17_4 - Comparing strength of locality of reference...

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Comparing strength of locality of reference – Popularity, majorization, and some folk theorems Sarut Vanichpun Armand M. Makowski Department of Electrical and Computer Engineering and the Institute for Systems Research University of Maryland, College Park College Park, Maryland 20742 Email: [email protected] [email protected] Abstract — The performance of demand-driven caching de- pends 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. Keywords: Locality of reference in request streams, Popu- larity, Majorization/Schur-concavity. I. I NTRODUCTION Web caching aims to reduce network traffic, server load and user-perceived retrieval latency by replicating “popular” content on (proxy) caches that are strategically placed within the network. This approach is a natural outgrowth of caching techniques which were originally developed for computer memory and distributed file sharing systems, e.g., [1, 2, 3] (and references therein). The performance of any form of caching is determined by a number of factors, chief amongst them the statistical properties of the streams of requests made to the cache. One important such property is the locality of reference present in a request stream whereby bursts of references are made in the near future to objects referenced in the recent past. The implications for cache management should be clear – Increased locality of reference should yield performance improvements for demand- driven caching that exploits recency of reference. In particular, under this form of cache management, we expect the following “folk theorem” to hold: The stronger the locality of reference in the stream of requests, the smaller the miss rate since the cache ends up being populated by Web objects with a higher likelihood of access in the near future. The notion of locality and its importance for caching were first recognized by Belady [4] in the context of computer memory. Subsequently, a number of studies have shown that request streams for Web objects exhibit strong locality of reference 1 [5, 6, 7]. Attempts at characterization were made early on by Denning through the working set model [8, 9].
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