ANALYTICAL MODELING OF PEER-TO-PEER FILE SHARING SYSTEMS. M. Meulpolder

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1 ANALYTICAL MODELING OF PEER-TO-PEER FILE SHARING SYSTEMS M. Meulpolder Parallel and Distributed Systems Group Department of Computer Science, Delft University of Technology, the Netherlands ABSTRACT In this paper, we present a survey of recent work in analytical modeling of peer-to-peer (P2P) file sharing systems. We focus on the modeling of three aspects of such systems. First, we describe several performance models regarding chunk-based protocols such as BitTorrent [1]. Secondly, we summarize a trace analysis and corresponding model of the workload and user behavior in a general file sharing context. Finally, we give a summary of a study into the effect of pollution attacks. 1. INTRODUCTION Peer-to-peer (P2P) file sharing systems have become very popular during the last decade. In the late 1990s, early systems such as Napster [10] and Kazaa [5] quickly obtained a large user base, and brought the concept of P2P under the attention of the large public. More recently, advanced protocols such as BitTorrent [1] have come to dominate traffic on the internet backbone [2]. Due to these developments, P2P systems and protocols have gained increasing interest in the research community. Several significant papers have been published, containing mathematical models, measurements, and simulation results. In this paper, we focus on the analytical modeling of P2P file sharing systems. We present a survey of recent, significant papers in this field. We give an overview of the models, the analysis and the main results presented in these papers. The choice of papers is centered around three different aspects of P2P file sharing systems: (1) the performance; (2) the workload and user behavior; (3) the effect of pollution attacks. Before covering these topics, we will give a short overview of relevant concepts in P2P file sharing systems. 2. OVERVIEW We focus on file sharing systems on the Internet, i.e., systems in which any user with an internet connection can decide to participate by downloading and/or sharing files. Users can decide to contribute to the system by sharing files altruistically to others, or they can try to freeride, i.e., downloading content without uploading. In general, end-user connections can be heterogeneous, varying from simple dial-up modem connections to high-speed broadband links. Nowadays, a large part of the users has access to ADSL, which has the property that the incoming bandwidth is usually considerably higher than the outgoing bandwidth. Currently, the most important protocol is the BitTorrent [1] protocol. In BitTorrent, files are divided into chunks (typically 256 KB), which are bartered among users interested in the file (tit-for-tat). Users that are in the process of downloading a file are called leechers, while users that provide the complete file to the system are called seeders. The collection of all seeders and leechers associated with a single file is called a swarm. The tit-for-tat protocol in BitTorrent forces an inherent balance between uploading and downloading, thereby making freeriding less attractive. This is the most important reason for the relatively high performance experienced by BitTorrent users, explaining its high popularity. Regarding performance modeling, we focus on chunkbased systems such as BitTorrent. In static performance models [6, 9], analysis focuses on a particular configuration with a constant number of downloaders and seeders. In dynamic performance models [11, 12], peer arrivals and departures are taken into account as well. Several different modeling strategies are applied in these papers, such as fluid modeling and linear programming. Unlike performance models, the workload and user behavior of a file sharing system can not be studied from an exclusively theoretical perspective. The popularity of files is strongly dependent on the interests, habits, and background of the users. However, certain characteristics of a system workload can be derived from the nature of the shared objects. In [4], a model is presented of the popularity of files, based on a real-life trace of Kazaa. In any file sharing system, malicious users can decide to consciously pollute the system with content that is injected under a false name. The effect of pollution attacks depends on the strategies of both attackers and normal users. In [7], a model is presented for pollution prolifer-

2 ation, based on different file popularity scenarios. In the coming sections, we will present the various papers on the three topics in more detail. 3. PERFORMANCE MODELING In this section, we summarize four insightful papers containing performance models of P2P file sharing systems. The first two papers focus on analysis in a static context, while the next two papers consider a system with dynamic peer arrivals and departures Optimal Peer-Assisted File Distribution: Single and Multi-Class Problems In [6], the authors analyze the distribution of a single file from a set of seeds to a set of leechers. A distinction is made between single-class and multi-class distribution. In the first case, all leechers are equal. In the latter case, second-class leechers are allowed to use the service, acting as rate amplifiers to increase the download speed of first-class leechers. In practice, this could resemble a situation in which some peers pay for service, while others are allowed to receive free service of lower speeds Single-class model The paper considers a heterogeneous set of nodes I = S L, with S the set of seeds, and L the set of leechers. Each node i has an upload bandwidth u i and a download bandwidth d i. The paper analyzes the minimum distribution time T min that is necessary to distribute a file of size F from the seeds to all of the leechers. A continuous fluid model is proposed, with fluid replication at the leecher nodes to account for the exchange of data between leechers. It is remarked that a fluid model has a certain error in approximating a discrete chunk-based model. However, this error is analyzed and shown to be roughly (log 2 L)/M, with M the number of chunks in the file. The main result of the paper is stated in the following theorem: Theorem 1 The minimum distribution time T min for a file of size F to L leechers in the general heterogeneous peer-assisted file distribution system is F T min = min{d min (L), u(i) (1) L, u(s)}, with d min (L) the minimum download bandwidth over all leechers, u(i) the aggregate upload bandwidth over all peers, and u(s) the aggregate upload bandwidth over all seeds. Figure 1: Comparison of the minimum distribution time for client-server and peer-assisted file distribution. In Figure 1, the minimum distribution time as predicted by the theorem is plotted against the distribution time of a client-server system. The paper argues that Theorem 1 can be used to benchmark the distribution time for any peer assisted file distribution protocol. A proof of the Theorem is presented. The paper as well benchmarks the Azureus BitTorrent client against the model Multi-class model In the second half of the paper, an extension of the model is presented in which the collection of leechers is divided into two subsets L 1 and L 2 : first-class leechers and second-class leechers. In this case, a large portion of the upload bandwidth of second-class leechers is used to increase the download speed of the first-class leechers. Since the leechers in L 2 can forward to the leechers in L 1 with a higher speed than with which they receive data from the seeds, they act as rate amplifiers for the first-class leechers (see Figure 2). The resulting minimum distribution time is given by: F T min = min{d min (L 1 ), u(i) L 1 u(l 2) L 2 1, u(s)}. (2) A comprehensive proof of the obtained expression is included in the appendix of the paper Optimal Scheduling of Peer-to-Peer File Dissemination In [9], the same problem of file dissemination from a seed to a number of leechers is studied. Special attention is given to the problem of scheduling which parts of a file are transferred between which users at a certain time. The authors present the results in [9] and [8]. Here, we will present the model and results of both papers as a whole.

3 scheduling, it is possible to enforce a schedule in which this balance is obtained Centralized scheduling with general capacities Figure 2: Schematic overview of the two classes of leechers in the multi-class model. The leechers in L 2 act as a rate amplifier for the leechers in L The uplink-sharing model The paper considers the dissemination of a file of M parts from a server to N end-users, and defines the corresponding uplink-sharing model. The network topology is a complete graph. The server has upload capacity C S and the N peers have upload capacities C 1,..., C N. It is supposed that any number of users can simultaneously connect to the server and any other number of users, the available upload capacity being shared equally among the open connections. A user can download more than one file part simultaneously from different sources. In the model, the upload capacities impose the only constraints on the rates at which file parts can be transferred. The paper addresses the problem of finding the minimum time to distribute the parts to all peers, and calls this time the minimal makespan. The scheduling (i.e., which part is transferred between which peers at which time interval) can be carried out either centralized or decentralized. Solutions for three different cases are presented: (1) centralized scheduling with equal upload capacities; (2) centralized scheduling with general upload capacities; (3) decentralized scheduling with equal capacities Centralized scheduling with equal capacities A solution for the uplink-sharing model with equal upload capacities is given by the following theorem: Theorem 1 Consider the uplink-sharing model with all upload capacities equal to 1. The minimal makespan T for all M, N is given by T = 1 + log 2 N M. (3) A proof is presented. The optimal strategy followed in the proof follows two principles: As many different peers as possible obtain file parts early on so that they can start uploading themselves, and the maximal possible upload capacity is used. Moreover, there is a certain balance in the upload of different file parts so that no part gets circulated too late. Due to the assumption of centralized The model for general capacities is extended in order to provide a solution for general upload capacities. The resulting problem is formulated as a mixed integer linear program (MILP). First, a solution is provided for the case with a large M where C 1 = C 2 =... = C N, but C S might be different. In this case, the minimal makespan is { 1 T N } = max,. (4) C S C S + NC 1 Next, a fluid limit solution is presented for arbitrary capacities C i. The model is also further extended to include the case where every user i has a file of size F i to disseminate to all other users. In this case, the minimal makespan is { T F1 = max, F 2,..., C N (N 1)F },. (5) C 2 C 2 F N C Here, F = N i=1 F i and C = N i=1 C i. A comprehensive proof is presented for the general case Decentralized scheduling with equal capacities In order to account for the more realistic assumption that there is no centralized scheduler, a decentralized solution is analyzed. The paper considers a straightforward randomized strategy and investigates the loss in performance that is caused by the lack of centralized control. The results suggest that the performance in the decentralized case is close to the performance in the centralized case, and that the obtained centralized results are therefore useful as performance benchmarks of real systems Discussion The paper concludes that the results provide realistic insight and solutions for file dissemination in a P2P system. It argues that it would be very interesting to assess real overlay networks to see how close the theoretical results are approximated. In practice certain overhead costs have to be taken into account, such as TCP overhead due to the many short connections corresponding to small file parts. Furthermore, it is suggested that the model could be extended to include dynamic arrival and departure of peers and freeriding Modeling and Performance Analysis of BitTorrent-Like Peer-to-Peer Networks In [11], a fluid model is presented of BitTorrent in a dynamic context. The scalability, performance, and effi-

4 ciency of the protocol are analyzed, and numerical results are presented based on both simulations and real traces A simple fluid model In the paper, a simple fluid model is derived from a Markov chain that represents a dynamic system with arriving and departing peers, peers that stay seeding, and peers that abort their download. The model is associated with one specific file. The continuous quantities ( fluids ) in the system are the number of downloaders, x(t), and the number of seeds, y(t). It is assumed that all peers have the same download bandwidth c, and the same upload bandwidth µ, with c µ. Peers arrive with rate λ and abort their download with rate θ. Seeds leave the system at rate γ. Furthermore, the parameter η is used to indicate the effectiveness of the file sharing, which is described an analyzed further on. The resulting model is analyzed in steady-state. First, the average number of downloaders and seeders are derived: x = ȳ = λ β(1 + θ β ) (6) λ γ(1 + θ β ) (7) with 1 β = max{ 1 c, 1 η ( 1 µ 1 γ )}. Furthermore, the average download time T is given as: T = 1 θ + β (8) It is observed that the average download time T is not related to the request arrival rate λ. The BitTorrent system therefore scales very well. Furthermore, it is observed that if the seed leaving rate γ is smaller than µ, the download bandwidth c determines the network performance, even though c may be much larger than µ. The effectiveness of file sharing, η, is determined by the ability of peers to upload to others. In practice, a peer i is connected to a limited set of k peers. The effectiveness η of file sharing is then determined by the probability that at least one of the connected peers is interested in a piece that i possesses. The paper shows that it is reasonable to approximate η by η 1 ( log N N )k, (9) where N is the number of pieces of the file. It is remarked that in practice, due to the large number of pieces in a typical BitTorrent file, η will be close to 1. In the presented model, the real BitTorrent system is approximated by a deterministic fluid model with constant x and ȳ in steady-state. However, in practice, these numbers are likely to fluctuate around the predicted values. In order to gain insight in this behavior, the paper presents a stochastic fluid differential equation, in which the components are independent Brownian motions. It can be derived from the resulting equations, that in steady-state the number of seeds and downloaders are distributed as Gaussian random variables. However, practical implications or examples are not included in the paper The BitTorrent incentive mechanism The paper also studies the BitTorrent incentive mechanism, which is meant to prevent freeriding. The basic idea is that each peer uploads to the peers from which it receives the highest downloading rates. In BitTorrent, a peer periodically uploads to a newly acquainted peer in an attempt to improve its selection (optimistic unchoking). Since the resulting peer selection in a practical system is hard to model due to the randomness of this mechanism, the paper presents a simplified model. It argues that this model approximates real BitTorrent. The model is then used to analyze the effect of the upload bandwidth on its received download bandwidth. The paper argues that under some conditions, a peer might lower the amount of upload bandwidth it donates, without losing the download bandwidth it receives. Therefore, it is not always necessary for a peer to donate its maximum available upload capacity Experimental results In order to validate the presented fluid model, the paper presents results of a series of three experiments. In the first two experiments, BitTorrent simulations are performed and compared with the fluid model. An important observation is that the higher the arrival rate, the more the simulation results resemble the model. In the third experiment, traces of a real BitTorrent system were collected. As visible in Figure 3, the general evolution of the number of seeds and the number of downloaders are quite accurately predicted by the model. However, the real traces (displayed in Figure 4) are far less similar to the model predictions. The paper explains this by arguing that the file was not very popular and the arrival rate was small. However, a trace analysis of a popular file is not presented by the paper Performance of Peer-to-Peer Networks: Service Capacity and Role of Resource Sharing Policies In [12], the performance of P2P networks is analyzed with a focus on the service capacity of the system. The paper focuses on a dynamic system and analyzes both the transient phase after introduction of a file, and the

5 (a) (b) Figure 3: The number of seeds (a) and number of downloaders (b) in a simulation and in the model. (a) (b) Figure 4: The number of seeds (a) and number of downloaders (b) in a real trace and in the model. stationary phase when the performance has more or less stabilized Service capacity model The model focuses on peer that are either sharing or downloading a single file. A peer can concurrently upload different parts of a file to other peers while it downloads other parts from its peers. Peers who finish downloads may randomly choose to leave the system or stay and continue sharing files (seeds). No peer leaves the system before completing its download. The upload bandwidth of a peer is considered to be limited, while the download bandwidth and network throughput are considered to be unlimited. As main performance metric of the system, the service capacity is defined. The paper gives two variants of this definition, based on the perspective of both the system and the user. The aggregate upload service capacity is the overall achievable throughput the system can offer to download peers interested in a given document, i.e., the effective upload bandwidth of both seeds and downloading peers. The per peer download throughput is the average download throughput achieved per individual peer Analysis In order to analyze the service capacity, the paper distinguishes two so-called regimes: the transient regime, and the stationary regime. The first denotes a phase with a large burst of requests, while the latter denotes a phase where the throughput performance of each peer is more or less stable. The distinction between these two regimes is based on observations of a trace of a BitTorrent system, in which the two phases can be observed after the introduction of a new document (see Figure 5). The paper does not describe any details regarding the nature or context of the mentioned trace. Transient analysis In order to analyze the system performance in the transient phase, a mathematical expression for the average download time is derived from the model. The derivation is based on regarding different rounds, during each of which every

6 peer exchanges data with every other peer in the system, and that a central mechanism ensures the proportional bandwidth allocation. Secondly, peerwise proportional fairness is analyzed, in which peers offer upload capacity to other peers based directly on the service they receive from them. Several examples are covered, which show that in some cases this policy leads to a suboptimal allocation of bandwidth. However, the exact reasoning behind this statement is not made explicitly clear in the paper. Figure 5: The transient and stationary phases after introduction of a new file into a BitTorrent system. peer uploads a piece it contains to another peer interested in this piece. The average delay d (m) for dissemination of a file of size s with m pieces to n peers with upload bandwidth b, is given as d (m) = s 2m 1 ( log n + ) s log n. (10) bm 2 bm It is concluded that the download time is therefore of O(log n). Surprisingly, the authors neglect a factor of order s/b here. If m is large with respect to (log n), i.e., (log n)/m 1, a far more accurate approximation would be d (m) s/b = O(1). This is in fact a highly reasonable assumption, e.g., for large files in BitTorrent m O(1000). Stationary analysis The analysis of the stationary phase is based on a different model. Here, a Markov chain is derived, entirely similar to the work in [11]. The system state is modeled as a pair (x, y) with x the number of peers downloading and y the number of seeders. New requests arrive according to a Poisson process. The corresponding equations and symbols are similar to those in [11]. The analytical results presented in the paper are validated by a trace of a BitTorrent network over several days Implication of fairness on performance Following the analysis of the upload capacity, the paper also covers an analysis of the implications of fairness policies on performance. It is argued that some seemingly straightforward criteria have unexpected outcomes and lead to unfair bandwidth allocation and poor performance. Various notions of fairness are analyzed mathematically. First, the policy where the bandwidth a peer receives is proportional to its own overall contribution to the system. This policy is shown to be effective. However, it is based on the unrealistic assumptions that every 4. WORKLOAD MODELING In this section we give a summary of a paper containing an insightful model of the workload and user behavior in P2P file sharing systems Measurement, Modeling, and Analysis of a Peerto-Peer File-Sharing Workload In [4], a trace of Kazaa traffic is analyzed. The fetch-atmost-once nature of file sharing request behavior is observed, and a model is presented that accounts for this behavior. Throughout the paper, the explicit distinction with regular World Wide Web object popularity is emphasized, noting that in many other papers on P2P systems such models are incorrectly applied Trace analysis The paper describes a 200-day trace of Kazaa peer-topeer file sharing traffic during It has been conducted at the campus of the University of Washington. The trace focused on request made by internal peers to download data stored on external peers, and captures the request behavior of a stable, complete user population over a period of time. Three properties of the user behavior are observed: 1. While users browsing the web are well-known to expect instant gratification, the trace results show incredible patience on the part of Kazaa users. Users are apparently willing to wait hours for small objects, and even days of weeks for large objects such as movies. This is related to the fact that downloads can be made in the background and are left until the download is finished. 2. New clients generate most of the load in Kazaa, and older clients gradually consume fewer bytes as they age. Probably older clients use the system less often, or ask for less when they do use the system. However, the relevance or objectiveness of these observations are not stated by the paper.

7 3. The measurements indicate that the distributions of the average session length and the activity fraction (i.e., the fraction of time a client is transferring content over the client s lifetime) are heavy-tailed. The median session length is in fact only a small fraction of the median download time for a small object. This apparent paradox is explained by the paper as an effect of failed transactions, server-unavailability, and transactions consisting only of chunks of a file. However, it is not made clear by the paper how a session is precisely defined. Furthermore, the characteristics of objects that are requested are observed. The paper states the workload of Kazaa is best described as a blend of workloads of three different object types: small objects (< 10 MB); medium objects ( MB); and large objects (>100 MB). It is shown that a dominant portion of requests (91%) is associated with the small objects, while a dominant portion of the transferred bytes (65%) is associated with the large objects. Furthermore, several observations are presented regarding the dynamics of object requests: (1) individual users fetch an object at most once; (2) the popularity of objects is often short-lived; (3) the most popular objects tend to be the ones recently injected; (4) in total however, most of the requests are for old objects. It is argued from these observations that the forces driving the Kazaa usage differ in many ways from those driving the Web. This is an important observation, since most work on P2P file sharing systems uses workload models derived from Web request traces. In many papers, the workload is modeled as a Zipf distribution (i.e., the popularity of the i-th most popular object is proportional to i α, with α the Zipf coefficient; a Zipf distribution forms a straight line when plotted on a log-log scale). However, the traces confirm the hypothesis that Kazaa workloads are not Zipf. The distribution observed in the traces differs from Zipf especially for the most popular objects (see Figure 6 (a)) Model Based on the observations from the traces, and the derived hypothesis, a model is presented of the workload in a fetch-at-most-once system. The popularity of objects is still based on Zipf s law, but it is corrected for the fetch-at-most-once behavior. In the model, subsequent requests from the same client obey distributions obtained by removing already fetched objects from the object set, and re-scaling the resulting distribution. When an object is born in the system, its popularity rank is determined by selecting randomly from a Zipf distribution. The model is validated by simulation, setting its parameters such that it resembles the popularity distribution measured in the trace (see Figure 6 (b)) Locality-awareness Following some observations regarding internal versus external bandwidth use, the paper explores how the locality of content can be exploited. By locality exploitation is meant the more effective use of content available within an organization to substantially decrease external bandwidth usage. It is observed from the trace, that 86% of the downloaded bytes already existed on other local clients at the time they were downloaded from external clients. The paper argues that substantial untapped locality exists in the Kazaa workload Paper conclusions The main results of the paper state that P2P file-sharing workloads are driven by consideraly different processes than the Web, due to the immutable nature of most objects and the corresponding fetch-at-once behavior of users. Therefore, the paper concludes that client births and object births are the fundamental forces driving the workloads. Finally, there is significant locality in the workload, and therefore substantial opportunity for caching to reduce wide-area bandwidth consumption. 5. POLLUTION MODELING In this section, an interesting paper on the effect of pollution attacks in P2P file sharing systems is summarized Fluid Modeling of Pollution Proliferation in P2P Networks In [7], a fluid model of pollution proliferation in P2P networks is presented. The model analyzes different types of user behavior and file popularity, and discusses possible attacks and corresponding counter-measures Pollution model The model described in the paper focuses on the proliferation of both good and polluted versions of a single title in the P2P network. Of each version, multiple copies can exist in the network. The network consists of two types of peers: benign peers and attacker peers. Peers possess at most one copy (good or polluted) at a time. Attacker peers provide copies of polluted versions to the network. Benign peers keep requesting and downloading copies until they have obtained a good version. After obtaining a good version, they start seeding the file for an indefinite amount of time. Peers are assumed to be homogeneous, having the same behavior. After a user queries for a title, it is assumed to receive a list of all versions of the title available in the network,

8 (a) (b) Figure 6: Comparison of the measured popularity of WWW objects versus Kazaa objects (a), and comparison of the model with the actual popularity (b). (a) (b) Figure 7: Evolution of the number of good and polluted copies in the copy centric model (a) and the version centric model (b). together with the number of copies per version. The paper covers two models for the selection distribution of the requests: 1. Copy centric model: Users select a copy at random, uniformly across all copies available in the network. 2. Version centric model: Users select a version at random, independently of the number of copies for each version. Pollution attacks and recovery are analyzed for both models Analysis When the system is under attack, the attacker uses its own peers to introduce polluted versions into the system. A discrete-state Markov process that describes a system with N attackers and M benign peers is presented. Since the Markov process is too complex to solve, a fluid flow approximation of the system is described. A non-linear system of differential equations is obtained for both models. For the copy centric model, a closed-form solution is derived and validated by simulation (see Figure 7 (a)). For the version centric model, an implicit solution is presented (see Figure 7 (b)). Figure 8: Proliferation of polluted copies with peer abandonment and freeloading More complex behavior In order to make the model more realistic, the paper analyzed a few more complex issues: Peer abandonment and freeloading In practice, it is reasonable to assume that peers can become frustrated with downloading polluted copies, and give up. Furthermore, some peers will not share their obtained good copies. An analysis is presented of a model in which after downloading a polluted copy, a peer quits with probability α, and in which a peer that obtains a good copy decides not to share with

9 probability γ. An system of differential equations is presented for this model, which is numerically solved with some specific parameters (see Figure 8). User bias In order to model the selection process in more detail, a generalization is presented in which an arbitrary bias towards versions with a larger fraction of copies can be realized. Results of some discrete event simulation with a slight bias are presented. It is observed that with a slight bias, both the extent of pollution and the time for every peer to get a good copy have noticeably decreased. In practice, attackers could respond to this bias by introducing fewer versions with many copies each. Blacklisting In order to provide countermeasures against attackers providing many polluted copies, users can blacklist polluted versions and exchange these lists with each other. Some results are presented for simulations with blacklisting. In this case, an attacker would have to change its strategy and present as many different polluted versions as possible, since a polluted version will never become really popular Paper conclusions The paper concludes that it is advantageous for the attacker to launch the pollution attack early in the process, that peer abandonment and freeloading significantly influence the spreading of good copies, and that an attack strategy is only successful if the number of polluted versions is high enough to circumvent blacklisting and low enough to obtain a certain version popularity. [4] K.P. Gummadi, R.J. Dunn, S. Saroiu, S.D. Gribble, H.M. Levy, and J. Zahorjan. Measurement, modeling and analysis of a peer-to-peer file-sharing workload. In Proc. of SOSP 03, October [5] Kazaa. [6] R. Kumar and K.W. Ross. Optimal peer-assisted file distribution: Single and multi-class problems. In Proc. of IEEE Workshop on Hot Topics in Web Systems and Technologies (HOTWEB 06), July [7] Rakesh Kumar, David D. Yao, Amitabha Bagchi, Keith W. Ross, and Dan Rubenstein. Fluid modeling of pollution proliferation in p2p networks. In SIGMET- RICS 06/Performance 06: Proceedings of the joint international conference on Measurement and modeling of computer systems, pages , New York, NY, USA, ACM Press. [8] J. Mundinger, R. R. Weber, and G. Weiss. Analysis of peer-to-peer file dissemination amongst users of different upload capacities. SIGMETRICS Perform. Eval. Rev., 34(2):5 6, [9] Jochen Mundinger, Richard R. Weber, and Gideon Weiss. Optimal scheduling of peer-to-peer file dissemination, [10] Napster. [11] D. Qiu and R. Srikant. Modeling and performance analysis of bittorrent-like peer-to-peer networks. In Proc. of ACM SIGCOMM 2004, Portland, Oregon, USA, August [12] Xiangying Yang and Gustavo de Veciana. Performance of peer-to-peer networks: service capacity and role of resource sharing policies. Performance Evaluation, 63(3): , CONCLUSIONS In this paper, we have presented a survey of analytical modeling in P2P file sharing systems. We have summarized various recent papers on performance modeling, workload modeling and pollution modeling of such systems. The main results of each paper are presented, offering an insightful and useful overview of system properties, protocol characteristics, and user behavior. References [1] BitTorrent. [2] CacheLogic. [3] D. Dumitriu, E. Knightly, A. Kuzmanovic, I. Stoica, and W. Zwaenepoel. Denial-of-service resilience in peer-topeer file sharing systems. In SIGMETRICS 05: Proceedings of the 2005 ACM SIGMETRICS international conference on Measurement and modeling of computer systems, pages 38 49, New York, NY, USA, ACM Press.