Optimal Peer Selection in a Free-Market Peer-Resource Economy

Transcription

1 Optimal Pee Selection in a Fee-Maket Pee-Resouce Economy Micah Adle, Rakesh Kuma, Keith Ross, Dan Rubenstein, David Tune and David D Yao Dept of Compute Science Univesity of Massachusetts Amhest, MA; micah@csumassedu Dept of Electical Engineeing, Polytechnic Univesity, NY; kuma04@utopiapolyedu Dept of Compute and Infomation Science, Polytechnic Univesity, NY; oss@polyedu Dept Of Electical Engineeing, Columbia Univesity, NY; dan@eecolumbiaedu Dept Of Compute Science, Califonia State Univesity San Benadino, CA; dtune@csusbedu IEOR Dept, Columbia Univesity, NY; yao@columbiaedu Abstact In a P2P fee-maket esouce economy a client pee may want to select multiple seve pees fo eithe downloading a file o steaming a stoed audio o video obect In geneal, multiple seve pees will make the obect available, with each pee offeing a diffeent pice The optimal pee selection poblem is to select fom a subset of the pees those pees that can povide the sevice at lowest cost In this pape we fomulate and solve the poblem of optimally selecting a subset of pees fo paallel downloading and fo paallel steaming I INTRODUCTION Today many pee computes paticipate in pee-to-pee file shaing applications in which the computes contibute stoage and bandwidth esouces File shaing, howeve, is not the only application that can exploit esouces made available by vast numbes of pee computes Othe impotant applications that could exploit a vast pool of pee esouces include pee-diven content distibution netwoks [1], globally distibuted achival stoage [5], [6], [12], and massively paallel computation [13] Of couse, applications can only haness the esouce pool if pees make available thei suplus esouces to them It is widely documented, howeve, that the P2P systems ae havens fo fee ides : a significant faction of uses do not contibute any esouces, and a minute faction of uses contibute the maoity of the esouces [3], [4] Clealy, to impove the pefomance of existing P2P file shaing systems, and to enable new classes of P2P applications, a compelling incentive system needs to be put in place Now suppose the existence of an online maket place whee entities - such as pees, companies, uses and so on - buy and sell suplus esouces In this maket place, a pee might puchase stoage and bandwidth fom a dozen othe pees fo the pupose of emotely backing up its files; a content publishe might puchase stoage and bandwidth fom thousands of pees to ceate a pee-diven content distibution netwok; a biotechnology company might puchase CPU cycles fom thousands of pees fo distibuted computation If such a flouishing esouce maket existed, individual pees would be incited to contibute thei esouces to the maket place, theeby unleashing the untapped esouce pool We envision a fee maket in which pees buy and sell esouces diectly fom each othe [1] In this fee maket, selling pees ae fee to set the pices of thei esouces as they please A client pee, inteested in puchasing a specific esouce, is pemitted to shop the diffeent seve pees and choose the pees that best satisfy its needs at the best pices The money paid by the client pees and eaned by the seve pees may be eal money, such as US dollas, o may be some pseudo-money akin to fequent flye miles Afte a selle eans money, it can late spend the money in the esouce maket, obtaining esouces fom othe selle pees Fo example, a pee Alice may ean money by tansfeing a potion of a video file to some pee Bob Pee Alice may then use this eaned money to puchase a diffeent video file fom a diffeent pee Claie O pee Bob may use the money to puchase a diffeent type of esouce, such as backup stoage o CPU cycles, fom pee Claie In the context of this fee-maket P2P esouce economy, this pape consides the poblem of optimal seve pee selection Specifically, we conside the following poblem A client pee wants to obtain an obect, such as a specific video file, diectly fom othe pees attached to the Intenet The client may obtain diffeent potions of the obect in paallel fom diffeent seve pees (as is cuently the case with KaZaA and othe file shaing systems) In this esouce economy, the seve pees chage client pees when they send files (o potions of files) to the client pees We suppose a fee maket economy, that is, seve pees may pice thei sevices as they wish Moe specifically, when a client pee wants to obtain a specific obect, the following steps ae taken: 1) The client fist uses a look-up sevice to discove seve pees that have a copy of the obect KaZaA is an example of such a lookup sevice; stuctued DHTs could also be used to ceate a lookup sevice 2) The client then queies the seve pees fo thei pices 3) The client may also use a eputation sevice to detemine the eliability of each of the seve pees (Reputation sevices ae beyond the scope of this pape; see [9], [10]) 4) Fom the subset of eputable seve pees offeing the obect, the client pee selects the seve pees fom which it will obtain the obect The client obtains diffeent potions of the obect fom each of the selected seve pees The client pee will natually choose the seve pees to minimize cost 5) Money is tansfeed fom the client pee to the seve

2 pees A potocol fo tansfeing money in a P2P esouce maket is descibed in [1] In this pape we exploe the optimal pee selection poblem fo two delivey schemes: (i) steaming, in which case the potions of the obect must aive in a timely manne, so that the client pee can ende the obect without glitches duing downloading; (ii) downloading, in which case the client wants to eceive the file as quickly and inexpensively as possible, but does not ende the file while downloading We will see that this two delivey schemes lead to diffeent fomulations of the poblem Fo both schemes, a content publishe may also be an active component of the system Fo example, CNNcom may contact with a lage numbe of pees to stoe chunks of video files When anothe pee Alice asks CNN to see a video, CNN may select the pees on Alice s behalf The selected pees would then eithe steam o download the video, depending on the delivey scheme II PRICING MODEL In this section we descibe ou picing model As mentioned in the intoduction, each seve is fee to set its own pices Conside a seve pee i As pat of a delivey session, the seve pee i will tansfe a potion of the bytes of some obect o to a client pee Fo such a delivey, the seve pee will fix an appopiate pice This pice should natually depend on: The obect itself: Fo example, fo video content, ecent videos may be moe expensive than olde videos The numbe of bytes tansfeed: The amount of bandwidth esouces consumed at the seve is popotional to the numbe of bytes tansfeed Rate of tansfe: The seve may want to chage moe fo tansfeing the bytes at a highe ate Let Ω i be the set of ates at which the seve pee can tansfe the obect potion; Ω i could consist of discete values o could be a continuous inteval of values Fo a paticula obect o, we conside picing functions of the fom pice = C i (b, x) whee b Ω i is the ate of the tansfe and x is the numbe of bytes tansfeed Note that if a client obtains x bytes at ate b bytes/sec, then the tansfe time is x/b seconds Also note that we ae taking the natual assumption that all bits have the same pice Befoe poceeding, let us examine moe caefully what it means fo a seve to be able to tansfe bytes at a specific ate b A seve i will have Intenet access with some upsteam ate u i At any given time, the seve pee i could be tansfeing files to multiple pees, with each file tansfe taking place at its negotiated ate In ode to meet its commitment, seve i, of couse, must ensue that the sum of all the ongoing tansfe ates does not exceed its upsteam access ate u i In today s Intenet (and in the foeseeable futue), the bottleneck is typically in the access and not in the Intenet coe In most boadband esidential connections today (including cable modem and ADSL), the upsteam ate is significantly less than the downsteam ate Thus, it is not uneasonable to assume that bandwidth bottleneck between seve and client is the seve s upload ate Thus, it is easonable to assume that a seve can povide an offeed ate b, as long as the sum of seve s committed ongoing ates is less than u i Thee will be situations, howeve, when the seve will not be able to hono its commitment due to unusual congestion o sevice failues in the coe In this case, the client pee may want some fom of a efund Futhemoe, eithe the seve o the client may be dishonest and claim that sevice was not eceived/endeed when indeed it was Thus, some fom of abitation - pefeably lightweight - may be needed in a P2P esouce maket [1] In Section 3, we will descibe a client stategy that allows any one of the the contacted pees to fail, eithe because of unfoeseen poblems beyond the seve s contol o because of dishonesty III OPTIMAL PEER SELECTION FOR DOWNLOADING In this pape we exploe the optimal pee selection poblem fo two delivey schemes: (i) steaming, in which case the potions of the obect must aive in a timely manne, so that the client pee can ende the obect without glitches duing downloading; (ii) downloading, in which case the client wants to eceive the file as quickly and inexpensively as possible, but does not ende the file while downloading In this section we conside the downloading poblem Fo the download poblem, we intoduce a simplification in ou picing model Specifically, we assume that Ω i = {b i }, that is, each seve i advetises a single download ate b i We also make the natual assumption that the seve s pice is popotional to the numbe of bytes tansfeed, that is, we conside picing functions of the fom pice = c i x i (1) Thus, unde this picing model, a seve i advetises a specific tansfe ate b i and a specific cost pe byte c i Natually, a client desiing a specific obect o would like to obtain the obect as quickly as possible and at least possible cost These two obectives will typically be conflicting, as seves that povide high tansfe ates will likely also demand high pe byte tansfe costs Thee ae many ways to fomulate an optimization poblem that takes into account these conflicting goals In this section, we conside one such natual fomulation: the client wants to select the pees in ode to minimize the total obect download time subect to a budget constaint fo a download We can now descibe the optimal pee selection poblem Conside a client pee that wants to download a file o Let F be the size of the file As descibed in the Intoduction, the client pee uses a location sevice to find a set of pees, denoted N = {1,, I}, that have a copy of the file Each seve pee i N offes the client pee a tansfe ate b i and cost pe byte c i We suppose that the client pee has a budget K fo this paticula download, that is, the client pee is pepaed to spend up to K units on this download Let x i be the numbe of bytes that the client tansfes fom seve pee i Because the client wants to obtain the entie file, we have x x I = F

3 Ou optimal pee selection poblem is to detemine x i, i N that minimizes the total download time subect to the budget constaint Because the client is downloading fom multiple seve pees in paallel, the total download time is the maximum download time fom each of the selected pees Because the download time fom pee i is x i /b i, the total download time is the maximum of x i /b i, i = 1,, I Thus, ou goal is to detemine optimum values of x i, i = 1,, I fo min max{ x 1, x 2, x I } (2) b 1 b 2 b I subect to c 1 x 1 + c 2 x c I x I K (3) x 1 + x x I = F (4) x i 0 i = {1,, I} (5) We now poceed to solve this optimization poblem Fist, we e-ode the seve pees so that 0 < c 1 < < c I (6) Note that all of the paametes b i, c i, F and K ae positive constants Wite the above optimization poblem as a linea pogam (LP): st min y (7) c i x i K, (8) i x i F, (9) i 0 x i b i y, i (10) The dual of the above is as follows, with the dual vaiables v and w coesponding, espectively, to the two constaints in (8) and (9), and z i coesponding to x i b i y in (10): min F w Kv (11) st w z i c i v 0, (12) b i z i 1, (13) i v 0, w 0, z i 0, i Below, we stat with deiving a dual feasible solution, which leads to a pimal feasible solution via complementay slackness Once these ae veified dual and pimal feasibility and complementay slackness, the poblem is completely solved To simplify notation, wite B := b i, β := i=1 b i c i ; (14) i=1 and wite B I and β I simply as B and β It is easy to veify that β /B is inceasing in, since β B β +1 B +1 iff β B c +1, and the last inequality follows fom (6) Letting the constaints in (12) and (13) be binding, we get: Then, the dual obective becomes z i = w c i v, (15) w = 1 + vβ B (16) F w Kv = F B + ( F β B K ) v Since we want to maximize the above, wt v, thee ae two cases: (i) If β B K F, then v = 0 β (ii) Othewise, conside, fo the time being, B > K F β I 1 Then v = v I := 1/(c I B β) Note in Case (ii), c I B > β follows fom (6); and fo all i, z i = 1 ( β ) B + B c i v I 0 follows fom v I 1 c i B β, i : c ib > β, since c n c i Also note that z n = 0 In case (i), the dual feasible solution esults in a dual obective value F/B The coesponding pimal feasible solution is: x i = F b i, i (17) B It is easy to veify that complementay slackness is satisfied: all pimal vaiables ae positive, and all dual constaints ae binding; all dual vaiables except v ae positive, and all pimal constaints except (8) ae binding Futhemoe, note that the pimal obective value is also F/B Fo case (ii), the dual feasible solution esults in a dual obective value as follows: F ( F β ) B + B K 1 c I B β = F c I K c I B β (18) Fo the coesponding pimal solution, conside the following: x i = b i y, i I; x I = F y ; (19) whee y is the pimal obective value, obtained via substituting the above solution into (8) and making the latte an equality: ( ) yβ I 1 + c I F y = K,

4 fom which we can obtain F c I K y = = F c I K c I β I 1 c I B β, (20) ie, the pimal obective value is equal to the dual obective value in (18) We still need to veify pimal feasibility and complementay slackness Fist note that y 0 follows fom (18): both tems on its LHS ae positive That x n 0 is equivalent to F F ( F β ) B + B K v I, which simplifies (with some algeba) to K F β I 1, the assumed condition in Case (ii) Othe aspects of pimal feasibility hold tivially Complementay slackness is eadily veified: all pimal vaiables ae positive, and all dual constaints ae binding; all dual vaiables except z n ae positive, and all pimal constaints except x I b I y, the I-th in (10), ae binding Next, in Case (ii), suppose K/F falls into the following ange: Then, the dual solution is: v = v I 1 := and The pimal solution is: β I 1 > K F β I 2 B I 2 1 (c I 1 β I 1 ), w = 1 + v I 1β I 1 ; z i = w c i v, i I 1; z I = 0 x = b i y, i I 2; x I 1 = F yb I 2, x I = 0; and y = F c I 1 K F c I 1 K = c I 1 B I 2 β I 2 c I 1 β I 1 Feasibility (pimal and dual) and complementay slackness can be veified as ealie To summaize, we have Poposition 1: The solution to the optimization poblem in (2) takes the following fom (with the costs odeed in (6) and the notation B, β, B, β defined in (14)) If K/F β/b, then x i = b i y fo all i, whee y = F/B Othewise, suppose fo some n, β B > K F β 1 B 1 (If β 1 /B 1 > K/F, ie, K < F c 1, then thee is no feasible solution to the optimization poblem in (2)) Then, whee x i = b i y, i 1; x = F yb 1 x +1 = = x n = 0; y = F c K c B β In both cases, y is the optimal obective value IV OPTIMAL SELECTION FOR STREAMING In this section we conside steaming encoded (compessed) audio o video The delivey constaints ae moe stingent than fo downloading: in ode to pevent glitches in playback, the seves must continuously delive segments of the obect on o befoe scheduled playout times An impotant paamete fo the steaming delivey is the obect s playback ate, denoted by Fo an obect of size F with playback ate, the viewing time is T = F/ seconds Suppose the use at the client begins to view the video at time 0 A fundamental constaint in the steaming poblem is that fo all times t with 0 t T, the client must eceive the fist t bytes of the obect We efe to this constaint as the noglitch constaint Thus, when selecting the seve pees and the obect potions to be obtained fom each seve pee, the client must ensue that this no-glitch constaint is satisfied A natual optimization poblem is, theefoe, to select the pees in ode to minimize the total steaming cost subect the no-glitch constaint Fo the steaming poblem, it is citical that the sevice continue even in one of the seve pees fails to povide its sevice Fo this eason, we conside the poblem of contacting an additional pee in such a manne that if any one seve pee fails, the client can continue to ende the obect without glitches In futue we will examine multiple pee failues In this section, we suppose that each seve pee advetises a set of ates Ω i, as descibed in Section I To simplify the analysis, we suppose that Ω i = [0, ] fo all i N Howeve, moe geneal ate sets can be handled as well Fo each advetised ate b Ω i, the seve advetises an associated cost ate of c i (b) Thus, seve i chages c i (b) t to tansfe bytes at ate b fo t seconds The client must not only select a subset of pee seves, but it must also detemine and schedule the specific potions of the file it is to download Thee ae two boad appoaches that can be taken to solving this poblem: time segmentation and ate segmentation In time segmentation, the video is patitioned along the time axis in distinct segments, and each seve is esponsible fo steaming only one of the segments in the patition This appoach equies client buffeing Futhemoe, in the optimal solution, the client will typically eceive the video fom all the selected seves at the beginning of the video and fom only one of the selected seves at the end of the video This means that the client must be able to download (at the beginning of the video) at a ate that is equal to the sum of the seve download ates, which will exceed the playback ate In ate segmentation appoach, each of the selected seves contibutes bytes fo each of the fames in the video, and at any instant of time the client downloads at the playback ate In this pape we conside only ate segmentation

5 A Poblem Fomulation Fo non-deceasing cost functions c i ( ), i = 1,, I, the ate segmentation poblem is min c 1(b 1) + + c I(b I) (21) st b, i = 1,, I i b i 0, i = 1,, I Note that the ate constaint ensues that client continues to eceive at ate at least even when one of the seves becomes unavailable The above poblem can be solved by fist solving the following poblem: fo any given y: 0 y, min c 1 (b 1 ) + + c I (b I ) (22) st b + y, 0 b i y, i = 1,, I Denote C(y) as the coesponding optimal value Then, solve the poblem min y C(y) To ustify that the poblem outlined in (22) is equivalent to the poblem outlined in (21), let Φ 1 be the set of all feasible solutions (b 1, b 2,, b I ) to (21) and Φ 2 (y) be the set of all feasible solutions (b 1, b 2,, b I ) to (22) fo paamete y Also let Φ 2 = Φ 2(y) 0 y It is staightfowad to show that Φ 1 = Φ 2 Thus optimizing c 1 (b 1 ) + + c I (b I ) ove Φ 1 is equivalent to optimizing the same function ove Φ 2 In paticula, an optimal solution fo (22) (optimized fo all 0 y ) is also optimal fo (21) B Convex Costs Suppose c i () is a convex function, fo all i = 1,, n Then, given y, (22) is geedily solvable via the following algoithm: Maginal Allocation: Stat with S := {1,, I} and b i = 0 fo all i S Each step identify i = ag min i S {c i(b i + ) c i (b i )}, whee > 0 is a pe-specified small incement (depending on equied pecision), and eset b i b i + Wheneve b i > y, eset S S {i} Continue until the constaint b + y is satisfied Note that the complexity of this algoithm is popotional to n( + y)/ To detemine the best y, we can do a line seach on C (y) = 0, fo y [ I 1, ] (If y < I 1, then (22) is infeasible) If C(y) is convex in y, then min y C(y) is itself geedily solvable: we can stat with y = n 1 ; incease y by a small incement each time, and solve the poblem in (22); stop when C(y) ceases to decease o y = is eached In this algoithm, when we go fom one y value to the next, say, y + δ, we do not have to do the maginal allocation that geneates C(y+δ) fom scatch (ie, stating fom all x values being zeo and S := {1,, I}) We can stat fom whee the pevious ound of maginal allocation the one that geneates C(y) fist hits a bounday, ie, b = y fo some, and continue fom thee O, if no b has hit the bounday in the pevious ound, then simply stat fom whee the pevious ound ends (ie, continue with the solution geneated by the pevious ound) [Recall, y [ I 1, ] As y inceases, the numbe of b values that can hit the bounday in the maginal allocation will decease Specifically, when y [ k 1, k 2 ], fo k = 3,, I, the numbe of b values that can hit the bounday cannot exceed k, since we have ky + y] The convexity of C(y), in tun, is guaanteed when c i ( ) s ae convex functions To see this, let (b i (y)) I i=1 denote the optimal solution to the poblem in (22), and conside two such poblems, coesponding to y = y 1 and y = y 2, espectively Fo any α (0, 1), we have αc(y 1 ) + (1 α)c(y 2 ) = α c (b (y 1 )) + (1 α) c (b (y 2 )) c (αb (y 1 ) + (1 α)b (y 2 )), whee the inequality follows fom the convexity of the c s Next, conside a thid vesion of (22), with y = αy 1 +(1 α)y 2 It is staightfowad to veify that αb (y 1 ) + (1 α)b (y 2 ), = 1,, I, is a feasible solution to this poblem Theefoe, we have and hence c (αb (y 1) + (1 α)b (y 2)) C(y), αc(y 1) + (1 α)c(y 2) C(y) = C(αy 1 + (1 α)y 2) That is, C(y) is a convex function To summaize, we have Poposition 2: Suppose fo each i = 1,, I, c i ( ) is a convex function Then, the optimal value in (22), C(y), is convex in y In this case, the steaming poblem in (21) is geedily solvable: each step incease y by a small incement I 1 (stating fom y = ), and apply the maginal allocation algoithm to geneate C(y); stop when C(y) ceases to decease o y = is eached V CONCLUSION We have fomulated and solved seveal pee selection poblems that aise in a fee-maket pee-esouce economy We have examined two delivey models, namely, downloading and steaming Fo both models, the selected seves tansfe the obect to the client in paallel Fo the download poblem, the total download time is the maximum download time fom each of the selected seves This gives ise to a min-max optimization poblem, fo which we obtain an explicit solution Fo the steaming poblem, we examine optimal ate patitioning fo which continuous playback is ensued even if one seve fails

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