On the move-to-front scheme with Markov dependent requests

In this paper we consider the operation of the move-to-front scheme where the requests form a Markov chain of N states with transition probability matrix P. It is shown that the configurations of items at successive requests form a Markov chain, and its transition probability matrix has eigenvalues that are the eigenvalues of all the principal submatrices of P except those of order N – 1. We also show that the multiplicity of the eigenvalues of submatrices of order m is the number of derangements of N – m objects. The last result is shown to be true even if P is not a stochastic matrix.