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The heaps process, libraries, and size-biased permutations

Published online by Cambridge University Press:  14 July 2016

Peter Donnelly
Peter Donnelly*
Affiliation:
Queen Mary and Westfield College, London
*
∗Postal address: School of Mathematical Sciences, Queen Mary and Westfield College, University of London, Mile End Road, London El 4NS, UK.
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Abstract

The heaps process (also known as a Tsetlin library) provides a model for a self-regulating filing system. Items are requested from time to time according to their popularity and returned to the top of the heap after use. The size-biased permutation of a collection of popularities is a particular random permutation of those popularities, which arises naturally in a number of applications and is of independent interest. For a slightly non-standard formulation of the heaps process we prove that it converges to the size-biased permutation of its initial distribution. This leads to a number of new characterizations of the property of invariance under size-biased permutation, notably what might be described as invariance under ‘partial size-biasing' of any order. Finally we consider in detail the heaps process with Poisson–Dirichlet initial distribution, exhibiting the tractable nature of its equilibrium distribution and explicitly calculating a number of quantities of interest.

Keywords

TSETLIN LIBRARIESMARKOV CHAINS ON PERMUTATIONSCOUPLING TECHNIQUESPOISSON–DIRICHLET DISTRIBUTIONGEM DISTRIBUTIONPARTIAL SIZE-BIASED PERMUTATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 2 , June 1991 , pp. 321 - 335
Copyright
Copyright © Applied Probability Trust 1991 

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