Limiting behavior of the search cost distribution for the move-to-front rule in the stable case

TL;DR

This paper investigates the limiting distribution of search costs in the move-to-front heuristic, establishing the behavior of moments as list size grows, based on recent theoretical extensions of Kingman's results.

Contribution

It links two recent studies to extend Kingman's results, providing a comprehensive analysis of the moments of the stationary search cost in the stable case.

Findings

Derived the limiting behavior of all moments of the search cost

Connected recent theoretical results to extend classical findings

Provided a rigorous framework for understanding search cost distribution

Abstract

Move-to-front rule is a heuristic updating a list of n items according to requests. Items are required with unknown probabilities (or popularities). The induced Markov chain is known to be ergodic. One main problem is the study of the distribution of the search cost dened as the position of the required item. Here we first establish the link between two recent papers that both extend results proved by Kingman on the expected stationary search cost. Combining results contained in these papers, we obtain the limiting behavior for any moments of the stationary seach cost as n tends to innity.