# Comparing Hybrid Peer-to-Peer Systems

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5 describe a simple query model that can be used to estimate the desired values. We assume a universe of queries Õ ½ Õ ¾ Õ. Wedefine two probability density functions over this universe: the probability function that describes query popularity. That is, µ is the probability that a submitted query happens to be query Õ. the probability density function that describes query selection power. In particular, if we take a given file in a user s library, with probability µ it will match query Õ. For example, if ½µ¼, and a library has 100 files, then we expect 50 files to match query Õ ½. Note that distribution tells us what queries users like to submit, while tells us what files users like to store (ones that are likely to match which queries). Calculating ExServ for the Chained Architecture. Using these two definitions we now compute ExServ, theexpected number of servers needed in a chained architecture (Section 3) to obtain the desired Ê MaxResults results. Let È µ represent the probability that exactly servers, out of an infinite pool, are needed to return Ê or more results. The expected number of servers is then: ÜËÖÚ ½ È µ ½ ½ È µ (1) where is the number of servers in the actual system. (The second summation represents the case where more than servers from the infinite pool were needed to obtain Ê results. In that case, the maximum number of servers,, will actually be used.) In [24] we show that this expression can be rewritten as: ÜËÖÚ ½ ½ É UsersPerServer FilesPerUserµ Here É Òµ is the probability that Ê or more query matches can be found in a collection of Ò or fewer files. Note that Ò UsersPerServer FilesPerUser is the number of files found on servers. To compute É Òµ, we first compute Ì Ò Ñµ, the probability of exactly Ñ answers in a collection of exactly Ò files. For a given query Õ, the probability of Ñ results can be restated as the probability of Ñ successes in Ò Bernoulli trials, where the probability of a success is µ. Ì Ò Ñµ can be computed as: Ì Ò Ñµ ½ µ Ò Ñ ¼ Thus, µµ Ñ ½ µµ Ò Ñ É Òµ can now be computed as the probability that we do not get ¼ ½ or Ê ½ results in exactly Ò files, or É Òµ ½ Ê ½ Ñ¼ Ì Ò Ñµ Calculating Expected Results for Chained Architecture. Next we compute two other values required for our evaluations, ExLocalResults and ExRemoteResults, againfor a chained architecture. ExLocalResults is the expected number of query results from a single server, while ExRemoteResults is the expected number of results that will be returned by a remote server. The former value is needed, for instance, to compute how much work a server will perform as it answers a query. The latter value is used, for example, to compute how much data a server receives from remote servers that assist in answering a query. Both values can be obtained if we compute Å Òµ to be the expected number of results returned from a collection of Ò files. Then, ExLocalResults is simply Å Ýµ, where Ý UsersPerServer FilesPerUser. Similarly, the expected total number of results, ExTotalResults,isÅ Ýµ. Then ExRemoteResults is ExTotalResults - ExLocalResults Å Ýµ Å Ýµ. In [24] we show that Å Òµ ½ Ê ½ Ò µ Ê µµ Ñ Ñ¼ Ñ ½ µµ Æ Ñ Ê Ñµ ¼ Calculating Expected Values for Other Architectures. So far we have assumed a chained architecture. For full replication, ExServ is trivially ½, since every server contains a full replica of the index at all other servers. Because ExServ is 1, all results are local, and none are remote. Hence ExRemoteResults ¼,andExLocalResults = ExTotalResults Å Æ µ, whereå is defined above, and Æ UsersPerServer FilesPerUser is the number of files in the entire system. For the unchained architecture, ExServ is trivially 1, since queries are not sent to remote servers. ExRemoteResults ¼ and ExLocalResults Å Òµ, where Ò UsersPerServer FilesPerUser is the number of files at a single server. Finally, for the hash architecture, again ExRemoteResults ¼ and ExLocalResults Å Æ µ, since all the results are found at the server that the client is connected to. ExServ is more difficult to calculate, however. The problem again takes the form of equation (1), but now the probability È µ that exactly servers are needed to to return Ê or more results is equal to the probability that exactly servers contain Û inverted lists, where Û is the number of words per query. Finding this probability is the same as finding the probability that Û balls being assigned to exactly bins, a well-known probability problem. We refer the reader to [24] for a complete solution. Distributions for and. While we can use any and distributions in our model, we have found that exponential distributions are computationally easier to deal with and provide accurate enough results in the music domain. Thus, we assume for now that µ ½. Since this function monotonically decreases, this means that Õ ½ is the most popular query, while Õ ¾ is the second most popular one, and so on. The parameter is the mean. If is small, popularity drops quickly as increases; if is large, pop-

9 Maximum Users Supported (per server) 9 x batch CHN incr. CHN 3 batch FR incr. FR 2 batch HASH incr. HASH 1 batch UNCH incr. UNCH Query/Logon Ratio Expected Number of Results (per query) batch CHN incr. CHN batch FR incr. FR batch HASH incr. HASH batch UNCH incr. UNCH Query/Logon Ratio Memory Required (bytes) 2.5 x batch CHN incr. CHN 1 batch FR incr. FR batch HASH 0.5 incr. HASH batch UNCH incr. UNCH Number of Users x 10 4 Figure 3: Overall Performance of Strategies vs. QueryLoginRatio Figure 4: Expected Number of Results Figure5: Memory Requirements memory required by these two strategies is shown in Figure 5 by the large o marks. Again, the amount of memory required by incremental CHN is far too large for our limit. However, looking at incremental CHN performance at the 4 GB limit, we find that the users supported by incremental CHN is greater than the 9190 users supported by batch CHN. Hence, incremental CHN is still the better choice, because within the limit, it still has better performance than batch CHN. 6.2 Beyond Music: Systems in Other Domains In Section 4, we describe a query model where, given distributions for query frequency and query selection power, we can calculate the expected number of results for a query, and the expected number of servers needed to satisfy the query. The model itself is completely general in that it makes no assumptions about the type of distributions used for and ; however, for the purposes of modeling musicsharing systems and validating the model, we made the temporary assumption that and are exponential distributions. Implied by this assumption is the additional assumption that and are positively correlated thatis, the more popular or frequent a query is, the greater the selection power it has (i.e. the larger the result set). While we believe that our music model can be used in many other domains, one can construct examples where the and distributions have different properties. Differences in and distributions can be caused by different characteristics of application domains, and/or varying expressiveness of queries. For example, if the system supported complex, expressive relational queries, an obscure query such as select * from Product where price 0 would return as many results as a common query such as select * from Product. In this case, there is very little, if any, correlation between the popularity of a query and the size of its result set. We say that such systems have no correlation distributions. Another example might be an archive-driven system, where users provide old, archived files online but keep their current, relevant information offline. This scenario might occur in the business domain where companies do not want their current data available to the public or their competitors, but might be willing to provide outdated information for research purposes. Because archives hold historical data, popular queries which refer to recent times return less data, whereas rare queries which refer to the past will return more. In this case, there is a negative correlation between query popularity and selection power. In [24], we show through detailed analysis how the behavior of systems with negative and no correlation distributions can in fact be approximated very closely by systems with positive correlation distributions. Although the input to the query model, the and distributions, are very different, the output of the model, values for ExServ and ExResults, are closely matched. In particular, no correlation distributions are closely approximated by positive correlation distributions where Ö ½, andnegative correlation distributions are closely approximated when Ö is very large. Thus, we can understand the behavior of systems with negative and no correlation distributions by studying the performance of systems with a wide range of positive correlation distributions. To illustrate, Figure 6 shows CPU performance of each strategy as Ö is varied. For comparison, the Ö value derived for the OpenNap system is shown by a dotted line in the figure. In addition, in [24] we presented several representative curves for negative and no correlation distributions, and determined the best-fit positive correlation approximations for each. For comparison, the Ö values derived for the approximations of the negative and no correlation curves are also shown in Figure 6 by dotted lines. From this figure, we can make several observations about the performance of strategies in different query models: With negative correlation distributions, or positive correlation with high Ö, incremental strategies are recommended. CPU performance of incremental strategies is usually bound by expensive queries (see [24]), but with negative correlation or high Ö, the expected number of results is very low, thereby making queries inexpensive. With no correlation distributions, or positive correlation with low Ö, batch strategies outperform incremental (except for FR). This is because when Ö is low, the expected number of results is very high, thereby making queries expensive and incremental performance poor. Batch FR performs so poorly because of expensive logins. CHN and HASH show greater improvement than FR as Ö increases. FR, which has poor login performance but

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