Abstract

Many distributed protocols arising in applications in online load balancing and dynamic resource allocation can be modeled by dynamic allocation processes related to the “balls into bin” problems. Traditionally the main focus of the research on dynamic allocation processes is on verifying whether a given process is stable, and if so, on analyzing its behavior in the limit (i.e., after sufficiently many steps). Once we know that the process is stable and we know its behavior in the limit, it is natural to analyze its recovery time, which is the time needed by the process to recover from any arbitrarily bad situation and to arrive very closely to a stable (i.e., a typical) state. This investigation is important to provide assurance that even if at some stage the process has reached a highly undesirable state, we can predict with high confidence its behavior after the estimated recovery time. In this paper we present a genera / framework to study the recovery time of discrete-time dynamic allocation processes. We model allocation processes by suitably chosen ergodic Markov chains. For a given Markov chain we apply path coupling arguments to bound its convergence rates to the stationary distribution, which directly yields the estimation of the recovery time of the corresponding allocation process. Our coupling approach provides in a relatively simple way an accurate prediction of the recovery time. In particular, we show that our method can be applied to significantly improve estimations of the recovery time for various allocation processes related to allocations of balls into bins, and for the edge orientation problem studied before by Ajtai et al.

Citations

284	Group representations in probability and statistics - Diaconis - 1988
250	Balanced Allocations - Azar, Broder, et al. - 2000
234	The Markov chain Monte Carlo method: an approach to approximate counting and integration - Jerrum, Sinclair - 1997
227	Random walks on finite groups and rapidly mixing Markov chains - Aldous - 1981
149	Path Coupling: A Technique for Proving Rapid Mixing - Bubley, Dyer - 1997
80	Urn models and their applications: an approach to modern discrete probability theory - Johnson, Kotz - 1977
80	How Useful is Old Information - Mitzenmacher - 1997
58	Faster random generation of linear extensions - Bubley, Dyer
56	Parallel randomized load balancing - Adler, Chakrabarti, et al. - 1995
55	On the Analysis of Randomized Load Balancing Schemes - Mitzenmacher
43	Load Balancing and Density Dependent Jump Markov - Mitzenmacher - 1996
32	Randomized allocation processes - Czumaj, Stemann - 1997
21	Studying balanced allocations with differential equations - Mitzenmacher - 1999
12	Allocating Weighted Jobs in Parallel - Berenbrink, Heide, et al. - 1997
11	J.H.: A fair carpool scheduling algorithm - Fagin, Williams - 1983
11	Lectures on the coupling method, wiley series in probability and mathematica statistics - Lindvall - 1992
7	walls: analysis of a combinatorial game - Disks - 1989
2	Beating the 2A bound for approximately counting colourings: A computer-assisted proof of rapid mixing - Bubley, Dyer, et al. - 1997
1	Delayed path coupling and its applications to analysing distributed stochastic processes for generating random permutations - Czumaj, Kanarek, et al. - 1998