On the Convergence of the TTL Approximation for an LRU Cache under Independent Stationary Request Processes

TL;DR

This paper provides a theoretical foundation for the TTL approximation in LRU cache analysis under independent stationary request processes, proving its asymptotic exactness and establishing convergence rates.

Contribution

It extends the TTL approximation's validity to a broader class of request models and offers convergence bounds, enhancing theoretical understanding of cache performance.

Findings

The TTL approximation is asymptotically exact as cache size and content number grow large.

The approximation holds for individual content hit probabilities.

Convergence rates of the approximation are quantified.

Abstract

The modeling and analysis of an LRU cache is extremely challenging as exact results for the main performance metrics (e.g. hit rate) are either lacking or cannot be used because of their high computational complexity for large caches. As a result, various approximations have been proposed. The state-of-the-art method is the so-called TTL approximation, first proposed and shown to be asymptotically exact for IRM requests by Fagin. It has been applied to various other workload models and numerically demonstrated to be accurate but without theoretical justification. In this paper we provide theoretical justification for the approximation in the case where distinct contents are described by independent stationary and ergodic processes. We show that this approximation is exact as the cache size and the number of contents go to infinity. This extends earlier results for the independent reference model. Moreover, we establish results not only for the aggregate cache hit probability but also for every individual content. Last, we obtain bounds on the rate of convergence.