Hydrodynamic limit of move-to-front rules and search cost probabilities

We study a hydrodynamic limit approach to move-to-front rules, namely, a scaling limit as the number of items tends to infinity, of the joint distribution of jump rate and position of items. As an application of the limit formula, we present asymptotic formulas on search cost probability distributions, applicable for general jump rate distributions.

💡 Research Summary

The paper investigates the move‑to‑front (MTF) rule—a classic self‑organizing list algorithm—through the lens of hydrodynamic limits, i.e., scaling limits as the number of items N tends to infinity. Each item i is assigned a jump (or request) rate λ_i, and requests arrive according to a Poisson process with intensity proportional to λ_i. When an item is requested it jumps to the front of the list, while the relative order of the remaining items is preserved. The authors model this microscopic dynamics as a continuous‑time Markov chain on the symmetric group S_N and then consider the empirical joint distribution of (position, jump rate) as N grows.

By rescaling the discrete position j∈{1,…,N} to a continuous variable x=j/N∈