Comparing strength of locality of reference - popularity, majorization, and some folk theorems

Abstract

Abstract-The performance of demand-driven caching depends on the locality of reference exhibited by the stream of requests made to the cache. In particular, it is expected that the stronger the locality of reference, the smaller the miss rate of the cache. For the Independent Reference Model, this amounts to a smaller miss rate when the popularity distribution of requested objects in the stream is more skewed. In this paper, we formalize this "folk theorem" through the companion concepts of majorization and Schur-concavity. This folk theorem is established for caches operating under a Random On-demand Replacement Algorithm (RORA). However, the result fails to hold in general under the (popular) LRU and CLIMB policies, but can be established when the input has a Zipf-like popularity pmf with large skewness parameter. In addition, we explore how the majorization of popularity distributions translates into comparisons of three well-known locality of reference metrics, namely the inter-reference time, the working set size and the stack distance.

Modeling Locality of Reference via Notions of Positive Dependence - Some Mixed News

“….. The next lemma identifies conditions on the request stream R for the limits (16) to exist; its proof can be found in [33]. …”

“…A natural step consists in relating locality of reference in a stream of requests to the skewness of its popularity pmf with the understanding that the more skewed the popularity pmf, the greater locality of reference. For instance, the notion of entropy [18] and the concept of majorization [22,32,33,35,36] have been used with some success precisely for that purpose. In [22,33,35] the authors then established a version of the folk theorem by showing (via majorization and Schur-concavity) that the more skewed the popularity pmf (thus, the stronger locality of reference), the smaller the miss rate of the cache.…”

“…For instance, the notion of entropy [18] and the concept of majorization [22,32,33,35,36] have been used with some success precisely for that purpose. In [22,33,35] the authors then established a version of the folk theorem by showing (via majorization and Schur-concavity) that the more skewed the popularity pmf (thus, the stronger locality of reference), the smaller the miss rate of the cache. This was done for various cache replacement policies under the standard Independent Reference Model (IRM) according to which the requests {R t , t = 0, 1, .…”

“…Temporal correlations and positive dependenceWe make a connection between the concepts of positive dependence in sequence of rvs [ We conclude in Section XII by explaining in what sense the news are indeed mixed! Many proofs have been omitted due to space limitations, but can be found in the thesis [33].…”

The Cache Inference Problem and its Application to Content and Request Routing

“…Vanichpun and Makowski [30], [31], [32] have presented a series of papers on the characterization of the miss stream of a cache under different input request models (independent reference model, higher-order Markov chain model, partial Markov chain model, LRU stack model). These works focus on the effect of locality of reference, either due to long term popularity or due to short term temporal correlation, on the miss-stream and the cache miss ratio and, therefore, examine caching in the inverse direction as compared to what we do; they are trying to characterize the output given the input, whereas we try to characterize the input (and what is in the box) given the output.…”