Online Procurement Auctions for Resource Pooling in ClientAssisted Cloud Storage Systems


 Suzanna Ramsey
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1 Onlne Procurement Auctons for Resource Poolng n ClentAsssted Cloud Storage Systems Jan Zhao, Xaowen Chu, Ha Lu, YuWng Leung Department of Computer Scence Hong Kong Baptst Unversty Emal: {janzhao, chxw, hlu, Zongpeng L Department of Computer Scence Unversty of Calgary Emal: Abstract Latest developments n cloud computng technologes have enabled a plethora of cloud based data storage servces. Cloud storage servce provders are facng sgnfcant bandwdth cost as the user populaton scales. Such bandwdth cost can be substantally slashed by explorng a hybrd cloud storage archtecture that takes advantage of underutlzed storage and network resources at storage clents. A crtcal component n the new hybrd cloud storage archtecture s an economc mechansm that ncentvzes clents to contrbute ther local resources, whle at the same tme mnmzes the provder s cost for poolng those resources. Ths work studes onlne procurement aucton mechansms towards these goals. The onlne nature of the aucton s n lne wth asynchronous user request arrvals n practce. After carefully characterzng truthfulness condtons under the onlne procurement aucton paradgm, we prove that truthfulness can be guaranteed by a prcebased allocaton rule and payment rule. Our truthfulness characterzaton actually converts the mechansm desgn problem nto an onlne algorthm desgn problem, wth a margnal prcng functon for resources as varables set by cloud storage servce provders for onlne procurement aucton. We derve the margnal prcng functon for the onlne algorthm. We also prove the compettve rato of the socal cost of our algorthm aganst that of the offlne VCG mechansm and of the resource poolng cost of our algorthm aganst that of the offlne optmal aucton. Smulaton studes drven by realworld traces are conducted to show the effcacy of our onlne aucton mechansm. I. INTRODUCTION Cloud storage servce (e.g., Amazon S3 [1], Dropbox [], Google Drve [3], SkyDrve [4]) has ganed wdespread recognton and adopton by Internet users, many of whom now routnely execute onlne storage and onlne backup tasks over the cloud. Cloud storage provders are facng two natural challenges. Frst, the storage servce hghly depends on centralzed datacenters n the cloud. Accdents and natural dsasters (fre, earthquake, power outage) wll have sgnfcant mpact on the servce performance. Second, the cloud storage provders face sgnfcant cost n purchasng or rentng real estate, cloud nfrastructures, hardware, power and network bandwdth. To address these two challenges, the clentasssted (or userasssted) cloud storage paradgm s a promsng choce to guarantee hgh data avalablty and relablty whle lowerng the operatonal cost [5]. The clentasssted desgn s a complementary technology for cloud storage servce that are based solely on servers n datacenters. Recent examples of clentasssted cloud storage archtecture nclude FSYou [6], AmazngStore [7], Wuala [8] [9], Trton [1], and Symform [11], whch have examned the techncal feasblty of the paradgm of clentasssted cloud storage servces. However, ths lne of work mostly gnore the need of an ncentve mechansm for users to contrbute ther resources, and assume that unutlzed resources n storage clents can be used for clentasssted cloud storage archtectures by default, or use an unattractable one by provdng more cloud storage space for clents who contrbute more local storage. The realty s perhaps less optmstc, as autonomous and selfsh users do not have nnerncentve for contrbutng ther resources (e.g., casual mode n Wuala [8]). Experence from prvate BtTorrent communtes shows that a well desgned ncentve mechansm can motvate users to contrbute ther resources and sgnfcantly mprove the overall system capacty and performance [1] [13]. To help fully make clentasssted cloud storage a realty, t s mperatve to desgn approprate ncentve mechansms for users to engage. Onlne procurement aucton s a natural canddate that can act as the fnancal catalyst for resource poolng transactons. The procurement form of the aucton s due to the multseller (cloud users) onebuyer (storage servce provder) nature of the aucton, and the onlne property s n lne wth asynchronous arrvals of user bds and requests. The goals nclude not only optmzed resource utlzaton across the network, but also fundamental changes n the ecosystem of cloud storage servces leadng to a wnwn strategy for both sdes. Users can receve monetary rewards for contrbutng ther otherwse dlng resources, whle cloud storage provders can cut ther cost substantally. The resource poolng market s modeled as an onlne procurement aucton market: cloud storage users are economcally motvated to contrbute ther local resources to the storage pool. The storage pool constructed by cloud users can help to reduce the network traffc as well as storage burden at servers n datacenters. Cloud storage users are ncentvzed to submt sellng bds to the cloud storage provder, ndcatng the amount of resources t plans to contrbute, the tme wndow when they are avalable, and the desred remuneraton. Upon recevng a bd, the storage servce provder makes a decson on the amount of resources to be procured, based on a prcng functon. Wth the onlne procurement aucton model for resource poolng market, we face the followng two questons: 1) How should the mechansm guarantee the
2 truthfulness property n such an onlne aucton? ) How well does our mechansm perform when compared wth an offlne VckreyClarkeGroves (VCG) aucton or an offlne optmal aucton? Our general soluton framework s based on convertng the onlne procurement aucton desgn problem nto an onlne algorthm desgn problem. Ths converson s based on constrants that provde condtons characterzng the onlne auctons truthfulness. Hence, we frst characterze the truthfulness for an onlne aucton startng from Myerson s prncple of truthfulness. We defne the allocaton monotoncty for onlne procurement auctons, and prove the exstence of a crtcal payment scheme to guarantee truthfulness n two dmensons, resource avalablty and resource margnal cost under the correspondng monotone allocaton rule. For convenent applcatons n onlne aucton, we prove the equvalence of a prcebased allocaton and payment rule n guaranteeng truthfulness. Wth the prcebased allocaton rule and payment rule, we convert the onlne procurement aucton desgn problem nto an onlne algorthm desgn problem wth margnal prcng functons as varables. An ntegral equaton s derved to solve the margnal prcng functon for guaranteeng a target compettve rato of γ n terms of the socal cost of our algorthm aganst that of the offlne VCG mechansm, and n terms of the resource poolng cost of our algorthm aganst that of the offlne optmal aucton, and dscuss the relaton between the optmal target compettve rato γ and resource pool capacty, number of bds. We summarze our contrbutons as follows. We prove the equvalence of a prcebased allocaton rule and payment rule wth those derved from Myerson s classc prncple of truthfulness, for guaranteeng truthfulness n our onlne aucton. Wth the margnalprce based allocaton and payment rule, we convert an onlne procurement aucton desgn nto an onlne algorthm desgn problem. We derve an ntegral equaton of the margnal prcng functon that can guarantee our mechansm s performance wth a target compettve rato γ of the socal cost of our algorthm aganst that of the offlne VCG mechansm, and of the resource poolng cost of our algorthm aganst that of the offlne optmal aucton, and yeld the range for the optmal target compettve rato γ. The rest of the paper s organzed as follows. Sec. II presents related work. Sec. III descrbes the system model for clentasssted cloud storage systems and the onlne procurement aucton for resource poolng. Sec. IV characterzes truthfulness for our onlne auctons. Sec. V presents the onlne algorthm desgn problem converted from the onlne aucton problem, and proves the compettve rato acheved by our algorthm. Sec. VI s performance evaluaton of our algorthm under dynamc onlne storage clents extracted from realworld traces. Sec. VII concludes the paper. II. RELATED WORK Clentasssted cloud storage paradgm desgn motvates much research work [6] [7] [8] [1]. Sun et al. [6] are the frst to desgn, mplement, and deploy a peerasssted sempersstent onlne storage system wth clent storage and bandwdth assstance. They make use of unstructured PP overlay constructon, sequental block schedulng mechansm, and server strateges. The desgn objectve s to acheve a tradeoff between fle avalablty and server cost. Yang et al. [7] propose a peerasssted onlne storage archtecture, AmazngStore, for cloudbased nfrastructures. They organzed nodes usng DHT and gave the number of requred replcas for ganng the requred data avalablty. Hgh fracton of data access requests are served by peers so as to reduce the server cost and mprove the data avalablty. Mager et al. [8] conduct a measurement study of Wuala, a popular PPasssted onlne storage and sharng system. Toka et al. [14] study the mpact of data placement and bandwdth allocaton on the tme requred to complete a backup and restore operaton and the clents costs. Davol et al. [1] propose solutons for acceleraton on data sharng, mprovng the consstency, and reducng latency for agreement protocol preventng concurrent accesses on shared data. Ths group of work proposes PPasssted onlne storage system desgn or conducts measurement work on such desgns based on an assumpton that peers contrbute ther resources by default. Only Mager et al. [8] and Toka et al. [14] menton the ncentve problem of peers contrbutng local resources. There are two modes for peers n Wuala. One type of peers are casual peers who contrbute lttle storage and the other s storage peers who trade local storage for ncreased cloud storage space or reduced cloud storage cost. In ths paper, we propose an onlne procurement aucton for tradng peers local resource, whch serves as an ncentve mechansm for peers to contrbute ther local resources. Aucton mechansms have been studed by some researchers [15] [16] and been used n some scenaros such as PP streamng [17], WF prcng [18], spectrum auctons [19] [], cloud computng prcng [1] [] [3]. Lav et al. [15] present compettve analyss for truthful onlne auctons. Ther method s based on a threatbased approach proposed by Yanv et al. [4]. Our compettve analyss s based on dervng and solvng an ntegral equaton, whch s qute dfferent from thers. Hajaghay et al. [16] study the onlne auctons for reusable goods. They propose smlar truthfulness condtons for onlne auctons, but they do not form an onlne algorthm desgn framework for the onlne aucton mechansm desgn. Fredman et al. [18] consder extendng the standard results of offlne mechansm desgn to apply to mechansm desgn for onlne problems such as WF Prcng for users arrvng over tme. Deek et al. [] study the onlne aucton for spectrum allocaton. They propose a 3D bnpackng based allocaton and tmesmoothed crtcal value based prcng scheme, and evaluate ts performance through experments. Zhang et al. [1] and Sh et al. [] study the onlne aucton for cloud computng resource allocaton. Zhang et al. [1] propose the
3 3 condtons of payment to ensure the truthfulness. Our method propose the allocaton monotoncty, whch s the necessary and suffcent condton to guarantee the truthfulness. Sh et al. [] apply the threatbased approach for the compettve analyss. Our onlne procurement aucton desgn framework can be seen as an optmal onlne aucton desgn, whch expands Myerson s framework on optmal aucton desgn [5]. III. MODEL FORMULATION In a clentasssted cloud storage system, the storage servce provder ams to procure ts clents unused resources to complment server resources n ts own datacenters. Such a hybrd, dstrbuted storage archtecture helps acheve hgher storage cost effcency and robustness aganst sngle pont of falure. The cloud servce provder needs storage and network bandwdth from clents to deploy such clentasssted storage servce. Resource s assumed as a combnaton of storage and network bandwdth at a calculated rato for fle avalablty and accessblty. Let S be the resource capacty that the servce provder ams to procure durng tme [, T ]. In the onlne procurement aucton mechansm M, the cloud storage provder acts as the auctoneer. Storage clents on the Internet are bdders who may sell ther own resources to the auctoneer. Clents are dynamc n that they may arrve and depart at any tme. The set of bdders s unknown to the auctoneer a pror, and a bdder learns ts type at the tme of bddng. Consequently, the storage provder receves a stream of bds, B = (b 1, b,..., b j,...), from dynamc clents, and needs to make decsons on each bd upon ts submsson. The bd specfes the avalable resource the bdder can contrbute and the monetary remuneraton asked n return. More specfcally, the jth receved bd s a tuple b j = (â j, ˆd j, ˆQ j, ĉ j ), where â j s announced arrval tme of avalable resources, ˆdj s announced departure tme of avalable resources, ˆQj s announced avalable resource capacty, and ĉ j s announced margnal cost, whch s the cost for bdder to provde one more unt of resource for one tme unt. The type of the bdder submttng bd b j (referred to as bdder j) s denoted by v j = (a j, d j, Q j, c j ), whch s prvate nformaton known to bdder j only. Let B j denote all other bds except the jth bd. When the auctoneer receves bd b j, t determnes the resource capacty to procure, q j (t, b j, B j ), q j ˆQ j durng tme t [â j, ˆd j ], and the correspondng payment, p j (t, b j, B j ). We call (q j, p j ) the allocaton rule and payment rule of the onlne procurement aucton. We wll use q j (t, b j ), p j (t, b j ) nstead of q j (t, b j, B j ), p j (t, b j, B j ) when there s no confuson. Bdder j s total utlty s the total payment t receves mnus ts total cost to offer the procured resources, ntegrated over the tme wndow [â j, ˆd j ]: U j (q j, p j ) = ˆd j â j u j (t, q j, p j )dt, where u j(t, q j, p j) = p j(t, b j) c jq j(t, b j). (1) Eqn. (1) ensures ndvdual ratonalty of bdders. We are nterested n mechansms wth the truthfulness property. We frst examne the bdders strategy space. For a j, d j, Q j, bdders can not report an earler arrval tme â j < a j, a later departure tme ˆd j > d j, a capacty larger than what s avalable ˆQ j > Q j, as such false reports are easly detected. Hence, the bdders strategy space s â j a j, ˆdj d j, ˆQ j Q j, ĉ j c j. The defnton of truthfulness for onlne aucton mechansms s as follows. Defnton 1 (Truthfulness). An onlne aucton mechansm M s truthful, f for any bdder j, regardless of the type reports of other bdders,.e., past and future bds, declarng a bd that reveals ts true type can maxmze ts utlty. I.e., for a bdder wth type v j = (a j, d j, Q j, c j ), every bd b j = (â j, ˆd j, ˆQ j, ĉ j ) of the bdder satsfyng â j a j, ˆdj d j, ˆQj Q j, ĉ j c j, we have U(q j (t, v j ), p j (t, v j )) U(q j (t, b j ), p j (t, b j )). () The defnton of truthfulness for our model s twofold: one s to reveal resource avalablty,.e., a j, d j, Q j, truthfully; the other s to reveal resource cost,.e., c j, truthfully. Let B t be the set of stream bds receved by tme t, B t = {b j â j t}. Wth B t, we can defne the servce provder s completeness rato for any tme t s resource poolng after recevng streamng bds B t, R(t, t), R(t j B, t) = t q j(t, b j) [, 1], t t T. (3) S When the requred capacty S s not acheved by tme t through clentasssted resources at tme t,.e., R(t, t) < 1, t wll need to use the storage and bandwdth from servers n the datacenters to compensate. Let c s be the margnal resource cost from servers. The cost of the cloud provder for provdng a capacty of S resource under a bddng sequence B under mechansm M : (q j, p j ) s denoted by C M (B). The resource poolng cost s the total payment he pays for resource poolng plus the server cost for compensatng the remanng resource. C M(B) = j B ˆdj â j p j(t, b j)dt + c s S[1 R(t, t)]dt. (4) The onlne procurement aucton mechansm desgn problem can be modeled as an onlne optmzaton problem as follows, mn C M(B) s.t. (1)()(3) The objectve functon C M (B) can be rearranged as follows: C M(B) = [ j B t p j(t, b j) + c s S[1 R(t, t)]]dt And the constrants about the allocaton rule and payment rule are also decoupled at dfferent tme ponts. The optmzaton problem s equvalent to mnmzng the followng problem for any tme t [, T ], mn p j(t, b j) + c s S[1 R(t, t)] j B t s.t. (1)() R(t, t) [, 1]. (5) Optmzaton problem (5) can be seen as a subproblem of resource poolng for tme t [, T ]. Table I summarzes mportant notatons for ease of reference.
4 4 TABLE I: Important Notatons S resource pool capacty. T tme span of resource. W M (B) the total socal cost. B stream of bds. M the onlne procurement aucton mechansm. B j all other bds except the jth bd. b j the jth bd n the bd stream. v j true type of the jth bd. a j true arrval tme of bdder j s resource. d j true departure tme of bdder j s resource. Q j true avalable resource of bdder j. c j true unt resource cost for unt tme of bdder j. q j (t, b j ) resource capacty procured from bdder j. p j (t, b j ) payment to bdder j for tme t s resource. U j (q j, p j ) bdder j s utlty. B t streamng bds receved by tme t. R(t, t) completeness rato of tme t s resource poolng based on procurement by t. c s unt resource cost for unt tme of storage provder. C M (B) cloud storage provder s total cost. IV. TRUTHFULNESS CHARACTERIZATION FOR ONLINE AUCTIONS A. Revstng Myerson s Prncple of Truthfulness Myerson [5] formulated the classc characterzaton of offlne truthful mechansms n a general Bayesan settng, for auctons of a sngle ndvsble good. Lemma 1 (Myerson, 1981). Let P (b ) be the probablty of bdder wth bd b wnnng an aucton. A mechansm s truthful f and only f the followngs hold for a fxed b : P (b ) s monotoncally nondecreasng n b ; Bdder bddng b s charged b P (b ) b P (b)db Gopnathan et al. [6] ponted out the two nterpretatons of Myerson s prncple: () there exsts a mnmum bd b such that wll wn only f t bds at least b, and () the payment charged to for a fxed b s ndependent of b. We wll frst present the truthfulness characterzaton for our onlne procurement aucton by adaptng and extendng Myerson s prncple of truthfulness. In our onlne procurement aucton, to guarantee truthfulness s to guarantee that bdders bd ther true types n both resource avalablty and resource margnal cost,.e., b j = v j (â j = a j, ˆd j = d j, ˆQ j = Q j, ĉ j = c j ), regardless of past and future streams of other bds. Myerson s prncple of truthfulness frst ponts out the monotoncty crteron of allocaton rule P (b ) under a Bayesan game model for auctons of a sngle ndvsble good. Our onlne procurement aucton s an onlne reverse aucton for a dvsble good wth quantty S. Hence, we frst defne a correspondng allocaton monotoncty for our aucton, whch s necessary and suffcent for the exstence of a payment rule elctng truthful bds. Defnton (Allocaton Monotoncty). Consder two bds b j = (a j, d j, Q j, c j ) and b j = (â j, ˆd j, ˆQ j, ĉ j ) n the onlne procurement aucton. If â j a j, ˆd j d j, ˆQj Q j, ĉ j c j,.e., bd b j s resource avalablty s a subset of that of bd b j, and bd b j s resource margnal cost s no smaller than that of bd b j, we say bd b j domnates bd b j, denoted by b j b j. An allocaton rule q s monotone f ˆdj â j q j(t, b j)dt d j a j q j(t, b j)dt j, f b j b j. (6) Let us examne the dfference between the allocaton monotoncty crteron between Myerson s prncple and our defnton. Myerson consders a bayesan game model for auctons of one sngle ndvsble good. Hence, the allocaton probablty to one bdder when havng ts bd b, P (b ), s used to represent the allocaton rule. Ths allocaton probablty s used because we consder other bds satsfy a probablty dstrbuton and the probablty dstrbuton of other bds s a pror. In our onlne procurement aucton for dvsble good wth quantty S, we use the allocaton quantty to one bdder when havng ts bd b j, ˆd j â j q j (t, b j )dt, to represent the allocaton rule, gven other bds are fxed. When other bds vary accordng to a pror probablty dstrbuton, the expectaton of the allocaton quanttes s stll monotone wth b j. Ths s consstent wth P (b ) s monotoncty n a sngle ndvsble good s aucton. Gven allocaton monotoncty, we present the payment scheme to mplement a truthful onlne procurement aucton n Theorem 1. Theorem 1. There s a payment rule p such that the mechansm M : (q, p) s truthful f and only f the allocaton rule q s monotone. And the payment scheme can be expressed as follows, p j(t, b j) = ĉ j q j(t, b j) + ĉ j q j(t, (â j, ˆd j, ˆQ j, x))dx. (7) Theorem 1 can be seen as an extenson of Myerson s prncple of truthfulness to onlne procurement auctons of a dvsble good. We prove the truthfulness of resource avalablty and resource margnal cost under condtons n Theorem 1. Detals of ts proof are n Appendx IXA. B. A Prcebased Allocaton and Payment Rule Theorem 1 s characterzaton of truthfulness stands n the vewpont of the allocaton rule. Once the allocaton rule s determned, the payments are correspondngly determned. Ths characterzaton s convenent for offlne auctons where all bds are collected smultaneously and then the allocaton decsons for all related bds are made at the same tme. In such cases, the allocaton rule s obvous. However, ths does not apply to onlne auctons where bds arrve at dfferent tmes, wth allocaton decsons made at realtme. We wll present an equvalent condton to guarantee truthfulness of onlne procurement aucton mechansms, n whch a margnal prcng functon wll be ntroduced. The allocaton and payment are based on the prcng functon. Prcebased Allocaton and Payment: Let Ψ(t, R) be the margnal prcng functon of the cloud storage provder for procurng tme t s resources from clents when the completeness rato of tme t s resource poolng s R. It s easy to see that Ψ(t, R) s bdndependent.
5 5 In an onlne procurement aucton for S resource capacty, when the cloud storage provder receves a bd b j = (â j, ˆd j, ˆQ j, ĉ j ), t procures q j (t, b j ) resource from the bdder and pays p j (t, b j ) to the bdder. q j (t, b j ) and p j (t, b j ) are determned as follows, q j(t, b j) =, f ĉ j Ψ(t, R); mn{ ˆQ j, S [Ψ 1 (t, ĉ j) R(t, â j )]}, f ĉ j < Ψ(t, R). p j(t, b j) = R(t,âj ) (8) R(t,â j ) S Ψ(t, R)dR (9) Here, R(t, â j ) = q k Bâ k (t, b k )/S s the completeness j rato for tme t s resource poolng, just before recevng bd b j. We have Theorem for the truthfulness of prcebased onlne procurement auctons. Theorem. When a cloud storage provder determnes ts allocaton and payment accordng to Eqn. (8) and (9) when recevng a bd, where Ψ(t, R) s nonncreasng n completeness rato R, the mechansm M : (q, p) s truthful. Proof: We frst prove monotoncty of the allocaton rule. Gven two bds b = (, ˆd, ˆQ, ĉ ), b = (a, d, Q, c ), b b from a bdder, we wll prove q (t, b ) q (t, b ). Let us examne the allocaton under a new bd b = (a, d, ˆQ, ĉ ). It s easy to verfy that b b b. As the resource avalablty of b domnates b, we have a. As bd b arrves later, the completeness rato of tme t s resource poolng before recevng bd b s smaller than or equal to that before recevng bd b,.e., S R(t, â ) = k Bâ q k (t, b k ) k B a q k (t, b k ) = S R(t, a ) The avalable resource and resource margnal cost s the same for b and b. Accordng to Eqn. 8, we have q (t, b ) q (t, b ). As for bd b and b, the resource arrval tmes are the same for the two bds. Hence, the completeness rato of tme t s resource poolng when recevng bd b or b are the same. As Ψ(t, R) s nonncreasng wth R, we can verfy that ts nverse functon Ψ 1 (t, c) s also nonncreasng wth c. Because ĉ c, we have Ψ 1 (t, ĉ ) Ψ 1 (t, c ) We also have ˆQ Q, ths results n q (t, b ) q (t, b ) accordng to Eqn. (8). In concluson, we proved q (t, b ) q (t, b ). The allocaton rule based on the margnal prcng functon Ψ(t, R)s monotone. Wth the monotone allocaton rule, based on Theorem 1, we know that the payment rule n Eqn. (7) truthfully mplements the allocaton rule. The prcebased payment Eqn. (9) can be derved by rearrangng Eqn. (7) based on the prcebased allocaton Eqn. (8). V. A COMPETITIVE ONLINE PROCUREMENT ALGORITHM The prcebased allocaton rule and payment rule work n concert to not only guarantee the truthfulness of an onlne aucton mechansm, but also help us convert an onlne aucton desgn problem nto an onlne algorthm desgn problem. The onlne algorthm desgn problem has the margnal prcng functon as ts varable. Usng an offlne VCG aucton and an offlne optmal aucton as benchmarks for socal cost and cloud storage provder s poolng cost respectvely, we derve an ntegral equaton for guaranteeng a target compettve rato γ, and further derve the margnal prcng functon from the ntegral equaton. A. Onlne Algorthm Desgn Problem Converted from Onlne Aucton Desgn By substtutng the prcebased payment rule (9) nto the optmzaton problem (5), we obtan the followng onlne algorthm desgn problem, mn s.t. j B t R(t,âj ) R(t,â j ) Ψ(t, R)S dr + c s S[1 R(t, t)] Ψ(t, R) s a nonncreasng functon Ψ(t, 1) c mn (1) Condton () of optmzaton problem (5),.e., property of truthfulness, can be guaranteed based on the prcebased allocaton and payment rule. Ψ(t, 1) c mn guarantees condton R(t, t) [, 1] of (5). Condton (1) s satsfed as the prcebased allocaton rule and payment rule only procure resource from clents wth resource margnal cost for tme t smaller than Ψ(t, R), and pay clents prces that are hgher than the resource margnal cost. Hence, for any bdder j, U j (q j, p j ). We summarze our algorthm framework for the onlne procurement aucton n Algorthm 1. B. Margnal Prcng Functon Dervaton and Compettve Rato Analyss To mplement Algorthm 1, we need to gve the margnal prcng functon. We wll see the prcng functon wll affect the compettve rato of our onlne procurement aucton. In our onlne procurement aucton desgn, we am at mnmzng the cloud provder s resource poolng cost, whch s the summaton of the payment of the cloud storage provder for procurng resource from clents and the cost of provdng complmentary resource from servers. Wth the socal cost of offlne VCG aucton as a lower bound of the resource poolng cost of the offlne optmal aucton, we analyze the compettve rato of the resource poolng cost aganst that of the offlne optmal aucton. Besdes the resource poolng cost, another mportant metrc s the socal cost,.e., the total cost for clents and servers to provde S resource. We benchmark our onlne algorthm aganst the offlne VCG aucton that acheves the mnmum socal cost.
6 6 Algorthm 1 Cloud Storage Provder s Algorthm for the Onlne Procurement Aucton Input: Margnal prcng functon Ψ(t, R), S, T. Output: q j (t, b j ), p j (t, b j ) 1: Intalze the completeness rato of tme t [, T ] s resource poolng R t = : whle Receve a bd b j do 3: for t [â j, ˆd j ] do 4: f R t = 1 then 5: Contnue 6: end f 7: f ĉ j < Ψ(t, R t ) then 8: f Ψ 1 (t, ĉ j ) > 1 then 9: Set Ψ 1 (t, ĉ j ) = 1 1: end f 11: Procure q j (t, b j ) resource from bdder j accordng to Eqn.(8) 1: Pay p j (t, b j ) to bdder j accordng to Eqn.(9) 13: Update R t = R t + qj(t,bj) S 14: end f 15: end for 16: end whle The socal cost for provdng tme t s S resource under our onlne algorthm s W M(t, B) = j B t c jq j(t, b j) + c s S[1 R(t, t)]. (11) The total socal cost s the ntegraton of W M (t, B) among tme [, T ],.e., W M (B) = W M(t, B)dt. Theorem 3. When the cloud storage provder uses the margnal prcng functon as Ψ(t, R) = c s c s (1 1 γ )e R γ, where R s the completeness rato of procured resource for tme t, our prcebased onlne procurement aucton wll acheve the resource poolng cost no larger than γ tmes that of an offlne optmal aucton,.e., C M (B) γ C opt (B) and acheve the socal cost no larger than γ tmes that of the offlne VCG aucton,.e., W M (B) γ W vcg (B). The optmal γ should be among the range of [γ,mn, γ,max ]. Here γ,mn and γ,max are the solutons to equatons (1 1 γ ) e 1 γ = 1 cmax c s, (1 1 γ ) e 1 γ = 1 c mn c s respectvely. C opt (B), W vcg (B) s the cloud storage provder s resource poolng cost under offlne optmal aucton and the socal cost under offlne VCG aucton respectvely. Proof: We frst check the socal cost under our onlne procurement aucton and offlne VCG aucton. Gven some bddng sequence B, let q j (t, b j ) be the quantty procured from bdder j for poolng tme t s resource, and Ψ(t, R(t, â j )) the lowest procurement prce for poolng tme t s unt resource of unt tme from bdder j. Let l be the last bdder wth resource procured, the correspondng lowest procurement prce for poolng tme t s resource from bdder l s Ψ(t, R(t, â l )). Let R(t, â l ) be the fnal completeness rato for tme t s resource procurement after procurng bdder l s resource. The poolng cost for tme t s resource s C M(t, B) = R(t,âl ) Ψ(t, R)S dr + c s [S S R(t, â l )] When the cloud provder procures resource, t pays more than the bdders cost. Hence, the socal cost for provdng S resource wll be no larger than the cloud provder s resource poolng cost,.e., W M (t, B) C M (t, B). An offlne aucton mechansm can collect all bds B = (b 1, b,..., b j,...), then make allocaton and payment decsons. As n our procurement aucton, servers resource can be utlzed as a reserved backup resource. Correspondngly, we add a new bd b wth announced cost c s and unbounded resource capacty. For procurng tme t s resource, let (b (vcg) 1, b (vcg), b (vcg) 3,..., b (vcg) x ) be the ascendng sequence of procured bds accordng to the announced cost under offlne VCG aucton. The prce pad to bdders s no smaller than the announced cost of bd b (vcg) x. As the offlne VCG aucton procure the total S capacty, ths means the total resource capacty of the bds procured n offlne VCG aucton s no smaller than the procured resource capacty from clents n our onlne procurement aucton. Hence, the announced cost of b vcg x s larger than or equal to Ψ(t, R(t, â l )). Otherwse, all bds n (b (vcg) 1, b (vcg), b (vcg) 3,..., b (vcg) x ) have announced cost smaller than Ψ(t, R(t, â l )). our onlne algorthm wll procure more than S resource capacty. Hence, the socal cost under offlne VCG aucton for provdng tme t s resource s W vcg (t, B) S c (vcg) x cost of bd b (vcg) S Ψ(t, R(t, â l )). Here c (vcg) x x. s the announced Wth the socal cost for provdng tme t s resource under our onlne procurement aucton and offlne VCG aucton, the compettve rato n terms of the total socal cost can be calculated as follows, W M(B) W = WM(t, B) vcg(b) T Wvcg(t, B) [ R(t,â l ) Ψ(t, R)S dr + c s (S S R(t, â l ))]dt [S Ψ(t, R(t, â l))]dt To guarantee a compettve rato of γ, we can let R(t,âl ) Ψ(t, R)S dr + c s (S S R(t, â l )) = γ (1) S Ψ(t, R(t, â l )) By solvng the ntegral equaton (1), we can derve the expresson of Ψ(t, R) as: Ψ(t, R) = c s c s(1 1 γ ) e R γ Let us consder the bound condton of the margnal prcng functon: () If Ψ(t, 1) < c mn, ths means Ψ(t, R) wll becomes equal to c mn at a pont R < 1, and the cloud storage provder wll stop procurng resources from bdders as the completeness rato does not reach 1. Ths may waste the chances to procure resource wth lower announced cost. use lower cost than server cost to buy comng bdders resources. Hence, ths s not optmal for settng target compettve rato, γ.
7 7 () If c max > Ψ(t, 1) > c mn, ths means the margnal prcng functon wll not be contnuous at R = 1. The cloud storage provder needs to set the prce as c mn when R = 1. But the useful scenaros for settng c max > Ψ(t, 1 ) > c mn s when the number of bdders s small enough or target resource pool capacty S s large enough. In such scenaros, R can not reach 1 due to scare resource from storage clents. Hence, the optmal choce s to accept as many bds as possble. The optmal bound condton s settng Ψ(t, 1) = c max. (1 1 γ ) e 1 γ = 1 c max c s. We can get the optmal target compettve 1 rato s γ,mn =, here, W W ( (x) s a Lambert cmax/cs 1 )+1 e W functon and W (x) 1. () If the number of bds s large enough or target resource pool capacty S s small enough, R can easly reach 1, the optmal soluton s settng Ψ(t, 1) = c mn. The optmal target 1 compettve rato γ,max =. W ( c mn /cs 1 e )+1 Summarzng the above analyss, we can see the optmal target compettve rato s n the range γ [γ,mn, γ,max ]. Next, let us see the resource poolng cost under our onlne procurement aucton and offlne optmal aucton. As the cloud storage provder pays more than bdders announced cost for procurng resource from clents, the resource poolng cost under the offlne optmal aucton should be larger than or equal to the socal cost under the offlne VCG aucton,.e., we have C opt (t, B) W vcg (t, B) S Ψ(t, R(t, â l )). Hence, the compettve rato of our onlne procurement aucton, n terms of resource poolng cost, s: C M(B) C = CM(t, B) opt(b) T Copt(t, B) [ R(t,â l ) Ψ(t, R)S dr + c s (S S R(t, â l ))]dt [S Ψ(t, R(t, â l))]dt whch equals the compettve rato γ of socal cost. VI. PERFORMANCE EVALUATIONS In ths secton, we study the socal cost and resource poolng cost acheved by our onlne procurement algorthm n the context of realworld traces. A. Experment Settngs The clents arrval/departure tme and avalable bandwdth are extracted from realworld traces [7]. We use the sesson 9 trace for clent level data collected from transamrt.net wth 5 mnute samplng ntervals, and plot the onlne statstcal nformaton n Fg. 1. By default, the cost of clents unt resource for a unt tme s generated randomly n the range of [.1,.5]$/GB per hour. The cloud storage provder s server cost for provdng resource s.1$/gb per hour. The total resource poolng capacty of the cloud storage provder s 5KB/s. We consder a tme range of 7 days. The number of bds s 7. We compare the socal cost acheved by our onlne procurement wth that of the offlne VCG aucton. We compare the resource poolng cost by our onlne procurement wth that of optmal aucton ndrectly. As the resource poolng cost of offlne optmal aucton should be larger than the socal cost acheved by offlne VCG aucton, we use the socal cost acheved by offlne VCG aucton as a lower bound for t. Hence, n our experments, the socal cost and resource poolng cost of our onlne procurement aucton are both compared wth the socal cost of the offlne VCG aucton. B. Performance of Our Onlne Procurement Mechansm Fg. compares the resource poolng cost and socal cost of our onlne procurement aucton mechansm wth offlne auctons. We make comparsons under dfferent resource poolng capacty, number of bds, and rato of server cost and average bddng cost. We use the target compettve rato γ,max n the margnal prcng functon. Fg. a vares the resource capacty among [1, 1]KB/s. Fg. b vares the number of bds among [1, 7]. Fg. c vares the rato between margnal server cost c s and average bddng cost among [, 6]. Our onlne procurement aucton acheves the socal cost and resource poolng cost wthn the target compettve rato. In most cases, our onlne procurement aucton s resource poolng cost s lower than that of the offlne VCG aucton. C. How Does Performance Change wth γ? Fg. 3 & Fg. 4 show the change of real cost rato wth the target compettve rato. The black star lnes n Fg. 3a & Fg. 3b are the poston of γ,mn. In Fg. 3a, S = 5KB/s, whch means clents resource s enough, we can see as the target compettve rato equals γ,max, our onlne procurement aucton acheves the optmal compettve rato. In Fg. 3b, S = 8KB/s, whch means clents resource s not enough, we can see the real cost rato s not senstve to the target compettve rato. These two fgures verfy the optmal target compettve rato s among [γ,mn, γ,max ]. Fg. 3c, Fg. 3d and Fg. 4 show the real rato for resource poolng (upper one) and the real rato for socal cost (lower one) under dfferent target compettve rato and changng resource pool capacty, number of bds, and rato of server cost and average bddng cost, rato of the maxmum and mnmum bddng cost. We can see when our algorthm acheves the best performance, the target compettve rato s no larger than γ,max. As the target compettve rato becomes small enough, the real cost rato may even become hgher, whch means the target compettve rato should not be too small. As S s large enough, or number of bds s small enough, the real performance s nsenstve to γ. VII. CONCLUSIONS Clentasssted cloud storage s emergng as a new, exctng hybrd archtecture for onlne storage systems that represent a potental wnwn soluton for both cloud storage provders and cloud users. Whle recent research along ths thread have focused on techncal challenges, ths work represents the frst effort that examnes the economc sde of the pcture. We desgn an onlne procurement aucton for resource poolng by the provder, whch serves as a fnancal catalyst for brngng
8 8 No. of Peers No. of Peers No. of Peers (a) Arrval Tme Dstrbuton (days) (b) Onlne Tme Dstrbuton (hours) (c) Resource Capacty Dstrbuton (MB/s) Fg. 1: Statstcal Informaton of Clent Onlne Behavor Cost($) W vcg C vcg γ W vcg W onlne C onlne S (KB/s) (a) Cost($) W vcg C vcg γ W vcg W onlne C onlne No. of Bds (b) Cost($) W vcg C vcg γ W vcg W onlne C onlne Server Cost/Average Bddng Cost Fg. : Performance of our onlne procurement aucton under varyng resource poolng capacty S, No. of bds, and rato among server cost and average bddng cost (c) 8 6 Real Rato for Socal Cost Real Rato for Pool Cost Target Compettve Rato 8 6 Real Rato for Socal Cost Real Rato for Pool Cost Target Compettve Rato Cost Rato 4 Cost Rato γ/γ,max (a) Real Cost Rato, S=5KB/s γ/γ,max (b) Real Cost Rato, S=8KB/s (c) Real Cost Rato under Varyng(d) Real Cost Rato under Varyng Poolng Capacty No. of Bds Fg. 3: Real cost rato wth varyng γ clentasssted cloud storage nto realty. The aucton proposed s truthful and guarantees a provable compettve rato aganst optmal offlne algorthms. VIII. ACKNOWLEDGEMENTS Ths work s supported by Hong Kong GRF grant HKBU 141. We thank the revewers for ther valuable comments. REFERENCES [1] Amazon S3, [] Dropbox, https://www.dropbox.com. [3] Google Drve, https://drve.google.com. [4] OneDrve, https://onedrve.lve.com. [5] X. Chu, H. Lu, Y.W. Leung, Z. L, and M. Le, UserAsssted Cloud Storage System: Opportuntes and Challenges, IEEE COMSOC MMTC ELetter, January 13. [6] Y. Sun, F. Lu, B. L, B. L, and X. Zhang, FSYou: PeerAsssted SemPersstent Onlne Storage at a Large Scale, n Proc. of INFOCOM, 9. [7] Z. Yang, B. Zhao, Y. Xng, S. Dng, F. Xao, and Y. Da, AmazngStore: Avalable, Lowcost Onlne Storage Servce Usng Cloudlets, n Proc. of IPTPS, 1. [8] T. Mager, E. Bersack, and P. Mchard, A Measurement Study of the Wuala Onlne Storage Servce, n Proc. of PP Computng, 1. [9] Wuala, https://www.wuala.com/. [1] A.Davol and A.Me, Trton: a PeerAsssted Cloud Storage System, n Proc. of PaPEC, 14. [11] Symform, [1] X. Chen, X. Chu, and Y. Jang, Measurements, Analyss and Modelng of Prvate Tracker, n Proc. of PP Computng, August 1. [13] X. Chu, X. Chen, A. L. JIa, J. A. Pouwelse, and D. H. J. Epema, Dssectng Darknets: Measurement and Performance Analyss, ACM Trans. Internet Technol., vol. 13, no. 3, pp. 7:1 7:5, May 14. [14] L.Toka, M.Amco, and P.Mchard, Onlne Data Backup: a Peer
9 9 (a) Real Cost Rato vs. Varyng Rato among Server Cost and Average Bddng Cost (b) Real Cost Rato vs. Varyng Rato among Maxmum Bddng Cost and Mnmum Bddng Cost Fg. 4: Real cost rato under varyng γ and rato among server cost and bddng cost Asssted Approach, n Proc. of PP Computng, 1. [15] R. Lav and N. Nsan, Compettve Analyss of Incentve Compatble OnLne Auctons, n Proc. of ACM EC,. [16] M.Hajaghay, R. Klenberg, M.Mahdan, and D. Parkes, Onlne Auctons wth Reusable Goods, n Proc. of ACM EC, 5. [17] X. Chu, K. Zhao, Z. L, and A. Mahant, AuctonBased OnDemand PP MnCost Meda Streamng wth Network Codng, IEEE Trans. Parallel Dstrb. Syst., vol., no. 1, pp , Dec. 9. [18] E. Fredman and D. Parkes, Prcng WF at Starbucks Issues n Onlne Mechansm Desgn, n Proc. of ACM EC, 3. [19] X.Y. L, P. Xu, S. Tang, and X. Chu, Spectrum Bddng n Wreless Networks and Related, n Proc. of COCOON, 8, pp [] L. Deek, X. Zhou, K. Almeroth, and H. Zheng, To Preempt or Not: Tacklng Bd and Tmebased Cheatng n Onlne Spectrum Auctons, n Proc. of INFOCOM, 11. [1] H. Zhang, B. L, H. Jang, F. Lu, A. Vaslakos, and J. Lu, A Framework for Truthful Onlne Auctons n Cloud Computng wth Heterogeneous User Demands, n Proc. of INFOCOM, 13. [] W. Sh, C. Wu, and Z. L, RSMOA: A Revenue and Socal Welfare Maxmzng Onlne Aucton for Dynamc Cloud Resource Provsonng, n Proc. of IWQOS, 14. [3] H. Fu, Z. L, C. Wu, and X. Chu, CoreSelectng Auctons for Dynamcally Allocatng Heterogeneous VMs n Cloud Computng, n Proc. of IEEE CLOUD, JuneJuly 14. [4] R.E.Yanv, A.Fat, R.Karp, and G.Turpn, Compettve Analyss of Fnancal Games, n Proc. of FOCS, 199. [5] R.B.Myerson, Optmal Aucton Desgn, Mathematcs of Operatons Research, vol. 6, pp , [6] A. Gopnathan, Z. L, and C. Wu, Strategyproof Auctons for Balancng Socal Welfare and Farness n Secondary Spectrum Markets, n Proc. of INFOCOM, 11. [7] B. Zhang, A. Iosup, and D. Epema, The PeertoPeer Trace Archve: Desgn and Comparatve Trace Analyss, TU Delft, Tech. Rep., Aprl 1. A. Proof of Theorem 1 IX. APPENDIX Proof: (a) We frst prove the f part. Let q be a monotone allocaton rule and consder a bd b j = (a j, d j, Q j, c j ). We show that the allocaton rule q n combnaton wth the payment rule p n Eqn. 7 consttute a truthful mechansm. We prove the truthfulness for resource avalablty and resource cost respectvely. () We frst prove the truthfulness for resource cost revelaton,.e., bdders bddng ther true cost wll maxmze ther utlty. We prove t by contradcton. If the mechansm s not truthful, there s a bdder, wth a true type v = {a, d, Q, c }, and a nontruthful bd b = (, ˆd, ˆQ, ĉ ), b v such that the utlty U[q (t, b ), p (t, b )] of bdder f t bds b s strctly greater than the utlty U[q (t, v ), p (t, v )], whch can be wrtten as follows, [ ] (ĉ c )q (t, b ) + q (t, (, ˆd, ˆQ, x))dx dt > a c ĉ q (t, (a, d, Q, x))dxdt As the allocaton rule s monotone, and a, ˆd d, ˆQ Q, we have q (t, (a, d, Q, x))dxdt a c c q (t, (, ˆd, ˆQ, x))dxdt Wth the above two nequaltes, we get the followng relatonshp, ĉ (ĉ c )q (t, b )dt > q (t, (, ˆd, ˆQ, x))dxdt Hence, f ĉ > c, by dvdng both sdes by ĉ c we get ˆd q (t, b )dt s strctly larger than the average of q (t, (, ˆd, ˆQ, x))dt over x [c, ĉ ], whch contradcts the monotoncty of q. If ĉ < c, by dvdng both sdes by ĉ c, we get ˆd q (t, b )dt s strctly less than the average of ˆd q (t, (, ˆd, ˆQ, x))dt over x [ĉ, c ], whch agan contradcts the monotoncty of q. Ths contradcton proves the truthfulness of the mechansm M : (q, p) for resource cost. () Secondly, we prove the truthfulness for resource avalablty. As we have prove the truthfulness for resource cost,.e., for any bd b = (, ˆd, ˆQ, ĉ ), the bd b = (, ˆd, ˆQ, c ) wth the same resource avalablty and the true cost value wll acheve a larger utlty. Next we wll prove the utlty under bd v wll be larger than that under bd b It s obvous that. U[q (t, v ), p (t, v )] = c a c q (t, (a, d, Q, x))dxdt q (t, (, ˆd, ˆQ, x))dxdt = U[q (t, b ), p (t, b )] c Hence, we prove the truthfulness for resource avalablty. (b) We then prove the only f part. We have a truthful mechansm M : (q, p). Consder a bdder and ts two possble types v = (a, d, Q, c ), v = (a, d, Q, c ). v v, and B are the same. If the scenaro s that v s bdder s true type, accordng to the truthfulness, we have a [p (t, v ) c q (t, v )]dt d a [p (t, v ) c q (t, v )]dt If the scenaro s that v s bdder s true type, we have d a [p (t, v ) c q (t, v )]dt a [p (t, v ) c q (t, v )]dt By addng the two nequaltes above and usng c > c, we have d a q (t, v )dt > d q a (t, v )dt. Therefore q s monotone.
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