Theoretical study of cache systems

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1 Theoretical study of cache systems Dmitry G. Dolgikh Samara State erospace University, Moscow sh., 34a, Samara, , Russia and ndrei M. Sukhov Samara Institute of Transport Engineering, Bezymyannyi per., 8, Samara, , Russia bstract The aim of this paper is a theoretical study of a cache system in order to find a way for optimization of proxy systems and to modernize construction principles. Two types of the correlations, Zipf-like distribution and normalizing conditions, play a role of fundamental laws. Corresponding system of equations allows to describe basic effects like ratio between construction parts, steady-state performance, optimal size, etc. Modification of the fundamental laws leads to a new effect of documents renewal in global network. Keywords: Cache proxy, Zipf-like distribution, cache optimization, steady-state performance, renewal of Web documents Inroduction Rapid development of computer technologies in the end of the last millennium leads to appearance of a virtual world with its own laws. Unfortunately, in this period too few attention was given to studying of fundamental principles of virtual life. The main constructions of the virtual world base on the concrete algorithms intended for providing vital functions. The easiest and obvious principles were used for constructing and modernizing of these algorithms. During their implementation weren t any time for studying and optimization. Often their low performance was compensated with growing power of computers. The main thing was that the new constructions of the virtual world allowed to manage the raising problems. Now days we can afford some respite and part of forces should be transferred to studying of fundamental laws and to optimization of vital systems on a base of recent knowledge. Frankly speaking, any research area may be considered as scientific field if and only if its basic principles are enveloped in a mathematical form and new rules may be predicted on a base of known facts. The aim of this paper is a theoretical study of a cache system in order to find a way for optimization of proxy systems and to modernize construction principles. Principles that should be used for a model is discussing as well. universal model can be easy generalized to any applications based on Zipf-like distri- corresponding author: sukhov@ssau.ru bution like Content distribution Networks (CDN) [8], peer-to-peer systems [, ], Internet search engine, etc. Model parameters In this section definitions of variables are given and basic approaches and terminus has been analyzed. ll network users are considered as one collective user. This user sends all requests to a global network as it is shown on the Fig.. Some documents are requested several times and therefore they should be held in a cache system. s it was reported in Refs. [, 3, 0] the relative frequency of requests to Web pages follows Zipf-like distribution [4]. This distribution states that the relative probability of a request for the i th most popular page is ϑ i = i α, () where = ϑ is the probability of the most popular item and α is positive exponential value less then unity. Let the users ask for K documents during time t: K(ν out, t) = Cν out t. () This result depends on aggregated bandwidth ν out, on time t and a constant C. The constant C from Eq. () is an inverse proportion of a mean size E(C) of documents received from the global network: C = E(C) (3) In Ref. [3] a number of received documents K is defined by request stream from a population of N users K(λ, N, t) = λnt, (4) where λ is average client request rate. Let p documents be unique and M be equal to quantity of documents that can be requested from the cache system as it has shown on the Fig.. Then an efficiency of the system or a hit ratio can be defined as H = k p M K = dx (5) xα

2 Figure : Scheme of the proxy cache the ratio of non-unique cacheable documents k p = k M x dx to their total number K. α Wolman et al. [3] examined steady-state or limiting performance of caching schemes. Their model of steady-state performance assumes that a cache can store all cacheable documents in the Web and there isn t any capacity misses in workload. The probability that a requested document is cacheable is p c, (k = p c K). With this assumptions the maximal rates of H is identical with p c, i.e. the ideal hit rate for cacheable documents H i = H/p c would approach 00%. key difference between the models presented by Breslau at al [3] and by Wolman et al [3] is that document rate of change has been incorporated into the model rather than assuming that documents are static. In order to describe this effect an additional multiplier has been included to a integral from Eq. (5): λnp i /(λnp i + µ), (6) where µ is an exponential parameter of interarrival times for object requests and updates, p i is proportional to /i α. This multiplier throws off one effective query of any document from cache during time T ch = /µ between its changes. Such an updating request must be redirected to the global network to get the renewed Web page. 3 The analytical laws and the special points n analytical model is constructed with a goal to predict behaviour of the investigated systems on the base of the analytical laws written in a mathematical form and to find ways for optimization. Mathematical constructions of computer models remind laws of the thermodynamics and the molecular physics in many respects. There are two directions of modeling: macroscopic and microscopic approaches. In the first case the corresponding laws are formulated for parameters describing cache systems as an Figure : Special points of cache system indivisible object. The following values should be considered as macroscopic parameters: hit ratio H, Zipf exponent α, the number of unique documents p, M, etc. The microscopic description assumes an analysis on a level of transmitting IP packets, of requests to the global network, i.e. huge number of the events. In this case investigators have to apply the theory of stochastic processes such as Poisson s distribution or Markov s chain. The main problem of this approach is a right interpretation of analytical results and their generalization, i.e in a limit passage from microscopic values to macrolaws. Usually the macroscopic dependencies operate with values that are measured directly. Unfortunately, corresponding laws are not yet found for the main part of the virtual world therefore a particular attention should be given to their search. Proxy cache is a pleasant exception to the general rule. Two types of correlations pretend to a role of fundamental laws described cache systems [5]: Zipf-like distribution, see Eq. () Normalizing conditions or a sum of the probability to request the universe of n k objects. Mentioned laws could be applied to the special points of Zipf distribution. s it is shown on the Fig. there are a few special points but only two of them, M and p, are used for construction of the theory. Then the Zipf-like distribution leads to k M α =, (7) k p α =. (8) Normalizing conditions for the first M and p documents from a cache give M x α dx = H i (9)

3 p dx = (0) xα Here H i is an ideal (steady-state) performance for cacheable documents. For a real system the Eq. (9) has been transformed to H = p c S k dx, () xα where S k is a number of cache objects requested repeatedly. Five Egs. (7-) allow to describe the basic effects, to predict parameters value, ways of optimization, etc. Modification of this system of five equations would lead to finding of new effects. 4 Elements of cache construction This section discusses which elements are compulsory for cache system. lso possible ratios between these elements should be investigated. Three main parts are distinguished in any cache system: kernel S k that contains popular documents with ϑ i. n accessory part S u that keeps unpopular documents requested from the Internet once, i.e. ϑ i =. managing part S m that contains statistics of requests and rules for replacement of cache objects. Now we can study mathematical correlations between cache elements S k, S u, S m. With help of the Eqs. (7, 8) p = /α M, () and with analyzing log files of proxy cache a mean lifetime t u can be calculated for those documents, which popularity is ϑ i =. Such a statistic determines also lifetime T eff of cache objects with a citing index ϑ i =, i.e. those items that have been stored in proxy cache, requested one time from a proxy and have been deleted subsequently. It is easy to see that a number of documents in a cache kernel S k is a value M(T eff ) corresponding to the statistic for time T eff, and S u depends on p, M, t u : S k = M(T eff ), (3) S u = p(t u ) M(t u ), (4) The following ratio between the part S k, S u has been found in Ref. [?] with help of Eqs. (), (): S k T eff = (5) S u ( /α )t u nother useful relation is given as a solution of a system of Eqs. (7,9) concerning variable M: M = ( α)h K (6) This means that only each 5th document should be stored in the cache among all documents requested from the global network (for typical values α 0.7, H 35%). The Eq. (5) gives a criteria for optimal using of a storage capacity. This criteria allows to plane an experiment: how does efficiently of cache replacement algorithm depend on correlation of the proxy parts? The analysis of log files from Samara State erospace University proxy [6] gives the equality T eff t u. It means that only 40% of storage capacity has been used as the kernel of the proxy cache. It should be noted this result as well as the experiment plan is the consequence of the theoretical study. 5 Steady-state performance In this section we examine steady-state or limiting performance of caching schemes. Wolman et al model [3] of steady-state performance assumes that a cache can store all cacheable documents in the Web and there aren t any capacity misses in the workload. With this assumption, they conceive that in the long term the hit rate H i for cacheable documents would approach 00%. However, considerable part of cacheable documents p M is requested from the global network only one time as it is shown on Fig. that illustrate Zipf-like distribution. More accurate expression for ideal hit ratio H i may be written after receiving correlation between p and k from system of Eqs.(8,0): k = p α, (7) which is used for experimental search of α. With help of Eqs.(,6) H i (α )/α, (8) that gives 80% for α = 0.75 or 84% for α = 0.8. The alternative expression for ideal hit ratio H i follows from Fig. : H i p M k = ( α)( /α ) (9) The hit rate H of web cache is considered to grow in a log-like fashion as a function of cache size [, 3, 4]. From analyzes of the Eq. () for cache performance the following dependence appears H H = ( S S that allows to talk about power fashion. ) α, (0)

4 6 Optimization of cache systems fine tuning of cache that increases hit ratios by only several percentage points would be equivalent to a several-fold increase in cache size. In order to determine an optimal cache size Kelly and Reeves [9] are guided on economical methods like monetary cost of memory and bandwidth. The aim of our theoretical investigation is a search of dependencies between cache size S = S k + S u and other macroscopic parameters like internal bandwidth ν int, maximum hit ratio H i, Zipf s exponent α, etc. s it is shown in Ref. [5] a ratio of effective size of cache system S eff to aggregated bandwidth of external links ν int can be considered as a constant: S eff ν int = τ. () Our goal is to calculate a lower limit τ(µ, α) when the performance mounts to the 35% level (H eff 35%) of well working system. Eqs.(8,9) give the following estimation for a real system with α 0.7, p c = 0.6: p c H i H eff 35% () Eqs.(9,) allow to calculate the correlation between documents number in a real cache kernel S k and a number M max corresponding to limiting performance H i : where S k = /((α )) M max (3) M max = ( α)p ch i ν out T ch (4) Parameter T ch = /µ u = 86 days is the time between renewal of unpopular documents [3]. If H eff = 0.35, then τ(µ u, α) = /α /((α )) α.6µ u. (5) It is easy to calculate τ = 6, 0 days for α = 0.8. Figure 3: The renewal effect The key formulas from Wolman et al. C N = C = n n ( Cx α + µcxα λn ) dx (6) dx (7) xα yield C N, the aggregate object hit ratio considering only cacheable objects. Document rates of change µ were considered to take two different values, one for popular documents µ p and another for unpopular documents µ u. In present paper we assume that the document rate of change µ(i) depends on its popularity i. Mathematical equivalent of this assertion is that the steady-state process is again described by Zipf distribution with a small α R as it is shown of Fig. 3. difference between limiting performance of a cache system and its real value has been conditioned by renewal of documents in the global network and can be investigated in an experimental way. We have to explore the log file collected for the time T st T eff from Eq. (3), so that M(T st ) S k. Then the ideal hit ratio H i corresponds to the upper line on Fig. 3 with exponent α, but the real hit ratio H determines α R as 7 Renewal of Web documents α R = M HK = p HK + p M (8) In the previous section we based our analysis on the model presented by Breslay et al [3] and extended by Wolman et al. [3] to incorporate document rate of change. The Wolman model yields formulas to predict steady-state properties of Web caching systems parameterized by population size N, population request rates λ, document rate of change µ, size of object universe n and a popularity distribution for objects. Now it is easy to find µ i µ(i) = ϑid i ϑ R i T st = (p/i)α (p/i) αr T st (9) It should be noted again that the experiment s plan is the consequence of the theoretical study. Finally, the system of Eqs. (7-) must be modernized by change of variables α on α R.

5 8 Discussion and feature work The aim of this paper is a theoretical study of a cache system in order to optimize proxy cache systems and to modernize construction principles. Two types of correlations, Zipf-like distribution and normalizing conditions, play a role of the fundamental laws. corresponding system of equations allows to describe the basic effects like ratio between construction parts, steady-state performance, optimal size, etc. modification of the fundamental laws leads to the new effect of documents renewal in the global network. The main result of any research of cache systems is finding a way for increasing hit ratio. Our theoretical study leads to the criteria for an optimal using of storage capacity. This criteria allows to plane several experiment on the base of the mathematical analysis. In order to find a ratio between the parts of cache construction we analyze the cache log files [6]. The mean lifetime t u has been calculated for those documents, which popularity is ϑ i =. Such statistics determine also the lifetime T eff for cache objects with the citing index ϑ i =, i.e. those items that have been stored in proxy cache, requested one time from the proxy, and deleted subsequently. nalysis of the experimental data leads to the facts that variables t u and T eff can be considered as coincided values. In other words the kernel and the accessory part are approximately correlated as 40:60 or only 40% of storage capacity is used for the basic goal of storing repeatedly requested documents. We see the two basic directions for increasing hit ratio. The first conjecture is the hit rate H increases with growth of S k /S u. The second one is based on the general feature of the existing cache algorithms that the only documents requested two times and more during the time t u are included in the kernel S k. Therefore we conclude that caching algorithm based on the rigid tie of the main cache parameters to the time t u is low effective. Our approach is to separate the requests statistics from the time t u, i.e. cache system operates with controlling information collected for the period longer then t u. The current situation can be resolved with the new construction principles for cache systems. Those principles are suggested in [7]. The most important one is that fixed ratio between cache parts must be guaranteed. The statistics of requests must be kept for long time for all cacheable documents including delayed ones. n inseparable part of the new construction is the replacement algorithm based on Zipf law. t this time we have prepared the corresponding application for an international patent. in: IEEE International Conference in Parallel and Distributed Information Systems, Miami Beach, Florida, US, December, 996 [] V. lmeida, M. Cesirio, R. Canado, W. Junior, C. Murta, nalyzing the behavior of a proxy server in the light of regional and cultural issues, 3rd International WWW Caching Workshop, Manchester, England, June 998 [3] L. Breslau, P. Cao, L. Fan, G. Phillips, S. Shenker, Web Caching and Zipf-like Distribution: Evidence and Implications, in: IEEE Infocom, vol. XX, no. V, 999, pp. -9 [4] P. Cao, S. Irani, Cost-aware WWW-proxy caching algoritms, Proceedings of the USENIX Symposium on Internet technology and Systems, December 997, pp [5] D.G. Dolgikh,.M. Sukhov, Parameters of cache system based on a Zipf-like distribution, Computer Networks, 37(6), 00, pp.7-76 [6] D. Dolgikh,. Sukhov, The measurements, parameters and construction of Web proxy cache, submitted to 7th International Workshop on Web Content Caching and Distribution (WCW), ddolgikh/constparameters.pdf [7] D. Dolgikh,. Sukhov, New principles of Web cache construction, submitted to HotNets-I0, ddolgikh/cachepatent.pdf [8] S. Gadge, J. Chase, M. Rabinovich, Web caching and content distribution: a view from interior, Computer Communications, 4(), 00, pp. -3 [9] T. Kelly, D. Reeves, Optimal Web cache sizing: scalable method for exact solutions, Computer Communications, 4(), 00, pp [0] N. Nishikawa, T. Hosokawa, Ya. Mori, K. Yoshidab, H. Tsujia, Memory-based architecture for distributed WWW caching proxy, in: The 7th WWW Conference, pril 998 [] M. Ripeanu, I. Foster,. Iamnitchi, Mapping the Gnutella Network: Properties of Large Scale Peerto-Peer Systems and Implications for System Design, matei/ppers/ic.ps [] K. Sripanidculchai, The popularity of Gnutella queries and its implications on scalability, February 00, kunwadee/research/pp/ gnutella.html [3]. Wolman, G. Voelker, N. Sharma, N. Cardwell,. Karlin, H. Levy, On the scale and performance of cooperative Web proxy caching, Operating Systems Review, 34(5), Dec.999, pp.6-3 [4] G.K. Zipf, Relativity frequency as a determinant of phonetic change, Reprinted from the Harvard Studies in Classical Philology, Vol. XL, 99 References [] V. lmeida,. Bestavros, M. Crovella, and. De Oliveira, Characterizing reference locality in the WWW,