Socially-optimal ISP-aware P2P Content Distribution via a Primal-Dual Approach

Transcription

1 Socially-optimal ISP-aware P2P Content Distribution via a Primal-Dual Approach Jian Zhao, Chuan Wu The University of Hong Kong {jzhao,cwu}@cs.hku.hk Abstract Peer-to-peer (P2P) technology is popularly exploite to enable large-scale content istribution (e.g., live an oneman vieo streaming) with low server costs. Most P2P protocols in place are network agnostic, where peers ownloa content chunks from each other regarless of their ISP belonging, leaing to significant amounts of inter-isp traffic. It has been a aunting challenge how to esign protocols that optimize the P2P content istribution topology, such that inter-isp traffic is minimize while the issemination performance is maximize, not to mention one that motivates peer s voluntary ISP-aware peer selection. In this paper, we formulate a social welfare maximization framework for ynamical construction of the P2P content istribution topology taking into consieration both peers gain ue to chunk ownloaing an inter-isp traffic that is incurre. Given its nature of an assignment problem, we resort to a primal-ual framework to esign the solution algorithm, which can be practically implemente as a set of istribute, interleaving auctions, where peers bi for banwith to ownloa chunks at other peers consiering the potential inter-isp traffic as a cost factor. Such an auction-base mechanism encourages peers to ownloa from neighbors with low network costs in between, in orer to succee in banwith acquisition for chunk ownloaing. We analyze an prove the social optimality achieve by the istribute auctions an verify the performance of our proposal using realistic emulation experiments with real P2P traffic. I. INTRODUCTION Forecasts from Cisco s Visual Networking Inex [1] reveal that P2P traffic is expecte to grow to more than 7 Petabytes per month by 214 more than the ouble of the amount of P2P traffic in 29. Most of the existing P2P protocols (for file sharing, live an on-eman meia streaming, etc.) are network agnostic, i.e., peers pick peers to ownloa from as long as the latter cache the content in nee, regarless of the ISP belonging of each other. This has resulte in significant amounts of inter-isp traffic that cost many ISPs early, leaing to their P2P traffic filtering, which in turn significantly eteriorates the performance of P2P applications. There have been many efforts on minimizing inter-isp traffic by connecting peers to nearby neighbors in the same AS or ISP. Aggarwal et al. [2] an Xie et al. [3] avocate collaboration between P2P applications an ISPs, where ISPs provie information of the unerlying network (e.g., banwith, istance) for a P2P application to make localize peer selection. Picconi et al. [4] propose a two-tier aaptive overlay structure, with highly clustere primary overlays built among nearby peers an a number of seconary links interconnecting the clusters. The seconary links are unchoke when necessary to enable global stream propagation. Peer selection is carrie out at a coarse level in the above work, without guarantee of the optimality of the resulting topologies in content istribution performance an inter-isp traffic reuction. Wang et al. [5] formulate an optimization problem for ISP-frienly rate allocation, aiming at guarantee QoS for users, reuce server loa an reuce ISP-unfrienly traffic. The rate allocation optimization problem is moele an solve in the flui level (optimal flow rate computation), an the results are translate into a packet-scheuling algorithm for implementation. This paper presents our initial attempt to esign P2P protocols that encourage ISP-aware peer selection, in a fully istribute algorithm framework. We first formulate a social welfare maximization framework for ynamical construction of the P2P content istribution topology (namely eciing who is ownloaing which content chunk from whom), where each peer s welfare is ecie by its gain for receiving the chunks an the network cost incurre ue to receiving chunks from ifferent ISPs. Instea of flow-level optimization, our optimization framework moels chunk-level content istribution among the peers, to enable straightaway implementation of the optimal chunk scheuling strategies in a real-worl P2P system, an to avoi loss of optimality ue to the flow rateto-packet scheuling translation. The optimization is a more ifficult integer optimization problem though. Nevertheless, given its nature of an assignment problem, we resort to a primal-ual framework propose by Bertsekas et al. [6] to erive the solution. The solution algorithm can be implemente as a set of istribute, interleaving auctions in the P2P system: Each peer acts as an auctioneer an hosts an auction to allocate its uploa banwith for serving chunks to requesting peers; each peer also bis in the auctions hoste by ifferent other peers for banwith to ownloa chunks they want. The biing price for a chunk cache at a neighbor is ecie by the utility gain the peer can obtain after acquiring the chunk, minus the network cost of receiving the chunk from the neighbor. Such an auctionbase algorithm framework encourage peers to ownloa from neighbors with low network costs in between, in orer to succee in banwith acquisition for chunk ownloaing. We also esign practical protocols for carrying out the auctions consecutively in a ynamic P2P system, where peers may come an go, an uploa banwith can be repeately sol to serve ifferent chunks to ifferent neighbors over time. We prove that the istribute auctions can collectively maximize

2 2 the social welfare of all peers in the system in Theorem 1. We implement an emulator of a large-scale, istribute P2P streaming system where content istribution is carrie out through the auctions. Extensive experiment evaluations verify the goo performance of the system in realistic environments with real P2P traffic. The remainer of the paper is organize as follows: Sec. III presents the P2P content istribution system moel an formulates the optimal chunk scheuling problem. Sec. IV presents a primal-ual auction algorithm to solve the problem an iscusses its practical implementation as a set of istribute auctions. Sec. V presents our experimental results an Sec. VI conclues the paper. II. RELATED WORK Developing ISP-aware P2P protocols has attracte significant attention from both the content istribution service proviers an the research community. These research work can be categorize into three camps. The first one is to achieve the network awareness through cooperation between ISPs an P2P systems. Aggarwal et al. [2] propose a cooperative mechanism between ISPs an P2P users for a better neighbor selection process as follows: the ISPs offer an oracle to the P2P users, the peers sen their lists of possible neighbors to the oracle, an then the oracle ranks the possible peer neighbors accoring to certain criteria, such as their proximity to the peer or higher banwith links in between. Xie et al. [3] propose P4P, the provier portal for applications. P4P provies a control plane that can provie network information, such as network policy, p4p-istance, capabilities, to peer users. Aitional infrastructures are necessary for P4P, which may not be easy to eploy. These mechanisms require trust between ISPs an P2P users. The secon group of work achieves network awareness through inferring network information by peers or base on peers self-aaptive protocols. Choffnes et al. [7] propose to use the information collecte from content istribution networks to guie the biase peer selection. The rationale is as follows: if two clients are ispatche to a similar set of replica servers, they are likely to be close to these servers an more importantly, to each other. This leas to clustere overlay with a topology following the unerlying physical one. Picconi et al. [4] propose an aaptive protocol for P2P live streaming with a large number of links between peers locate in ifferent ISPs. It buils a highly clustere primary overlay with ynamically unchoke seconary inter-cluster links. The clustere primary overlay can reuce the unnecessary inter-isp traffic. The ynamically unchoke seconary inter-cluster links can ensure that the QoS is not impacte. These mechanisms use heuristic self-aaptive protocols to reuce the inter-isp traffic an keep goo performance. Our algorithm explores the optimality of peers utility in an auction base on a primalual optimization framework. The thir camp of work exploits rate control on inter-isp links. Wang et al. [5] propose an formulate an optimization problem for rate allocation among peers in a P2P VoD system. The rate allocation optimization problem is moele in the flui level. It then translates the flui-level rate allocation algorithm into an implementable packet-level scheuling algorithm. Our approach formulates a packet-level optimization problem for the rate allocation problem irectly an avois the loss of optimality ue to the flow rate-to-packet scheuling translation. A sequence of work by Bertsekas et al. [8] [6] theoretically stuy the auctions base on a primal-ual optimization framework for the assignment problems an transportation problems. Such auction-base optimization has also been applie in improving the performance of P2P content istribution [9] [1] [11]. These papers o not take ISP-awareness into consieration. To the authors knowlege, our paper is the first in applying an auction for achieving the social optimality of ISP-aware P2P content istribution. A. System Moel III. PROBLEM MODEL We consier a mesh-base P2P content istribution system, e.g., a P2P VoD streaming system, eploye over the networks of M Internet Service Proviers (ISPs). Let P m enote the set of peers in ISP m [1,M]. There exist a number of tracker servers, from which the peers can obtain a set of neighbors which may potentially cache the content they want, upon joining the system. Let N n () enote peer s neighbor set in ISP n. The set of all neighbors of peer is hence M n=1n n (). Each content (a file or a vieo stream) in the system is ivie into multiple equal-size chunks an istribute. The system works in a time slotte fashion over t =,1,2,...,T, where T is a potentially larger integer. Each peer maintains a moving winow of interest, specifying the chunks it wishes to ownloa in each time slot (e.g., the chunks to be playe next in a streaming system). A peer exchanges buffer maps of chunk availability with its neighbors, an requests chunks of interest from the neighbors which cache the chunks. Let R t () enote the set of chunks that peerintens to ownloa from neighbors at time slot t. A request in the system can be represente by a three-tuple (I,I u,c), where I is the i of the ownstream peer which issues the request, I u is the i of the upstream peer being requeste, an c is the ientifier of the requeste chunk. We use B(u) to enote the uploa banwith of peer u, which represents the number of chunks peer u can uploa in a time slot (suppose one unit of banwith is use to uploa one chunk). We assume that peers uploa banwith reners the banwith bottleneck in the system, while the ownloa banwith is much more sufficient comparably. Let v (c) () enote peer s valuation for receiving chunk c, i.e., the value chunk c brings to peer. Let u be the inicator of whether request (I,I u,c) is serve, i.e., u = 1 if the request is serve by the corresponing upstream peer u, an u = otherwise. The network cost for peer to receive a chunk from peer u is w u, which has ifferent values between peers in ifferent pairs of ISPs. Such a network cost can represent network latency for sening a chunk between peers, or the possibility that the chunk is

3 3 TABLE I IMPORTANT NOTATION M No. of ISPs P m Peers in ISP m N n() Peer s total neighbor set in ISP n B(u) # of chunks peer u can uploa in a time slot I I of request source peer I u I of request estination peer c Inex of requeste chunk R t() set of peer s intereste chunks at time t N n (c) () set of peer s neighbors in ISP n with chunk c u inicator of whether request r receives the banwith allocation v (c) () valuation for peer receiving chunk c w u network cost for transmitting a chunk from u to λ u ual variables for peer u s uploa banwith ual variables for request (I,c) being blocke ue to filtering of egress/ingress P2P traffic at one ISP. The net utility peer receives by ownloaing chunk c from peer u is v (c) () w u. The important notation in this paper is summarize in table I for ease of reference. B. Social Welfare Maximization Problem In each time slot, we seek to ecie the optimal chunk scheuling strategy, u, for all the chunk requests issue by all the peers, M m=1p m,c R t (),u M n=1n n (c) (), to maximize the social welfare, i.e., the total utility of peers from chunk ownloaing, as follows: max s.t. M m=1 Pm c R t() u M n=1 N(c) n (),c:u M n=1 N(c) n (),c R a(c) t() u M n=1 N(c) n () a(c) u [v(c) () w u ] u B(u), (1) u M m=1p m, (2) u 1, M m=1p m,c R t (), (3) u {,1}, M m=1p m, (4) c R t (),u M n=1n n (c) (). Objective function (1) is to maximize the total utility of all peers. Constraint (2) states that the total number of chunks a peer uploas to its neighbors shoul not excee its uploa banwith limit. Constraint (3) specifies that a peer will ownloa a chunk from no more than one neighbor. The problem in (1) is an integer linear program. We will esign an efficient primal-ual auction algorithm to solve this integer linear program. Introucing ual variables λ u, to constraints (2) an (3) respectively, the ual problem of (1) can be formulate as follows: min u M m=1 Pm λ u B(u)+ M m=1 Pm c R t() (5) s.t. λ u + v (c) () w u, M m=1p m, (6) c R t (),u M n=1n n (c) (), λ u, u M m=1p m, (7), M m=1p m,c R t (). (8) Note that we omit the integrality constraint (4) in the primal problem when formulating the ual. Nevertheless, we will show in the next section that our auction algorithm exactly solves the primal an ual problems, with optimal binary solutions to the primal problem. IV. THE PRIMAL-DUAL AUCTION ALGORITHM A. Conversion to An Assignment Problem The social welfare maximization problem in (1) can be treate as a transportation problem [6]. In a transportation problem, a set of source noes are connecte to a set of sink noes in a bipartite graph. The set of matching eges between the sources an the sinks are being sought such that each source is connecte to no more than α sinks an each sink is connecte to no more than β sources, an the total weight on the selecte eges is the largest. In our problem, a request for a specific chunk c from a peer, i.e., (I,c), can be treate as a source. Each peer u is a sink. An ege connects a source to a sink if the corresponing sink (peer u) is a neighbor of the requesting peer an caches chunk c. The weight on an ege is v (c) () w u. By solving problem (1), we wish to fin the subset of eges between the sources an the sinks, such that each source is connecte to no more than one ege in the set (constraint (3)) an each sink (peer u) is connecte to no more than B(u) eges (constraint (2)). A transportation problem is an assignment problem in nature. In Bertsekas et al. s work [6] [8], an auction-like primalual algorithm is esigne to solve the classical assignment problem, where X istinct objects are to be assigne to Y persons, such that each person receives one object, an the total weight of the person-object matchings is the largest. The transportation problem can be converte to an assignment problem by replacing each source (sink) with α (β) copies of persons (objects). To convert our problem to an assignment problem, each sink (peer u) is replace by B(u) units of uploa banwith (treating one unit of uploa banwith as one object). Each of the B(u) objects connects to the sources that sink u connects to in the original problem, an the same weight as on the original ege is applie on the new eges. An illustration of our problem in the transportation problem moel, as well as its conversion to the assignment problem, is given in Fig. 1. Base on this conversion, we are able to esign a primal-ual auction algorithm to solve problem (1), base on the iea of the auction algorithm in [6]. B. The Primal-Dual The main iea of the primal-ual algorithm is as follows: Each peer maintains a unit price λ u for one unit of its uploa banwith, which correspons to the ual variable in the ual

4 4 v ( c) ( ) wu 1 ( I, c) u ( I, c) 1 v ( c) ( ) wu2 u 2 v v ( c) ( c) ( ) ( ) 1 wu u 1 wu2 Fig. 1. An illustration of our transportation problem (a) an its conversion to the classical assignment problem (b). problem (5). Peer u upates the price iteratively, accoring to the level of competition for its uploa banwith, i.e, the relationship between the number of requests it receives,c:u M n=1 N(c) n (),c R a(c) t() u an its overall uploa banwith B(u), an allocates its uploa banwith for serving chunks to peers (i.e., computes u ) accoring to the utility that each chunk can bring to the corresponing requester, v (c) () w u. When the iterative process converges, we are able to show that the optimal binary solution u to the primal problem an optimal solution λ u to the ual problem are achieve. The optimal values of the other ual variables, s, are ecie by η(c) = max u M n=1 N n (c) () {v(c) () w u λ u}, which is the minimum value (in orer to minimize the objective function of the ual problem) that satisfies constraint (6). Base on the above iea, we esign a set of istribute, interleaving auctions to carry out the primal-ual algorithm. In each time slot, each peer u is an auctioneer, hosting an auction to sell its B(u) units of uploa banwith. The peers who wish to acquire one unit of uploa banwith at peer u for retrieving one chunk c that u caches, are the biers in this auction. We next escribe the biing strategy of the biers an the allocation strategy at the auctioneers, respectively. Biing of Peer : Peer etermines the set of chunks to ownloa in this time slot, R t (), an values each chunk in R t (), i.e., computes v (c) (), c R t () (e.g., accoring to the playback ealine of a chunk in a P2P streaming system). Base on exchange bitmaps with neighbors, peer can ecie the set of neighbors which cache chunk c, i.e., N n (c) (),n = 1,...,M. It then ecies the following: (1) From which neighbor to bi for one unit of uploa banwith for retrieving chunk c. The net utility peer can acquire by ownloaing c from peer u M n=1n (c) n () is ecie by u 2 v (c) () w u λ u (the banwith price λ u at peer u is consiere). Let u be the upstream peer that provies the largest utility, i.e., u = argmax u M n=1 N n (c) () v(c) () w u λ u. Peer will sen the request for chunk c to peer u. (2) How much peer shoul bi for one unit of uploa banwith at peer u. Let ϕ(,c,u ) = v (c) () w u λ u enote the largest net utility peer can obtain by ownloaing chunk c. Suppose û is the neighbor which can provie the secon largest net utility ϕ(,c,û) = v (c) () wû λûm if peer was to ownloa c from û. Peer bis b(,c,u ) = λ u + ϕ(,c,u ) ϕ(,c,û) = wû w u + λû for one unit of banwith to ownloa chunk c from u. If bi b(,c,u ) = λ u, peer will not sen a bi to auctioneer u, since the bi will nevertheless be unsuccessful, accoring to the banwith allocation mechanism below. Instea, peer waits until the banwith prices at the upstream peers change such that its optimal bi becomes larger than a respective banwith price. Banwith Allocation at Peer u: Peer u maintains an assignment set containing the requests (I,c) with the highest bis, to which it will allocate one unit of its uploa banwith (corresponing to u = 1). The maximum size of the assignment set is B(u). At the beginning of each time slot, the set is empty an the initial unit banwith price is set to λ u =. Upon receiving a bi b(,c,u), if the price b(,c,u) λ u, peer u rejects the bi. Otherwise, if its assignment set is not full, peer u irectly as the request (I,c) to the set; if the set is full, the request whose biing price is the lowest among all the requests in the assignment set (which equals λ u ), is remove from the set (i.e., set the respective u = ), an request (I,c) is ae. If the size of the assignment set is B(u) (i.e., all B(u) units of its uploa banwith are allocate), Peer u upates λ u to the smallest biing price among all accepte requests in the current assignment set, an informs its neighbors this upate banwith price. A bi at an upstream peer u can be unsuccessful ue to concurrent bis from other peers which push the price λ u up, or can be accepte first but remove from the assignment set later on, ue to the arrival of higher bis. In these cases, the bier can compute its new bi accoring to the upate prices from the upstream peers, an bi again either to the same upstream peer u (if v (c) () w u λ u is still the largest), or to another upstream peer u (if v (c) () w u λ u becomes the largest). The auction in each time slot repeats iteratively, until the biing process converges, i.e., no auctioneer u wishes to change its banwith allocation u s an price λ u, an no bier wishes to bi again. Then the chunks corresponing to the winning bis are transmitte to the respective biers, using the acquire banwith. The auction algorithm for one time slot is summarize in Alg. 1. The fully istribute auction algorithm achieves maximize social welfare, as given in the following theorem. Theorem 1: Uner the assumption that the uploa ban-

5 5 with an the istribution of peers cache chunks in the system can satisfy peers ownloaing requirements in each time slot, Alg. 1 terminates an gives the optimal solution u to the primal problem (1) an λ u to the ual problem (5) upon termination. The proof is given in Appenix A. Algorithm 1 The in Time Slot t ***At Bier Peer :*** 1: exchange buffer maps with neighbors an ecie R t () 2: for each chunk c in R t () o 3: calculate net utility v (c) () w u λ u for all neighbors which cache c, an select neighbor u proviing the largest net utility 4: sen bi b(,c,u ) = wû w u +λû to neighbor u, whereûis the neighbor proviing the secon largest net utility 5: en for 6: upon failure to acquire a unit of banwith for a chunk c at a neighbor u an price upates from neighbors 7: repeat Lines 3 an 4 with upate prices ***At Auctioneer Peeru: *** 1: Initialization: λ u =, assignment set A = 2: while a bi b(,c,u) is receive o 3: if b(,c,u) λ u then 4: reject the bi 5: else 6: if size of A equals B(u) then 7: fin a request(i,c ) inawhose bi is the lowest, A A {(I,c )} 8: en if 9: A A+{(I,c)} 1: if size of A equals B(u) then 11: upate λ u to the smallest bi among all requests in A, an inform neighbors the new price 12: en if 13: en if 14: en while C. Implementation Issues in a Dynamic P2P System Over time, the auctions accoring to Alg. 1 repeat in each time slot, an uploa banwith at each peer is repeately sol to serve ifferent chunks to ifferent neighbors. The time slot in our moel can be treate as the biing cycle, accoring to which peers ecie the next batch of chunks to ownloa an seek the upstream peers which can serve the chunks through the auctions. The actual chunk transfers happen as soon as the auction algorithm converges in each time slot (i.e., when the optimal chunk scheuling is ecie), an can be finishe into the next time slot (i.e., biing for banwith for retrieving the next batch of chunks can happen concurrently with the transfer of the previous batch of chunks). Peers may come an go in a ynamic P2P system. When an auctioneer peer u receives new bis from newly joine peers in the mile of a time slot, it elays hanling of these bis until the start of the next time slot, such that the convergence of the auction process in this time slot is not isturbe. Upon eparture of a peer when the auction algorithm is still running in a time slot, the algorithm can hanle it smoothly an converge to the maximum social welfare where the eparte peer is exclue. If an auctioneer peer eparts when chunk transfers have starte in a time slot, the unaffecte chunk transfer scheules remain an the impact of the peer eparture will be taken into consieration in the next time slot. V. PERFORMANCE EVALUATION To evaluate our auction base content istribution algorithm, we emulate an efficient multi-threae P2P VoD system in Java an eploy it on a cluster of 6 high-performance blae servers with a 16-core Intel Xeon E5-26 processor an 8GB RAM. Each peer in the system is emulate by one process. Real network traffic is sent between peers in the system. The program at each peer inclues the following components: a neighbor manager for upating the peer s neighbors; a buffer manager for retaining chunks an exchanging bitmaps with the neighbors; a biing moule for calculating bis an sening them to auctioneers; an allocator moule for etermining the allocation of uploa banwith; a transmission manager for transmitting chunks to the winning biers. We emulate 5 ISPs. Peers of the same ISP are eploye in the same server. There is a track server which keeps track of online peers an bootstraps new joining peers with a list of neighbors with close playback positions. We set up the experiments to emulate a realistic P2P vieo streaming system, such as YouTube [12], YouKu [13]. We use short vieo files just like most vieos on YouTube, an the size of a vieo file is aroun 2 MB. The playback bitrate of a vieo is 64 Kbps, which is similar to the bitrate of a YouTube 36p vieo. We choose the chunk size of 8 KB just as the size of a sub-piece in PPStream [14]. There are 1 vieos in the system. Our emulator supports ynamic peer joins an epartures. Peers join the system as a poison process with rate 1 peer per secon, an are istribute in the 5 ISPs evenly. When a peer joins the system, it will select vieo i (1 i 1) to watch accoring to the Zipf-Manelbrot istribution p(i) = 1 (i+q) α 1 1,α =.78,q = 4 [15]. The efault number of i=1 (i+q) neighbors α for each peer is 3. A peer tries to pre-fetch 1 secons of the vieo, i.e., it tries to ownloa the next 1 chunks in avance of the playback position. In each ISP, for each vieo, there are 2 see peers with a uploa banwith that is 8 times of the streaming rate, which cache the complete vieo. We know that there are ifferent types of Internet connection services with ifferent levels of uploa banwith [16]. As the technology evelops, both the streaming rates of Internet vieos an the uploa banwiths of peers are moving to higher levels. Hence, we set the uploa capacity of peers following the uniform istribution within the range of [1, 4] times of the streaming bitrate in our system.

6 6 α We use a ealine-base valuation function, log(β +), for chunk evaluation at the peers [9], emphasizing how urgent a chunk is for playback. Here is the time to the playback ealine of the chunk, α an β are constants with efault values α = 2 an β = 1.2. Hence, the ealine-base valuation is within the range of [.8,8]. We use network latency as the network cost in our experiments. The inter-isp link elay costs an intra-isp link elay costs follow truncate normal istributions [17]. The istribution of inter-isp link costs has a mean 5 an a stanar variance 1, truncate within range [1, 1]. The istribution of intra-isp link cost has a mean 1 an a stanar variance 1, truncate within range [, 2]. A. Convergence of the Banwith Price We first stuy the evolution of the price for one unit of uploa banwith, λ u, with our auction algorithm, in a static network with 5 peers. Each time slot lasts 1 secons. During one time slot, a peer keeps biing in orer to acquire the banwith to receive the 1 chunks it wants next. Fig. 2 plots the evolution of the price λ u at a representative peer. For better illustration, we only show the evolution of the price in the time slots between 15 secons an 25 secons, an the evolution of the price is similar in other time slots. We can see that the price converges after aroun 5 secons in each time slot. This verifies the convergence of our auction-like primal-ual algorithm. λu Time (secons) Fig. 2. The evolution of a peer s price λ u. In the following subsections, we compare the social welfare, inter-isp traffic an chunk miss rate uner our auction algorithm with a simple locality-aware chunk scheuling algorithm, as follows: each ownstream peer requests chunks from upstream neighbors with the lowest network costs in between as much as possible; for banwith allocation at an upstream peer, it always prioritizes to transmit chunks with more urgent ealines. B. Social Welfare Fig. 3 shows the evolution of the system s social welfare in each time slot in a ynamic P2P network, where peers arrive following the ynamic moel escribe at the beginning of this section, an stay until they finish watching the respective vieo. We can see that as more peers join the system over time, larger social welfare per time slot can be achieve with our auction algorithm. However, the social welfare achieve by the simple locality-aware algorithm rops ue to more inter-isp traffic incurre with more peers in the system. The negative values of the social welfare with this algorithm are because it oes not consier peers chunk valuation when scheuling chunk transmissions (such that v (c) () w u can be negative). Social welfare Time(secons) Fig. 3. C. Inter-ISP Traffic Comparison of social welfare. Fig. 4 shows the percentage of inter-isp traffic incurre in all the traffic in the system in each time slot in a static network of 5 peers. We can see that the percentage of inter- ISP traffic is smaller with our auction algorithm, since with our algorithm, a peer only ownloas a chunk from an ISP with a large network cost in between when its valuation for the chunk is large enough, reucing unnecessary inter-isp traffic as much as possible. % of Inter ISP traffic Time(secons) Fig. 4. D. Chunk Downloa Performance Comparison of inter-isp traffic. Fig. 5 plots the average chunk miss rate of all peers in a static network of 5 peers, which is the percentage of chunks which fail to be ownloae before the respective playback ealines. With our auction algorithm, the average chunk miss rate is smaller. This verifies the efficiency of uploa banwith allocation in our auction algorithm, which takes ownstream peers valuation of the chunks into consieration. E. Comparison uner Peer Dynamics Fig. 6(a), 6(b), an 6(c) show the social welfare, inter-isp traffic an chunk miss rate with our algorithm an the simple locality protocol, in a ynamic P2P network where peers arrive following the ynamic moel escribe at the beginning of this section, an epart at any time with probability.6. We can see that our algorithm still performs better in general than the simple locality-aware algorithm in case of peer ynamics.

7 7 Miss rate.1.5 Fig. 5. Social welfare % of Inter ISP traffic Miss rate Time(secons) Comparison of the chunk miss rate Time(secons) (a) Social welfare Time(secons) (b) Inter ISP traffic Time(secons) (c) Chunk miss rate Fig. 6. Comparison of social welfare, inter-isp traffic an chunk miss rate uner peer ynamics. the algorithm s efficacy in reucing ISP-unfrienly traffic an maintaining goo chunk ownloa performance. This work represents our initial attempt to esign an auction-like mechanism to encourage peers voluntary ownloa from neighbors with low network costs in between. We are improving the auction mechanism esign to enforce truthfulness of the bis in cases of selfish peers that may manipulate the mechanism, in our ongoing work. ACKNOWLEDGEMENT The research was supporte in part by a grant from Hong Kong RGC uner the contract (Ref: HKU71871E). REFERENCES [1] I. Cisco, Cisco visual networking inex: forecast an methoology, , White Paper. [2] C. S. V. Aggarwal, A. Felmann, Can ISPs an P2P Users Cooperate for Improve Performance? ACM SIGCOMM Computer Communication Review, vol. 37, pp. 31 4, 27. [3] H. Xie, Y. Yang, A. Krishnamurthy, Y. Liu, an A. Silberschatz, P4P: Provier Portal for Applications, in Proc. of ACM SIGCOMM, August 28. [4] F. Picconi an L. Massoulie, ISP Frien or Foe? Making P2P Live Streaming ISP-Aware, in Proc. of IEEE ICDCS, June 29. [5] J. Wang, C. Huang, an J. Li, On ISP-frienly Rate Allocation for Peer-assiste VoD, in Proc. of ACM Multimeia, October 28. [6] D. P. Bertsekas an D. A. Castanon, The for the Transportation Problem, Annals of Operational Research, vol. 2, pp , [7] F. E. B. D. R. Choffnes, Taming the Torrent A Practical Approach to Reucing Cross-ISP Traffic in Peer-to-Peer Systems, in Proc. of ACM SIGCOMM, August 28. [8] D. P. Bertsekas, The : a Distribute Relaxation Metho for the Assignment Problem, Annals of Operational Research, vol. 14, pp , [9] C. Wu, Z. P. Li, X. J. Qiu, an F. C. M. Lau, Auction-base P2P VoD Streaming: Incentives an Optimal Scheuling, ACM Transactions on Multimeia Computing, Communications an Applications, vol. 8, pp. 14:1 14:22, 212. [1] C. Wu, B. Li, an Z. Li, Dynamic Banwith Auctions in Multioverlay P2P Streaming with Networking Coing, Proc. of IEEE TPDS, vol. 19, pp , 28. [11] X. Chu, K. Zhao, Z. Li, an A. Mahanti, Auction-Base On-Deman P2P Min-Cost Meia Streaming with Network Coing, Proc. of IEEE TPDS, vol. 2, pp , 29. [12] YouTube, [13] YouKu, [14] J. Jia, C. Li, an C. Chen, Characterizing PPStream across Internet, in IFIP Network an Parallel Computing Workshop, September 27. [15] J. Dai, B. Li, F. M. Liu, B. C. Li, an H. Jin, On the Efficiency of Collaborative Caching in ISP-aware P2P Networks, in Proc. of IEEE INFOCOM, April 211. [16] C. Huang, J. Li, an K. W. Ross, Can Internet Vieo-on-Deman Be Profitable? in Proc. of ACM SIGCOMM, October 27. [17] H. Jiang an C. Dovrolis, Passive estimation of TCP roun-trip times, Proc. of ACM SIGCOMM Computer Communications Review, vol. 32, pp , 22. VI. CONCLUSIONS This paper aresses the optimal chunk issemination topology construction problem in a P2P content istribution system, with the objective of social welfare maximization with minimal inter-isp traffic. We utilize a primal-ual optimization framework an esign an efficient auction algorithm to achieve the optimal issemination topology in a fully istribute manner. Our experiments uner realistic settings base on emulator implementation of a P2P streaming system verify

8 8 APPENDIX Proof: The proof consists of two parts: (i) The auction algorithm terminates in a finite number of iterations, an (ii) upon termination, the complementary slackness of the primal an ual optimization problems in (1) an (5) is satisfie. (i) The termination of the auction algorithm can be prove by way of contraiction. Suppose it never terminates. Then the number of units of allocate banwith is non-ecreasing, because one unit of banwith, once allocate, remains allocate throughout the auction. The total number of units of allocate banwith is upper-boune by the overall banwith eman from ownloaing peers in the system. Uner the assumption that the overall uploa banwith is sufficient to serve each chunk, there exist units of uploa banwith that are never allocate. Therefore, since the algorithm oes not terminate, we can infer that there exists a peer wanting to ownloa a chunk, which bis for one unit of banwith at an upstream peer u 1 whose banwith has all been allocate an whose price is growing unbounely, rather than bis at another peer u 2 with a unit of unallocate banwith an banwith price. This implies that the valuation of ownloaing the chunk from peer u 2 with banwith price is negative infinity (this is the only possibility when u 1 is always selecte rather than u 2 ), contraicting the fact that the valuation shoul be finite. (ii) We first list the complementary slackness conitions of the primal an ual problems: λ u >,c:u M n=1 N(c) n (),c R a(c) t() u = B(u), u M m=1p m, u > λ u + = v (c) w u, M m=1p m,c R t (),u M n=1n n (c) (). > u M n=1 N(c) n () a(c) u = 1, M m=1p m,c R t (). The first conition means that the uploa banwith at a peer u with a non-zero banwith price must have all been allocate. This is obviously true with our algorithm, since if there is one unit of unallocate banwith, the banwith price at peer u shoul be λ u =. The secon conition states that when a request (I,c) obtains a unit of banwith from peer u, the optimal solution shoul be equal to v (c) w u λ u. Recall that the optimal value of is compute as = max u M n=1 N(c) n () {v(c) () w u λ u }. Since the net utility v (c) w u λ u for peer to ownloa chunk c from peer u is the largest among the net utilities from all neighbors that can provie chunk c to peer, the secon conition is satisfie. The thir conition states that when a request (I,c) s achieve maximum utility is larger than, it efinitely has acquire a unit of uploa banwith. This is obviously true with the algorithm, since if a request oes not receive a unit of uploa banwith, its achieve maximum utility will be.