Asymptotic Approximation of the Move-to-front Search Cost Distribution and Least-recently Used Caching Fault Probabilities

Consider a finite list of items n = 1;2; : : : ;N, that are requested according to an i.i.d. process. Each time an item is requested it is moved to the front of the list. The associated search cost CN for accessing an item is equal to its position before being moved. If the request distribution converges to a proper distribution as N→∞, then the stationary search cost CN converges in distribution to a limiting search cost C. We show that, when the (limiting) request distribution has a heavy tail (e.g., generalized Zipf ’s law), P’R = n“ ∼ c/nα as n→ ∞, α > 1, then the limiting stationary search cost distribution P’C > n“, or, equivalently, the least-recently used (LRU) caching fault probability, satisfies