On the asymptotics of fault probability in least‐recently‐used caching with Zipf‐type request distribution

Abstract

ABSTRACT:We consider the least-recently-used cache replacement rule with a Zipf-type page request distribution and investigate an asymptotic property of the fault probability with respect to an increase of cache size. We first derive the asymptotics of the fault probability for the independentrequest model and then extend this derivation to a general dependent-request model, where our result shows that under some weak assumptions the fault probability is asymptotically invariant with regard to dependence in the page request process. In a previous study, a similar result was derived by applying a Poisson embedding technique, where a continuous-time proof was given through some assumptions based on a continuous-time modeling. The Poisson embedding, however, is just a technique used for the proof and the problem is essentially on a discrete-time basis; thus, it is preferable to make assumptions, if any, directly in the discrete-time setting. We consider a general dependent-request model and give a direct discrete-time proof under different assumptions. A key to the proof is that the numbers of requests for respective pages represent conditionally negatively associated random variables.

A Fluid Limit for Cache Algorithms with General Request Processes

“…We find that asymptotic characteristics of LRU are often preserved in our fluid limit. Specifically, we will see that, as K → ∞,p (∞) is asymptotically insensitive to particular dependencies in the request processes when the request rates have a heavy tail (Theorem 3), which agrees with the findings forp in [3], [18]. We also find that the asymptotic analysis ofp (∞) appears to be simpler than a corresponding analysis ofp.…”

“…, N are conditionally independent given J(·). Note that Ψ is stationary, which is also assumed in [3], [9], [11], [18], so that the stationary miss probability exists (see Lemma 2.1 from [18]) and agrees with the average miss probability.…”

“…In this section, we study the request processes that are similar to those studied in [3], [9], [11], [18]. Specifically, let J(·) be a stationary and ergodic semi-Markov chain on a finite state space that determines the request rate for e i at time t with λ i (J(t)) for 1 ≤ i ≤ N .…”

“…Jelenković and Radovanović [9] and Sugimoto and Miyoshi [18] show that, when the request rates have a heavy tail, the miss probability for LRU is asymptotically insensitive to the types of dependencies in the request process studied in Coffman and Jelenković [3] as K → ∞. Jelenković et al [11] characterize the critical cache sizes where the miss probability for LRU becomes insensitive to the dependencies.…”

The persistent‐access‐caching algorithm

“…These demonstrated performance improvements, both analytical and experimental, as well as the simplicity of implementation, suggest a potential use of the proposed PAC policy for practical purposes. Given the analytic approach established in our recent work on the analysis of the LRU policy in the presence of dependent requests [12] and variable page sizes [13] (see also [17]), it can be shown that analogous results hold for the PAC algorithm as well. In the context of the ordinary MTF with lighter tailed request distributions, the asymptotic fluid limits of the search cost derived in [10] could be analogously extended to the PAC policy as well.…”