Asymptotic approximation of the move-to-front search cost distribution and least-recently used caching fault probabilities

Abstract

Consider a finite list of items $n = 1, 2,\dots, N$, 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 $C^N$ for accessing an item is equal to its position before being moved. If the request distribution converges to a proper distribution as $N \to \infty$, then the stationary search cost $C^N$ 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), $\mathbb{P}[R = n] \sim c/n^{\alpha}$ as $n \to \infty, \alpha > 1$, then the limiting stationary search cost distribution $\mathbb{P}[C > n]$, or, equivalently, the least-recently used (LRU) caching fault probability, satisfies $$\lim_{n \to \infty} \frac{\mathbb{P}[C > n]}{\mathbb{P}[R > n]}= (1 - \frac{1}{\alpha}) [\Gamma(1 - \frac{1}{\alpha})]^{\alpha} \nearrow e^{\gamma}$ \text{as}$\alpha \to \infty,$$ where $\Gamma$ is the Gamma function and $\gamma(=0.5772\dots)$ is Euler's constant.

When the request distribution has a light tail $\mathbb{P}[R = n] \sim c \exp(-\lambdan_{\beta})$ as $n \to \infty (c, \lambda, \beta > 0), then $$\lim_{n \to \infty} \frac{\mathbb{P}[C_f > n]}{\mathbb{P}[R > n]} = e^{\gamma},$$ independently of $c, \lambda, \beta$, where $C_f$ is a fluid approximation of C.

We experimentally demonstrate that the derived asymptotic formulas yield accurate results for lists of finite sizes. This should be contrasted with the exponential computational complexity of Burville and Kingman's exact expression for finite lists. The results also imply that the fault probability of LRU caching is asymptotically at most a factor $e^{\gamma} (\approx 1.78)$ greater than for the optimal static arrangement.