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On the matrix occurring in a linear search problem

Published online by Cambridge University Press:  14 July 2016

R. M. Phatarfod
R. M. Phatarfod*
Affiliation:
Monash University
*
∗Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
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Abstract

In this paper we consider the Markov chain formed by the operation of the move-to-front scheme. We show that the eigenvalues of the transition probability matrix are of the form pi, pi + pj, ···, where pi is the probability of selecting the ith item and N is the number of items; further, that the multiplicity of the eigenvalues of the form Σpi where the summation is over m items is equal to the number of permutations of N – m objects, ordered in some way, such that no object is in its natural position. Finally, we show that the Markov chain is lumpable – many times over.

Keywords

EIGENVALUESMATCHING PROBLEMSLUMPABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 2 , June 1991 , pp. 336 - 346
Copyright
Copyright © Applied Probability Trust 1991 

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