbing times via fluid approximations

Part of: Ergodic theory Extremal combinatorics Markov processes

Published online by Cambridge University Press:  08 September 2017

Nicolas Gast  and
Bruno Gaujal
Nicolas Gast*
Affiliation:
Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LIG, 38000 Grenoble, France
Bruno Gaujal*
Affiliation:
Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LIG, 38000 Grenoble, France
*
* Postal address: Institute of Engineering, Université Grenoble Alpes, Inria, Bâtiment IMAG, 700 avenue Centrale, 38400 St Martin d'Heres, France.
* Postal address: Institute of Engineering, Université Grenoble Alpes, Inria, Bâtiment IMAG, 700 avenue Centrale, 38400 St Martin d'Heres, France.
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Abstract

In this paper we compute the absorbing time Tn of an n-dimensional discrete-time Markov chain comprising n components, each with an absorbing state and evolving in mutual exclusion. We show that the random absorbing time Tn is well approximated by a deterministic time tn that is the first time when a fluid approximation of the chain approaches the absorbing state at a distance 1 / n. We provide an asymptotic expansion of tn that uses the spectral decomposition of the kernel of the chain as well as the asymptotic distribution of Tn, relying on extreme values theory. We show the applicability of this approach with three different problems: the coupon collector, the erasure channel lifetime, and the coupling times of random walks in high-dimensional spaces.

Keywords

Markov chainfluid approximationextreme valuecoupon collectorcoupling time

MSC classification

Primary: 05D40: Probabilistic methods
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 37A25: Ergodicity, mixing, rates of mixing
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 768 - 790
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Anceaume, E., Busnel, Y. and Sericola, B. (2015). New results on a generalized coupon collector problem using Markov chains. J. Appl. Prob. 52, 405–418. CrossRefGoogle Scholar
[2] Benaïm, M. and Le Boudec, J.-Y. (2008). A class of mean field interaction models for computer and communication systems. Performance Evaluation 65, 823–838. Google Scholar
[3] Berenbrink, P. and Sauerwald, T. (2009). The weighted coupon collector's problem and applications. In Computing and Combinatorics, Springer, Berlin, pp. 449–458. CrossRefGoogle Scholar
[4] Brayton, R. K. (1963). On the asymptotic behavior of the number of trials necessary to complete a set with random selection. J. Math. Anal. Appl. 7, 31–61. Google Scholar
[5] Chaintreau, A., Le Boudec, J.-Y. and Ristanovic, N. (2009). The age of gossip: spatial mean field regime. ACM SIGMETRICS Performance Eval. Rev. 37, 109–120. Google Scholar
[6] Chazottes, J.-R., Collet, P. and Méléard, S. (2016). Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes. Prob. Theory Relat. Fields 164, 285–332. Google Scholar
[7] Doumas, A. V. and Papanicolaou, V. G. (2012). The coupon collector's problem revisited: asymptotics of the variance. Adv. Appl. Prob. 44, 166–195. Google Scholar
[8] Doumas, A. V. and Papanicolaou, V. G. (2016). The coupon collector's problem revisited: generalizing the double dixie cup problem of Newman and Shepp. ESAAIM Prob. Statist. 20, 367–399. Google Scholar
[9] Erdős, P. and Rényi, A. (1961). On a classical problem of probability theory. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6, 215–220. Google Scholar
[10] Flajolet, P., Gardy, D. and Thimonier, L. (1992). Birthday paradox, coupon collectors, caching algorithms and self-organizing search. Discrete Appl. Math. 39, 207–229. Google Scholar
[11] Ganesh, A. et al. (2012). Load balancing via random local search in closed and open systems. Queueing Systems 71, 321–345. Google Scholar
[12] Gast, N. and Bruno, G. (2010). A mean field model of work stealing in large-scale systems. ACM SIGMETRICS Performance Eval. Rev. 38, 13–24. Google Scholar
[13] Kang, W. and Ramanan, K. (2012). Asymptotic approximations for stationary distributions of many-server queues with abandonment. Ann. Appl. Prob. 22, 477–521. Google Scholar
[14] Kulkarni, D., Schmidt, D. and Tsui, S.-K. (1999). Eigenvalues of tridiagonal pseudo-Toeplitz matrices. Linear Algebra Appl. 297, 63–80. Google Scholar
[15] Kurtz, T. G. (1978). Strong approximation theorems for density dependent Markov chains. Stoch. Process. Appl. 6, 223–240. Google Scholar
[16] Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. Society for Industrial and Applied Mathematics, Philadelphia, PA. Google Scholar
[17] Le Boudec, J.-Y. (2013). The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points. Networks Heterogeneous Media 8, 529–540. CrossRefGoogle Scholar
[18] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. American Mathematical Society, Providence, RI. Google Scholar
[19] May, R. (2008). Coupon collecting with quotas. Electron. J. Combinatorics 15, 31. Google Scholar
[20] Neal, P. (2008). The generalised coupon collector problem. J. Appl. Prob. 45, 621–629. Google Scholar
[21] Newman, D. J. and Shepp, L. (1960). The double dixie cup problem. Amer. Math. Monthly 67, 58–61. Google Scholar
[22] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252. Google Scholar
[23] Shank, N. and Yang, H. (2013). Coupon collector problem for nonuniform coupons and random quotas. Electron. J. Combinatorics 20, 33. Google Scholar