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May 1999 Asymptotic approximation of the move-to-front search cost distribution and least-recently used caching fault probabilities
Predrag R. Jelenković
Ann. Appl. Probab. 9(2): 430-464 (May 1999). DOI: 10.1214/aoap/1029962750

## 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.

## Citation

Predrag R. Jelenković. "Asymptotic approximation of the move-to-front search cost distribution and least-recently used caching fault probabilities." Ann. Appl. Probab. 9 (2) 430 - 464, May 1999. https://doi.org/10.1214/aoap/1029962750

## Information

Published: May 1999
First available in Project Euclid: 21 August 2002

zbMATH: 0940.60044
MathSciNet: MR1687406
Digital Object Identifier: 10.1214/aoap/1029962750

Subjects:
Primary: 60F99
Secondary: 68P05 , 68P10

Keywords: heavy-tailed distributions , least-recently-used caching (paging) , light-tailed distributions , move-to-front searching , Self-organizing lists

Rights: Copyright © 1999 Institute of Mathematical Statistics

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