Relative Reduced Hops

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1 GreedyDual-Size: A Cost-Aware WWW Proxy Caching Algorithm Pei Cao Sandy Irani y 1 Introduction As the World Wide Web has grown in popularity in recent years, the percentage of network trac due to HTTP requests has steadily increased. Recent reports show that Web trac has constituted 40% of the network trac in 1996, compared to only 19% in Since the majority of Web documents requested are static documents (i.e. home pages, audio and video les), caching at various network points provides a natural way to reduce web trac. A common form of web caching is caching at HTTP proxies. Key to the eectiveness of proxy caches is a document replacement algorithm that can yield high hit ratio. Unfortunately, conventional wisdom which has been gained in the context of le caching and virtual memory page replacement does not necessarily transfer to Web caching. There are three primary dierences between Web caching and conventional paging problems. First, web caching is variable-size caching. Due to the restriction in HTTP protocols that allows only whole- le transfers, a cache hit only happens if the entire le is cached, while web documents vary dramatically in size. Second, web pages take dierent amount of time to download. A proxy that wishes to reduce the average latency of web accesses may want to adjust its replacement strategy based on the downloading latency. Third, access streams seen by the proxy cache is the union of web access streams from tens to thousands of users, instead of coming from a few programmed source as in the case of operating system paging. In addition, proxy caches are in a unique position to aect web trac on the Internet. Since the replacement algorithm decides whether a document gets cached and which documents are replaced, it affects which future requests will be cache hits. Thus, if Department of Computer Science, University of Wisconsin- Madison, cao@cs.wisc.edu. yinformation and Computer Science Department, University of California-Irvine, irani@ics.uci.edu. the institution employing the proxy must pay more on some network links than others, the replacement algorithm can favor expensive documents (i.e. those travelling through the expensive links) over cheap documents. If it is known that certain network paths are heavily congested, the caching algorithm can retain more documents which must travel on congested paths. Finally, the proxy cache can reduce its contribution to the network router load by caching more documents that travel many hops than those that travel few hops. Web cache replacement algorithms can incorporate these considerations by associating an appropriate network cost with every document. and minimizing the total cost incurred over a particular access stream. In this paper, we introduce a new algorithm, called GreedyDual-Size, which combines locality, size and latency/cost concerns eectively to achieve the best overall performance. GreedyDual-Size is a variation on a simple and elegant algorithm called GreedyDual (designed by Neal Young [You91b]), which handles uniform-size variable-cost cache replacement. Using trace-driven simulation, we show that GreedyDual- Size with appropriate cost denitions outperforms existing web caching algorithms in a number of performance aspects, including hit ratios, latency reduction, and network cost reduction. Due to space limitation we only report a subset of our results here; more details can be found in [CI97]. 2 Algorithm GreedyDual-Size The original GreedyDual algorithm is proposed by Young and Tarjan [You91b]. It is concerned with the case when pages in a cache have the same size, but incur dierent costs to fetch from a secondary storage. The algorithm associates a value, H, with each cached page p. Initially, when a page is brought into cache, H is set to be the cost of bringing the page into the cache (the cost is always non-negative). When a replacement needs to be made, the page with the

2 Algorithm GreedyDual: Initialize L 0. Process each request document in turn: The current request is for document p: (1) if p is already in memory, (2) H(p) L + c(p)=s(p). (3) if p is not in memory, (4) while there is not enough room in memory for p, (5) Let L min q2m H(q). (6) Evict q such that H(q) = L. (7) Bring p into memory and set H(p) L + c(p)=s(p) end Figure 1: Greedy Dual Algorithm. lowest H value, min H, is replaced, and then all pages reduce their H values by min H. If a page is accessed, its H value is restored to the cost of bringing it into the cache. Thus, the H values of recently accessed pages retain a larger portion of the original cost than those of pages that have not been accessed for a long time. By reducing the H values as time goes on and restoring them upon access, the algorithm integrates the locality and cost concerns in a seamless fashion. To incorporate the dierence sizes of the document, we extends the GreedyDual algorithm by setting H to cost=size upon accesses to a document, where cost is the cost of bringing the document, and size is the size of the document in bytes. We called the extended version the GreedyDual-Size algorithm. The denition of cost depends on the goal of the replacement algorithm: cost is set to 1 if the goal is to maximize hit ratio, it is set to the downloading latency if the goal is to minimize average latency, and it is set to the network cost if the goal is to minimize the total cost. At the rst glance, GreedyDual-Size would require k subtractions when a replacement is made, where k is the number of documents in cache. However, a different way of recording H removes these subtractions. The idea is to keep an \ination" value L, and let all future setting of H be oset by L. Figure 1 shows an ecient implementation of the algorithm. Using this technique, GreedyDual-Size can be implemented by maintaining a priority queue on the documents, based on their H value. Handling a hit requires O(log k) time and handling an eviction requires O(log k) time, since in both cases the queue needs update. 3 Simulation Results We compare the performance of GreedyDual-Size with, Size, Hybrid, and LRV. Size, Hybrid, LRV are \champion" algorithms on improving hit ratio and reducing average latency, as reported in [WASAF96, LRV97, WA97]. Size replaces the largest document in the cache rst. Hybrid is designed to minimize average downloading time, and it replaces the document with the lowest value of the function c s + W b b s z p (n p ) Wn, where c s is the estimated time to connect to the server, b s is the estimated bandwidth to the server, z p is the size of the document, n p is the number of times the document has been referenced, and W b and W n are constants. LRV replaces the document with the lowest value of the following function: V (i; t; s) = P1 (s)(1? D(t)) if i = 1 P i (1? D(t)) otherwise where i is the number of times the document has been accessed, t is time of last reference to the document, s is the size of the document, P i is the conditional probability that a document is reference i + 1 times given that it has been accessed i times, P i (s) is the conditional probability specic to documents of size s, and D(t) is calculated by: D(t) = :035 log(t + 1) + :45 1? e?t 2e6 : Essentially, LRV tries to improve hit ratios by taking into account of a document's reference frequency, time of last reference, and size. We consider four aspects of web caching benets: hit ratio, byte hit ratio, reduced latency, and reduced hops. By reduced latency, we mean the sum of downloading latency for the pages that hit in cache as a percentage of the sum of all downloading latencies. By reduced hops, we mean the sum of the network costs for the pages that hit in cache as a percentage of the sum of the network costs of all Web pages. We model the network cost associated with each document as \hops". In our simulation, each Web server has a hop value of 1 or 32, and we assign 1=8 of the servers with hop value 32, and 7=8 with hop value 1. The hop value can be thought of either as the number of network hops travelled by a document, or as the monetary cost associated with the document. We present the results on a subset of Digital Equipment Cooperations WWW proxy traces [DEC96](results on other proxy traces can be found in [CI97]). The subset includes HTTP requests made by users 0-512

3 Trace Total Total Hit Byte Reduced Reduced Requests GB Rate HR Latency Hops 8/29-9/ /5-9/ /12-9/ /19-9/ Table 1: Benets under a cache of innite size for each trace, measured as hit ratio, byte hit ratio, reduced latency, and reduced hops. The table also lists the total requests and the total data set size in GB for each trace. during each week, for the three and half week period from 8/29/96 to 9/22/96. Table 1 shows the performance benets under an innite-size cache for these traces. We consider GreedyDual-Size with three cost definitions. sets the cost for each document to be 1, thus trying to maximize hit ratio. GD- Size(packets) sets the cost for each document to 2 + size=536, i.e. estimated number of network packets sent and received if a miss to the document happens (1 packet for the request, 1 packet for the reply, and size=536 for extra data packets, assuming a 536-byte TCP segment size). It tries to maximize both hit ratio and byte hit ratio. Finally, GD-Size(hops) sets the cost for each document to the hop value of the document, trying to minimize network costs. Figures 2 and 3 show the hit ratio, byte hit ratio, reduced latency and reduced hops for various algorithms. The cache sizes are 0.5%, 5%, 10%, and 20% of the total data set sizes in each trace. The data are shown as a percentage of the corresponding value under the innite cache. The graphs show the averaged results on the four traces. The results show that provides the highest hit ratio and reduced latency, provides the highest byte hit ratio and relatively high hit ratio, and GD-Size(hops) provides the highest reduced hops. and GD-Size(hops) approaches the maximal achievable benets very fast, having over 95% of the maximal hit ratio or reduced hops when the cache size is only 5% of the total data set size. Some observations can help explain our results. GreedyDual-Size combines the locality, size and cost of a document by using (cost=size) (i.e. savings per byte) as the H value of a cached document. It then subtracts the H value of each document as it makes replacements, and re-instates the H value if a document is accessed again. Thus, a document with high H is initially cached, but if it is not accessed for a long time, it will be replaced. By subtracting and re-instating H, GreedyDual-Size combines locality, cost and size considerations without using any weight function or parameters. In contrast, does not consider document sizes and Size does not consider locality. LRV consider all relevant factors, but combines them in a complicated function that makes its performance trace-dependent. Hybrid performs poorly because it ignores the locality in reference streams, and thus has a very low hit ratio, which in turn results in low latency savings. In summary, if a proxy wants high hit ratio or low average latency, is the appropriate algorithm. If the proxy desires high byte hit ratio as well, then achieves a good balance among the dierent goals. If documents have associated network or monetary costs that do not change over time, or change slowly over time, then GD-Size(hops) is the appropriate replacement algorithm. 4 Conclusion In this paper, we introduce GreedyDual-Size web cache replacement algorithm, and show that it incorporates locality, size and cost considerations eectively to achieve the best performance. We also show that under GreedyDual- Size, web caches have the potential to regulate Web trac to avoid congested links. References [LRV97] P. Lorenzetti, L. Rizzo and L. Vicisano. Replacement Policies for a Proxy Cache. Manuscript. [CI97] Pei Cao and Sandy Irani. Cost-Aware Web Proxy Caching. Technical Report CS-1343, Computer Science Department, University of Wisconsin-Madison. [DEC96] Digital Equipment Cooperation, Digital's Web Proxy Traces ftp://ftp.digital.com/pub/dec/traces/proxy/webtraces.html.

4 LRV LRV Relative Hit Ratio Relative Byte Hit Ratio Figure 2: Relative hit ratio and byte hit ratio for, Size, LRV,, and. Hybrid GD-Size(hops) Relative Reduced Latency Relative Reduced Hops Figure 3: The left gure shows the relative reduced latency for, Size, Hybrid, and GD- Size(packets), and the right gure shows the relative reduced hops for, Size, and GD-Size(hops).

5 [WASAF96] S. Williams, M. Abrams, C.R. Standbridge, G.Abdulla and E.A. Fox. Removal Policies in Network Caches for World-Wide Web Documents. In Proceedings of the ACM Sigcomm96, August, 1996, Stanford University. [WA97] R. Wooster and M. Abrams. Proxy Caching the Estimates Page Load Delays. To appear in the 6th International World Wide Web Conference. [You91b] N. Young. The k-server dual and loose competitiveness for paging. Algorithmica, To appear. Rewritten version of \On-line caching as cache size varies", in The 2nd Annual ACM- SIAM Symposium on Discrete Algorithms, , 1991.