Analytical Framework for Optimizing Weighted Average Download Time in Peer-to-Peer Networks

Analytica l F ramew ork fo r Optimizi ng W eigh ted Av erage Do wnlo ad Time i n P eer-to-P eer Net w orks Bik e Xie, Mihaela v an der Sc haa r and Ric har d D. W esel Departmen t of Electrical Engineering, Univ ersit y of California, Los Angeles, CA 90095- 1594 Email: xbk@ee.ucla.edu , mihaela@ee.ucla.edu, we sel@ee.ucla.edu Abstr act — This pap er pro poses a n a nalytical framework for peer-to -peer (P2P) ne tw o r ks and in tr oduces schemes for building P2P netw orks to approach the minimum w eig h ted av erag e download t ime (W ADT). In the considered P2P framework, the server, w hich ha s the info r mation of all the download bandwidths a nd upload bandwidths o f the peers, minimizes the we ighted average download time by deter- mining the optimal transmission rat e from the server to the peers and from the peers to the other p eers. This pap er first defines the static P2P netw ork, the hierar c hical P2P netw ork and t he str ictly hiera rchical P2P netw ork. An y static P2P netw o rk can b e decompo sed into a n equiv alent netw ork of sub-peers that is strictly hierarchical. Thi s pap er shows that con vex optimiza tion can m inimize the W ADT for P2P net works by equiv alently m inimizing the W ADT for strictl y hierarchical netw orks of sub-p eers. This pap er then g ives an upper b ound for mini m izing W ADT by con- structing a hiera rchical P2P netw ork, and low er b ound by weak ening the constraints of t he con vex problem. Both the upper b ound and the lowe r b ound are v ery t ight. This pa- per al so provides several subopti mal solutions for minimiz- ing the W ADT for str ictly hiera r c hical netw orks, in which peer selection algori thms and c hunk selectio n algori t hm can be lo cally designed. Index T erms — P2P netw or k, weighted av era ge download time, hierarchical P2P netw or k, strictl y hi erarchical P2P netw ork I. Introduction P2P applications (e.g, [1], [2], [3]) ar e increasingly po p- ular and repre sent a large ma jorit y of the traffic currently transmitted ov er the In ter net. A unique feature of P2 P net works is their flexible and distributed nature, where each pe e r ca n act a s b oth server and client [4]. Hence, P2P netw o rks pr ovide a cost-effective a nd easily deploy- able framework for dissemina ting large files w itho ut relying on a centralized infrastructure [5]. Thes e features of P 2P net works hav e made them p opula r for a v ar iety o f broad- casting and file-distribution applications [5 ] [6] [7] [8 ] [9]. Spec ific a lly , chun k-based and data-driven P2P broadca s t- ing systems s uc h as Co olStreaming [6 ], Overcast [10] and Chainsaw [7] ha ve been develope d, which adopt pull-based techn iques [6], [7]. In these P2P sys tems, the p eers poss ess several c hunks a nd these chu nks ar e shared by p eers that are in terested in the same conten t. An imp ortant pro b- lem in such P2P systems is ho w to transmit the c h unks to the v a rious peer s and crea te relia ble and efficient connec- tions b etw een p eers. F or this, v arious appro aches hav e been prop osed including tree-based and data-dr iven approa c hes (e.g. [8] [11] [1 2] [13] [14] [15] [16]). Besides these prac tica l appr oaches, some research has beg un to a nalyze P2P netw orks from a theoretic p ersp ec- tive to quantify the achiev a ble p erformance. The p erfor - mance, scalability and ro bus tnes s of P 2 P ne tw orks using net work co ding are studied in [17] [18 ]. In these inv es- tigations, ea ch p eer in a P 2P netw o r k randomly chooses several p eers including the server as its par ents, and also transmits to its children a random linear co m bination of all packets the p eer has received. Netw o rk co ding, w or king as a perfect c hunk selec tio n algor ithm, g uarantees every pack et transmitted in a P 2P netw or k has new information for its r e ceiver, which makes elegant theoretica l ana lysis po ssible. Other resear c h studies the steady-state b ehavior of P 2P netw or ks with homogenous p eers using fluid mo dels [19] [20] [21]. Most pap ers providing theoretical analys is for P2P netw orks ass ume dynamic sy stems with homogenous pee rs. This pap er establishes an analytica l framework for opti- mizing weighted av era ge download time (W ADT) for P 2P net works with hetero geneous p eers, i.e., p eer s with differ- ent download bandwidths and upload bandwidths. This pap er fo cuses on static P 2P netw ork s with a single server and a fixed num b e r o f p eers. In other words, no p eer leaves or joins the P2P netw ork. In the scheme of building the P2P net work, the server collects all the download band- widths and uploa d bandwidths of the p eers, and minimizes the W ADT by determining the optimal transmissio n rate from the server or any p eer to any other p eer. A dynamic P2P sys tem can also be mo deled as a s e quence of static P2P sy stems. Therefore, this study of static P2 P netw or ks is the first step to a complete a nalytical framework. W e leav e the study of dynamic P2P netw or ks with heteroge- neous p eers fo r future work. A static P 2P netw or k is a dir ected graph which has one ro ot, the server, and at lea s t one dire cted path from the ro ot to ea ch of the other nodes , whic h are p eers . In a P2P netw o r k, peer s are placed int o levels accor ding to the top ological distances b etw een these p eers a nd the server. A peer is in level K if the length of the longest directed acyclic path fro m the server to the p eer is K . A hierar ch ical P2P netw o rk is a P2P netw o rk in which ea ch p eer can only download from p eers in the low e r levels and upload to pee rs in the higher levels. Peers in the same lev el cannot download or uplo a d to each o ther. A strictly hierarchical P2P netw o r k is a P2P netw o rk in which each peer in le vel K can only download fro m peers in level K − 1 and upload to p eers in level K + 1. This pap er shows that any sta tic P2P netw or k ca n be de- comp osed into an equiv a lent net work of sub- p eer s that is strictly hierarchical. Ther efore, con vex optimization can minimize the W ADT for P2P net works by equiv alently minimizing the W ADT for str ictly hiera rchical net works of sub-p eers . This pap er then g ives an achiev able uppe r bo und fo r minimizing W ADT by constructing a hier archi- cal P2P net work, a nd low er b o und by weak ening the c on- straints o f the con vex pro blem. Both the upper bo und and the low er b ound ar e very tigh t. The str ictly hiera rchical P2 P net work is pr actical for pro- to col design beca use p eer selectio n algor ithms a nd ch unk selection algor ithms can b e lo cally designe d level by level instead of globally designed. Minimizing the W ADT for strictly hiera r chical net works is a 0-1 conv ex optimization problem. How ever, if we have assigned all p eers each to a level, then the globa l bandwidth allo ca tion problem de- comp oses into lo cal ba ndwidth allo cation problems at each level, which have water-filling solutions. Several sub op- timal p eer as s ignment algor ithms are provided and simu- lated. Some of these suboptimal but pr actical s chemes ca n be use d for conten t distribution sys tems, e.g. Overcast [10]. This pape r is organized a s follows. In Section II, def- initions and notatio n for P2P netw orks are introduced. Section I I I pr ovides and discusses a taxonomy of over- lay net works. In Section IV, the problem of minimiz- ing the weigh ted av erage download time is formulated and solved. Section V prese nts the sim ulation results. Section VI prese nts the co nclusions. I I. S etup and Problem Definition Consider a scenario where millions of p eers w ould lik e to download conten t from a s erver in the In ternet. The server has sufficient bandwidth to ser ve tens or hundreds of peer s, but no t millions. In the absence o f IP multicast, one solu- tion is to form the se r ver and the pe e r s in to a P2P o verlay net work and distribute the cont ent using application lay er m ulticast [17] [22]. In this scenario , the con tent in the server is par titioned in to ch unks . Peers not o nly do wnload ch unks fro m the ser ver a nd other p eers but also uploa d to some other peers that are in ter ested in the conten t. This pape r focus es on cont ent distribution applications (e.g, BitT orr ent , Overcast [10]) in whic h peer s are only int erested in conten t at full fidelit y , even if it means that the conten t do es not beco me av aila ble to a ll p eers at the same time. The key iss ue for these P2 P a pplications is to minimize download times for pe e rs. Since it usually takes s everal hour s or da ys for a peer to do wnload conten t in full fidelity , our w ork is less concerned with in teractive resp onse times and transmission delays in buffers and in the netw o rk. This pap er studies a scheme to minimize the weigh ted av er age do wnload time for a static P2P netw o rk. In this scheme, the server first collects the infor mation of p eers’ weigh ts, download bandwidths, and uplo ad bandwidths and then p erforms a cent ralized optimization to find the bes t static P2P netw ork, i.e., the o ptimal tra ns mission rates of the tra ns mission flo ws from the server or an y p eer to any other peer to minimize the weigh ted av erage do wn- Peer 1 S 2 3 4 1 o s r 1 2 o r 1 3 o r 3 1 o r 4 1 o r 1 4 o r 2 1 o r Fig. 1 The peer model load time. The se r ver passes the optimal solution to the pee rs, and the peer s build the c onnections according to the optimal solution. The rest of the pap er will fo cus on the cen tralized o ptimization algorithm for determining the optimal static P2P netw or k. In a s ta tic P2P netw ork, the ser ver with ba ndwidth S has a file, whose size is 1 unit for simplicity . There are N pee rs who wan t to share the file in the net work. Ea ch p eer has download ba ndwidth d i and uploa d bandwidth u i , for i = 1 , 2 , · · · , N . These download ba ndwidths and upload bandwidths a re usually determined a t the application lay e r instead o f the physical lay er b ecause an Internet user can hav e several downloading tasks a nd these tasks share the ph ysical download and uploa d bandwidth of the user. It is reas onable to assume tha t d i ≥ u i for e ach 1 ≤ i ≤ N . F or the c a se of d i < u i for p eer i , we just use the pa r t of the upload bandwidth whic h is the same a s the download bandwidth and leav e the rest of the uplo ad ba ndwidth. Denote the transmiss ion rate from the ser ver to pee r j as r s → j and the tr ansmission rate fr om p eer i to peer j as r i → j . The total download rate o f p eer j , denoted a s r j , is the summatio n of r s → j and r i → j for a ll i 6 = j . Since the total do wnload rate is constrained b y the download bandwidth, we hav e r j = r s → j + P i 6 = j r i → j ≤ d j for all j = 1 , · · · , N . Since the total uplo ad rate is constrained b y the upload bandwidth, we a lso hav e P i 6 = j r j → i ≤ u j for all j = 1 , · · · , N . One example of the p eer mo del is s hown in Fig. 1. The do wnlo a d bandwidth and uplo ad bandwidth of Peer 1 a re d 1 and u 1 resp ectively . Thus, the total download rate r s → 1 + P 4 i =2 r i → 1 is le s s than or equal to d 1 . The tota l upload ra te P 4 i =2 r 1 → i is less than or equa l to u 1 . Since o ur work is less concer ned with in teractive resp onse times and transmission delays in buffers and in the netw or k, the download time for each p eer is dominated by 1 /r i and the weigh ted av erage download time is P N i =1 w i /r i where w i ≥ 0 is the weigh t o f p eer i . In a P2P netw ork, if peer i forwards conten t to p eer j , then peer i is a parent of pee r j and p eer j is a child of peer i . Each no de can hav e s everal parents and several children. A primary goal of this pap er is to minimize the W ADT computed as P N i =1 w i /r i . In the W ADT computation r i could refer to the actual download rate r ( a ) i or the bud- S 1 2 3 4 Fig. 2 A hierarchical P2P n etwo rk with 4 peers geted download rate r ( b ) i . In genera l, r ( a ) i ≤ r ( b ) i bec ause the parents o f a p eer mig h t not alwa y s have new conten t for sharing. The net work coding strategy can be used as a per fect ch unk selection alg orithm to g ua rantee that a par- ent alwa ys has new co n tent for its children [17]. This pap er uses the netw ork co ding strategy a s a ch unk selection al- gorithm in P 2 P netw o rks and so r ( a ) i = r ( b ) i . There fore, in the re st of the pa per , we use r i for b oth budge ted download rate and a ctual download rate. I I I. T axonomy of Overla y Networks This s ection s tudies graph s tructures of P 2P overla y net- works. Definition 1: static P2P net w ork : A static P 2P net- work is a directed gra ph which has one ro ot and a t least one dir e c ted path fr o m the r o ot to ea ch of the other no des. The ro o t no de is the server which ha s the conten t to share. All the o ther no des a re the p eers that are interested in the co n tent. In a P2P netw or k , p eers ar e placed into levels acco r ding to the top olo g ical distances betw een these pee rs and the server. Definition 2: level of p eer : A pee r/no de(use p eer or no de?) is in level K if the length of the lo ngest directed acyclic path from the s erver to the p eer is K . The server is defined in level 0. Definition 3: level of P2P net work : A P2P net work has level K is the maximum lev el of peers is K . A. The hier ar chic al P2P net work Definition 4: hierarc hical P2P net w ork : A hierarchi- cal P 2P netw ork is a P2P net work in which each p eer can only download from p eers in the low e r levels and upload to pee rs in the higher levels. Fig. 2 shows a hierarchical P2P net work with 4 peer s, in which p eer i is in level i . Note that in this example, each peer do wnloads fro m all p eers in the low er levels and uploads to a ll p eers in the higher le vels. Howev er, by defi- nition, p eers a re not required to download from all p eers in the low er levels or upload to /from? a ll p eers in the higher levels. F or example, the netw ork shown in Fig. 2 will still be a hiera rchical P2P net w ork if the edge 1 → 4 v a nishes. How ever, p eers are not allowed to do wnload from any p eer in a higher level or upload to /from? any peer in a low er level. Also p eers in the same level canno t download o r upload to each other . L emma 1: A hierarchical P2P netw or k contains no di- rected cycle. Pr o of :(by contrapos ition) Supp ose there is a directed cy- cle A 1 → A 2 → · · · → A n → A 1 , then some A i → A i +1 or A n → A 1 violates the r equirement that p eers in a hier- archical P2P network canno t uplo ad to a pee r in a lower level. The r efore, the net work con taining the directed c y cle cannot b e a hierarchical P2P netw or k. Q.E.D. L emma 2: A dir ected acyclic P2P netw ork is a hiera r- chical P2P net work. Pr o of :(by contrapo sition) Supp ose a P2P net work is not hierarchical, i.e., there exits a no de A in level m and a no de B in level n such that m ≥ n and A → B . Let S → A 1 → · · · → A m − 1 → A be the longes t directed acyclic path from S to A and S → B 1 → · · · → B n − 1 → B be the long est directed acyclic path from S to B . Since S → A 1 → · · · → A m − 1 → A → B is a directed path from S to B with length m + 1 > n , this path m ust contain a directed c ycle, and hence, the P2P netw ork is not a directed acyclic gra ph. Therefore, a dir e cted acyclic P2P netw or k is a hierarchical P 2P netw ork . Q.E.D. The or em 1: T he set of all hierarchical P2P netw o rks is the set of all directed acyclic P2 P net works. It is a direct co ns equence of Lemma 1 and Lemma 2. B. The st rictly hier ar chic al P2P network Definiation 5: strictly hierarc hical P2P ne t work : A strictly hierarchical P2P net work is a P2P net work in which each p eer in lev el K can only download from p eers in level K − 1 and upload to p eers in level K + 1. Fig. 3 shows a s trictly hier archical P2P netw ork with 3 le vels. In a strictly hierar chical P 2P netw ork, p eers in level K work together as a vir tua l se rver and upload to pee rs in lev el K + 1. In Fig. 3, p eer 1 a nd 2 form the virtual server in level 1, denoted as S 1 . Peer 3, 4 and 5 form the v irtual s erver in level 2, denoted as S 2 . Since a ll transmission flows are be tw een tw o consecutive levels, p eer selection a lgorithms and chu nk se le ction algor ithms c a n b e lo cally desig ned lev el by lev el. The relatio nships among the P2 P netw ork , the hierar ch i- cal P2P netw or k and the strictly hie r archical P2P netw ork are co nc luded in Fig. 4. C. Net work of s ub-p e ers A P 2P overla y netw or k is divisible if peer s can be di- vided into sub-pe ers, and it is indivisible if p eers ca nnot be divided. The division of peers can b e p erformed at the ap- plication lay er [4] [2 3]. Even for indivisible P2P net work, pee rs can be conceptually divided into virtual sub-pee r s for theoretica l analysis . A simple exa mple of p eer division is given in Fig. 5. Fig. 5(a) shows the origina l P2 P net- work and Fig. 5(b) shows the netw o rk of sub-p eers after the divisio n of p eer 1 in to sub-p eer 1 A and sub-p eer 1 B . A peer division is equiv a lent if the tra nsmission ra tes from the s e r ver to ea c h p eer and from each p eer to each S 1 2 4 5 6 3 S 1 S 2 Fig. 3 A st rictl y hierarchical P2P network P 2 P  n e t w o r k s h i e r a r c h i c a l  P 2 P n e t w or k s  =  d i r e c t e d a c y c l i c  P 2 P  n e t w o r k s s t r i c t l y  h i e r a r c h i c a l P 2 P  n e t w o r k s Fig. 4 Conclusion on t axonomy of overla y netw orks other p eer are inv a riant. The netw ork of sub-p eers is equiv - alent to the or iginal P 2P netw ork if the p eer division is equiv alent. The p eer division in Fig. 5 is equiv a lent if r s → 1 = r s → 1 A , r 1 → 2 = r 1 A → 2 and r 3 → 1 = r 3 → 1 B . The or em 2: Any P2 P netw o rk with N peers and K levels can be deco mpo sed into an e quivalent netw ork of sub-p eer s that is strictly hierarchical, and has at most K levels, each of which contains at most N sub-p eers. Pr o of of The or em 2: In or der to prove the theor em, it suffices to construct a strictly hier archical netw ork o f sub- pee rs which is equiv alent to the origina l P2P netw or k. F or any P 2P net work with N p eer s and K lev els, deno te the server S a s node 0 and the pee r s a s no de 1 , 2 , · · · , N . Let K i = { k : ∃ a directed acyclic path fro m S to p eer i with length k } , (1) and so | K i | ≤ K is the ca rdinality of K i . Divide p eer i int o | K i | sub-p eers, which are deno ted a s s ub-pe e r ( i, k ) for k ∈ K i . The sub-p eer ( i, k ) is the part of p eer i in level k . In the netw o rk of sub-pee r s, there is an edge from the server S to sub-p eer ( i, 1) if and only if S → i in the original net work. W e as sign the transmis sion r ate r s → ( i, 1) = r s → i . There is an edge from sub-p eer ( i, k ) to sub-p eer ( j, k + 1) if and only if there exits a directed acyclic path with length k + 1 suc h that S → · · · → i → j in the or iginal P2P S 1 2 3 S 1 A 2 3 1 B 1 ( a ) ( b ) Fig. 5 Peer division 3 2 3 2 1 S ( 1 , 1 ) ( 2 , 1 ) ( 2 , 2 ) ( 3 , 2 ) ( 1 , 3 ) ( 3 , 3 ) 1 S ( 1 , 1 ) ( 2 , 1 ) ( 2 , 2 ) ( 3 , 2 ) ( 1 , 3 ) ( 3 , 3 ) ( 1 , 3 ) ( 2 , 3 ) ( 3 , 1 ) ( a ) ( b ) Fig. 6 Constructio n of the network of sub-peers net work. W e assign the transmission r ates level by level with r ( i,k ) = X ( i ′ ,k − 1) → ( i,k ) r ( i ′ ,k − 1) → ( i,k ) , (2) r k i → j = r i → j − X m ≤ k r ( i,m − 1) → ( j,m ) , (3) r ( i,k ) → ( j,k +1) = min( r ( i,k ) − X j ′ m , and hence X k ≤ K r ( i,k − 1) → ( j,k ) = X k ≤ m r ( i,k − 1) → ( j,k ) = r i → j . (11) Finally , w e need to show that the assigned tra nsmission rates in the net work o f sub-p eers a re feasible, i.e., the down- load rate of each sub-p eer is grea ter than o r equa l to it upload ra te. Plugging j = N int o (4), r ( i,k ) → ( N ,k +1) ≤ r ( i,k ) − X j ′ √ w i R d i if √ w i R > d i (18) where R is c hosen appro priately such that P N i =1 ( r i − u i ) = S − max i ( u i ). This upp er b ound is almos t o ptimal. Consider the fol- lowing low er b ound which is very clo se to the upp er b ound. Since S ≥ P N i =1 ( r i − u i ) is a relaxe d constraint, an low e r bo und is the solution to the optimization pr oblem Minimize N X i =1 w i /r i (19) Sub ject to d i ≥ r i ≥ u i , i = 1 , 2 , · · · , N , N X i =1 r i − u i ≤ S. Note that this optimization pro ble m (19) differs from the problem (17) only by replacing S with S − max i ( u i ). Th us, the low er b ound is P N i =1 w i /r i with r i =    √ w i R if u i ≤ √ w i R ≤ d i u i if u i > √ w i R d i if √ w i R > d i (20) where R is c hosen appro priately such that P N i =1 ( r i − u i ) = S . Since S ≫ max( u i ), one has S ≃ S − ma x i ( u i ). The upper bo und and the low er b ound are almost the same. Therefore, b oth o f these t wo bounds are very tig h t, and the upper b ound is almost an a nalytical so lution to minimizing the W ADT. Hence, it is sufficient to build a hierarchical P2P netw o r k to achieve the minim um W ADT. C. Pr actic al Solut ions Compared with building netw orks of sub-p eer s and hi- erarchical P2P net works, it is more practical to build a strictly hierarchical P2P netw ork. First, in a strictly hier- archical P2 P net work, p eers do not need to b e decomp osed int o sub-peer s. Thus, it can have a muc h simpler proto col in netw or k lay er than a net work of sub-p eers. Second, in a strictly hier archical P2P netw o rk, each level usually con- tains o nly tens or h undreds of p eers. Therefore, one can lo cally design p eer s election algor ithms and ch unk selection algorithms level by level, which only depend o n a small col- lection of pe ers in the P2P netw o rk. The lo cally desig ned pee r and ch unk selection algor ithm might be mu ch simpler than the netw or k co ding stra tegy a nd other global designed pee r and c hunk selection algorithms. The following theo- rem shows that there exists a stric tly hierarchical net work of sub-p eer s which achiev es the upper bo und in the previ- ous subsection, and is very close to a strictly hier archical P2P netw o r k. The or em 3: T he r e ex ist multiple upp er- bo und-achieving net works of sub-p eers in which at most K − 1 p eer s need to be decomp osed int o 2 s ub-pee rs, where K ≪ N is the nu mber of levels. Pr o of : Equation (18) gives r i , i = 1 , · · · , N , for the hierarchical P2P net work which achiev es the upper b ound. Conv er t the hierarchical P2P netw ork to mult iple optimal strictly hierarchical netw orks of sub-pee rs b y the follo wing algorithm. This a lg orithm indicates tha t a t most K − 1 peer s need to b e divided into 2 s ub-pe e r s a nd each level of the co n- structed netw o rk of sub-p eers c ont ains at mo s t 2 sub-p eer s. Since there are m ultiple choices in step (2), this alg orithm provides multiple str ictly hierar chical netw orks of sub-p eers which are very close to a stric tly hierarchical P2P netw o rk. Q.E.D. Another practical solution is to solve the problem of min- imizing the W ADT dir ectly for str ictly hierarchical P2P net works. This problem can b e formulated a s a 0-1 co n- vex optimization. The complexity to solve a 0-1 conv ex optimization pro blem is exp onential to the problem size . F ortunately , this problem has an analytical sub optimal solution. Suppose there are N peers and K levels in a Algorithm 1 Peer Placement Algorithm 1: Init ialize lev el l = 1 ; 2: Init ialize the rest ser ver bandwidth for the current level s = S ; 3: Init ialize the set of the re s t o f the p eers G = 1 , 2 , · · · , N ; 4: while G is not empt y do 5: Cho ose any p eer i fro m the set G ; 6: if s ≥ r i then 7: Put Peer i in the current level l ; 8: Set G = G \{ i } ; 9: Set s = s − r i ; 10: else 11: Decomp ose Peer i into 2 sub-p eers with r (1) i = s , r (2) i = r i − s , u (1) i = min( u i , r (1) i ), u (2) i = u i − min( u i , r (1) i ); 12: Put the s ub-pee r (1) in level l and the sub-p eer (2) in level l + 1; 13: Set s = ( P Peer j in le vel l u j ) − r (2) i ; 14: Set l = l + 1; 15: Set G = G \{ i } ; 16: end i f 17: end w hile strictly hierarchical P 2P netw o rk and the le vel lo ca tion of each p eer is already g iven. T hen the g lobal optimiza- tion problem can b e deco mpos ed int o K lo cal optimization problem and a ll o f them hav e a n ana lytical “W a ter Filling” solution. In par ticular, supp ose the bandwidth of the server is S = S 0 . The download bandwidth, uploa d bandwidth and the allo cated do wnload rate of P eer i a re d i , u i and r i re- sp ectively . S j , the bandwidth of the virtual server in Lev el j , is equal to the summation o f the upload bandwidths of the p eers in Lev el j . If S j is grea ter or equal to the sum- mation of the uploa d bandwidth of all p eers in lev el j + 1, then o ne has d i ≥ r i ≥ u i and S j +1 = P Peer i in level j u i . If S j is less than the summation of the uplo a d bandwidth of all pe e r s in level j + 1, one has 0 ≤ r i ≤ u i and S j +1 = S j = P Peer i in level j r i . Therefore, no ma tter ho w to allo ca te download r ates to peer s in eac h le vel, the band- width of the virtual ser ver in each level is fixed as long as the level p ositions of peer s a re fixe d. Th us, the global minim um weigh ted av erage transmission rate pr o blem can be decompo sed to K lo cal optimization pro blem. Each of them has a fixed server bandwidth and fixed num b er o f pee rs in the lev el. Thus, if the virtua l ser ver S j is greater or equal to the summation o f the upload bandwidth of a ll pee rs in level j + 1, then the lo cal optimization pro ble m is Minimize N j +1 X i =1 w i /r i Sub ject to d i ≥ r i ≥ u i , i = 1 , 2 , · · · , N j +1 N j +1 X i =1 r i ≤ S j where N j +1 is the num b er of p eers in level j + 1, for j = 0 , · · · , K − 1. By the Karush-K uhn-T uck er conditions, the optimal so- lution is r i =    √ w i R if u i ≤ √ w i R ≤ d i u i if u i > √ w i R d i if √ w i R > d i (21) where R is chosen appropr ia tely suc h that P N j +1 i =1 r i = S j . If the virtual server S j for level j + 1 is le s s than the summation of the upload bandwidth of all p eers in level j + 1 , then the loc al optimization pr oblem is Minimize N j +1 X i =1 w i /r i Sub ject to u i ≥ r i ≥ 0 , i = 1 , 2 , · · · , N j +1 N j +1 X i =1 r i ≤ S j where N j +1 is the num b er of p eers in level j + 1, for j = 0 , · · · , K − 1. By the Karush-K uhn-T uck er conditions, the optimal so- lution is r i =  √ w i R if 0 ≤ √ w i R ≤ u i u i if u i < √ w i R (22) where R is chosen appropr ia tely suc h that P N j +1 i =1 r i = S j . V. Resul ts In this sectio n, we sim ulate and ev alua te the p e r for- mances of the metho ds provided in Section IV. Note that in this pa per the download bandwidth d and the upload bandwidth u are decided a t the application lay er instead of the physical lay er . Thus, the download ba ndwidth d can be contin uously distributed in a large range such as 1 0 kbps to 100 Mbps. W e a ssume that the download bandwidths d i , i = 1 , · · · , N ar e independently ident ically distributed (i.i.d.) with uniform distribution ov er [ β , 2 − β ], wher e β is a very small p ositive num b er such as 10 − 2 , 10 − 4 . This distribution has the normalized mean v alue 1 and lar ge maximum to minimum ra tio (2 − β ) /β . In practice, this parameter β ca n be determined as the minimum do wn- load bandwidth which is required by the P2P s y stem. W e also assume that the uplo ad bandwidth u i is unifor mly dis- tributed ov er [ αd i , d i ], where 0 < α < 1 is the minimum upload to download bandwidth r e quired by the P 2P sys- tem. A. A sm al l network simulation In this small simulation, the bandwidth of the server is 10 and the num b er of the p eers in the net work is N = 100. The download bandwidths d i , i = 1 , · · · , 100, are i.i.d. with uniform distribution over [0 . 01 , 1 . 99 ], i.e., β = 10 − 2 . F or ea ch peer i , the upload bandwidth is uniformly distributed over [0 . 1 d i , d i ], i.e., α = 0 . 1 . The weigh ts for all the peers ar e equal and norma lized to b e 1 / N = 1 / 100, and so P i w i = 1. Thus, the W ADT is P i w i /r i . W e will compare 7 methods in this e x per imen t. They are • 1. O ptimal solution to the conv ex optimization program with level K = 30. The conv ex prog ram solver is CVX [26]. • 2. The upper bound (17) by constructing a hier archical P2P netw o r k. • 3. The lower b ound (19) by relaxing the co nstraints. • 4. The sub optimal s olution to the 0- 1 conv ex optimiza- tion pr o blem for str ictly hiera rchical P2 P net works. The nu mber of the levels are set to b e K = 30 and w e randomly put aro und N /K p eers in each lev el. • 5. The sub optimal s olution to the 0- 1 conv ex optimiza- tion problem for str ic tly hierar chical P2P netw o rks. W e conv er t the upper -bo und-achieving hierar chical P2P net- work (Method 2) to a ne tw ork of sub-pe e rs by Algorithm 1. The p eers are placed into levels a ccording to the con- structed net work of sub-p eers, which is very close to a strictly hiera rchical P2P net work. • 6 This is a trivial upper bound suc h that the r ate o f each pee r is the same as the upload ba ndwidth. • 7 This is a trivial lo wer b ound such that the ra te o f ea ch pee r is the same as the download bandwidth. The distr ibution o f the w eighted av era ge download times of 500 exp e riments are shown in Fig. 8. The distribution of the difference and the rela tive difference betw een Method 3 and other methods are shown in Fig. 9 and Fig. 10. The weigh ted average download times o f Metho d 1 is conce n- trated in the range of [1 . 7 , 4 . 1] with mea n v alue 2 .877. Metho d 2 and Metho d 3 provide the upper bo und and low er b ound for the minimum W ADT. Fig. 8, 9, a nd 10 show that these tw o b ounds ar e v ery tight. In this cas e , the low er bo und is almost alwa ys the same as the optimal so lu- tion since the distribution curves of Method 1 and Method 3 p erfectly match. The distribution c urve of Metho d 2 is slightly different from that o f Metho d 1, whic h means that the upp er b ound is only slightly larg er than the optimal solution for mo st of the exp eriments. In this simulation, the bandwidth of the ser ver is 1 0, which is no t muc h larger than the maximum of the upload bandwidths. That is wh y the upp er b ound is still slightly different fro m the optimal solution. The distribution of Metho d 5 sho ws that in most of the expe riments, the W ADT of Metho d 5 is close to the optimal s olution, how ever, it is s ometimes muc h la r ger than the optimal solution. This is b ecause the p eer placement in Metho d 5 is usually very go o d but so metimes bad. Note that it is NP c o mplete complex to find the best peer place- men t. In order to improve Metho d 5 without incr easing the complexity , we need provide a b etter but still lo w-complex pee r place men t alg orithm. The per formance o f Metho d 4 is m uch worse than the p erformance of Method 5 b ecause the p eer placement a lg orithm is Metho d 4 is muc h worse that in Method 5. The bandwidth usage is the ratio of the to tal transmis- sion ra te and the tota l download bandwidth o f all the peer s in the netw ork . F or a fixed W ADT, it is clear that the lower bandwidth usage the b etter. Ho wever, it is a trade o ff to 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 Weighted average download time Distribution Experiment A. Method 7: trivial lower bound Method 1: optimal Method 2: tight upper bound Method 3: tight lower bound Method 5: strictly hierarchical with suboptimal placements Method 4: strictly hierarchical with random placements Method 6: trivial upper bound Fig. 8 Distribution of the W ADT for different methods. −0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 absolute distance of weighted average download time Distribution Experiment A. Method 7: trivial lower bound Method 1: optimal Method 2: tight upper bound Method 3: tight lower bound Method 5: strictly hierarchical with suboptimal placements Fig. 9 Distribution of the differences of the W ADT between Method 3 and other m ethods. decrease the bandwidth usage and to decrea se the W ADT, which is also v erified in Fig. 11. Fig. 11 shows the dis tri- bution of the bandwidth usage of different metho ds. The distribution of Method 1, the optimal solution, is exa c tly the same as the distribution of Metho d 3, the low er bo und. The ba ndwidth usage of Metho d 2, the upp er b ound, is slightly less than that of Method 1 and the bandwidth us- age of Method 5 is slig h tly les s than tha t o f Metho d 2. It is verified tha t the metho d with s maller W ADT alwa ys has higher bandwidth usage. The mean v alues of the weigh ted av er age download times and the bandwidth usage of differ- ent methods are listed in T able I. F or one typical exp eriment, the plots o f the vir tual server bandwidths v er sus the levels for differen t metho ds are shown in Fig. 12. F or Metho d 1, the n um b er of the lev els K is ma n ually chosen. In this exp eriment, K is 30. One ca n −0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Experiment A. relative difference of weighted average download time Distribution Method 7: trivial lower bound Method 1: optimal Method 2: tight upper bound Method 3: tight lower bound Method 5: strictly hierarchical with suboptimal placements Fig. 10 Distribution of the rela tive d ifference of the W ADT between Method 3 and other methods. 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 bandwidth usage Distribution Experiment A. Method 1: optimal Method 2: tight upper bound Method 3: tight lower bound Method 5: strictly hierarchical with suboptimal placements Method 4: strictly hierarchical with random placements Method 6: trivial upper bound Fig. 11 Distribution of the bandwidth usage for diff erent m ethods. T ABLE I The mean v a lues of the weighted a verage download times (W.A.D.T.), the normalized weighted a verage do wnload times (N.W.A.D.T.)and the bandwid th usage (B.U.)of different methods for experiment A. Metho d W.A.D.T. N.W.A .D.T. B.U. 7 2.468 0.854 1.000 3 2.877 1.000 0.650 1 2.877 1.000 0.650 2 2.947 1.025 0.633 5 3.461 1.200 0.621 4 5.338 1.849 0.471 6 6.361 2.194 0.551 0 5 10 15 20 25 30 35 −2 0 2 4 6 8 10 level bandwidth of virtual servers Experiment A. Method 1: optimal Method 2: tight upper bound Method 5: strictly hierarchical with suboptimal placements Method 4: strictly hierarchical with random placements Fig. 12 Bandwidt h of vir tual ser vers for differen t meth ods. see that the bandwidths o f the virtual servers from level 5 to level 30 are linearly decreasing to 0 for Method 1 . The bandwidth of the v irtual server g enerated by the last level is almost 0 . In other w ords, all the upload bandwidths of the server and the pe ers a re fully used. This is the rea son why the per formance of Metho d 3, the tigh t low er b ound, is almo st the same as that of Method 1 . F o r Metho d 2, the nu mber of the levels needed is automatically solved. It is 15 in this exp eriment. B. A lar ge n etwork simulation In this simulation, the bandwidth of the server is 50 and there are 4000 p eers in the net work. The download ba nd- widths d i , i = 1 , · · · , 4000 , are i.i.d. with uniform distri- bution ov e r [0 . 01 , 1 . 99 ]. The uploa d bandwidth u i is uni- formly dis tributed ov er [0 . 1 d i , d i ]. The weigh ts for all the pee rs a re equal and nor malized to be 1 / N = 1 / 40 00, and so P i w i = 1. The distr ibution o f the w eighted av era ge download times of 80 0 exp eriments ar e shown in Fig. 1 3. The distribu- tion of the difference and the rela tiv e difference b etw een Metho d 3 and other metho ds a r e shown in Fig. 14 and Fig. 15. These figures show that the low er bound and the upper bound provided by Metho d 3 and Metho d 2 are very close. The p erformance of Metho d 1 should b e betw een these tw o bo unds, a lthough we don’t simulate Metho d 1 in this case. In this simulation, the bandwidth of the server is 50 , which is muc h larger than the maximum of the up- load bandwidths. That is why S ≃ S − max i ( u i ) and the upper bo und is almost the same as the low er bo und. The per formance o f Metho d 5 is worse than that of Metho d 2 but still a lot b etter that of Metho d 6 a nd 7. The bandwidth us a ges o f these metho ds a re shown in Fig. 16. The distribution of Metho d 2 is a lmost the same as the distribution o f Metho d 3, which verifies the almost same pe rformance of Method 2 a nd 3. The mean v alues of 2 3 4 5 6 7 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Weighted average download time Distribution Experiment B. Method 7: trivial lower bound Method 2: tight upper bound Method 3: tight lower bound Method 5: strictly hierarchical with suboptimal placements Method 4: strictly hierarchical with random placements Method 6: trivial upper bound Fig. 13 Distribution of the W ADT for different m ethods. −2 −1 0 1 2 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 difference of weighted average download time Distribution Experiment B. Method 7: trivial lower bound Method 2: tight upper bound Method 3: tight lower bound Method 5: strictly hierarchical with suboptimal placements Method 4: strictly hierarchical with random placements Method 6: trivial upper bound Fig. 14 Distribution of the differences of the W ADT between Method 3 and other methods. the weight ed av erage do wnload times and the bandwidth usage o f different metho ds a re listed in T able II . Co m bining the simulation results, the comparis on of these 7 metho ds in different criteria a re listed in T a ble I I I. VI. Conclusions This pap er pro p os es a n analytical framework for peer -to- pee r (P2P) net works and in tro duces schemes for building P2P net works to a pproach the minimum w e ighted av erage download time (W ADT). In the considered P2P framework, the server, which has the information of all the download bandwidths and upload bandwidths of the pee r s, minimizes the weigh ted av era ge download time b y determining the optimal transmission rate from the server to the pe ers and from the peers to the other p eers. This pap er first defines the static P2 P netw or k, the hi- −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 relative difference of weighted average download time Distribution Experiment B. Method 7: trivial lower bound Method 2: tight upper bound Method 3: tight lower bound Method 5: strictly hierarchical with suboptimal placements Method 4: strictly hierarchical with random placements Method 6: trivial upper bound Fig. 15 Distribution of the rela tive diff erence of the W ADT between Method 3 and other methods. 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 bandwidth usage Distribution Experiment B. Method 2: tight upper bound Method 3: tight lower bound Method 5: strictly hierarchical with suboptimal placements Method 4: strictly hierarchical with random placements Method 6: trivial upper bound Fig. 16 Distribution of the bandwidth usage for diff erent m ethods. T ABLE I I The mean v alues of the weighted a verage download times (W.A.D.T.), the normalized weighted a verage do wnload times (N.W.A.D.T .)and the bandwid th usa ge (B.U.)of different methods for ex periment B. Metho d W.A.D.T. N.W.A.D.T. B.U. 7 2.659 0.6898 1.000 3 3.854 1.000 0.562 1 - - - 2 3.870 1.041 0.562 5 4.666 1.210 0.552 4 6.709 1.740 0.438 6 6.804 1.765 0.550 T ABLE I I I Comp arison of the methods in the criteria of the weighted a verage do wnload time (W. A.D.T.), the b andwidt h usage (B.U.), the complexity of the comput a tion in the ser v er (S . Comp.) and the complexity of the algorithms in peers (P. Comp.) Metho d W.A.D.T. B.U. S. Comp. P . Comp. 1 Optimal Highest O ( K 3 N 3 ) High 2 Almost Opt. V e ry High O ( N ) Low 5 Go o d High O ( N ) Low 4 Bad Low O ( N ) Low 6 Bad Low O ( N ) Low erarchical P2P netw or k and the s trictly hierarchical P2P net work and studies the graph structures o f these P 2P net- works. The main result is that an y static P 2P netw ork can be decomp osed in to an equiv a lent netw ork o f sub-p eers that is stric tly hierarchical. T her efore, conv ex optimization can minimize the W ADT fo r P2P net works by equiv ale n tly minimizing the W ADT for strictly hierar chical netw orks of sub-p eers . This pap er then g ives an achiev able upp e r bo und fo r minimizing W ADT by constructing a hier archi- cal P2P net work, a nd low er b o und by weak ening the c on- straints o f the con vex pro blem. B oth the upper bo und and the low er b ound ar e very tigh t. The str ictly hiera rchical P2 P net work is pr actical for pro- to col design beca use p eer selectio n algor ithms and ch unk selection algorithms can b e lo cally designed level b y level instead of globally designed. Minimizing the W ADT for strictly hierarchical netw orks is a 0-1 conv ex optimization problem. How ever, if we have assigned all p eers each to a level, then the globa l bandwidth allo cation pr oblem de- comp oses into lo cal ba ndwidth a llo cation problems at ea c h level, which have w ater-filling solutions. 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