Onlne Auctons n IaaS Clouds: Welfare and roft Maxmzaton wth Server Costs aox Zhang Dept. of Computer Scence The Unvety of Hong Kong xxzhang@cs.hku.hk Zongpeng L Dept. of Computer Scence Unvety of Calgary zongpeng@ucalgary.ca Zhy Huang Dept. of Computer Scence The Unvety of Hong Kong zhy@cs.hku.hk Francs C.M. Lau Dept. of Computer Scence The Unversty of Hong Kong fcmlau@cs.hku.hk Chuan Wu Dept. of Computer Scence The Unvety of Hong Kong cwu@cs.hku.hk ABSTRACT Aucton desgn has recently been studed for dynamc resource bundlng and VM provsonng n IaaS clouds, but s mostly restrcted to the one-shot or offlne settng. Ths work targets a more realstc case of onlne VM aucton desgn, where: () cloud use bd for resources nto the future to assemble customzed VMs wth desred occupaton duratons; () the cloud provder dynamcally packs multple types of resources on heterogeneous physcal machnes (serve) nto the requested VMs; () the operatonal costs of serve are consdered n resource allocaton; (v) both socal welfare and the cloud provder s net proft are to be maxmzed over the system runnng span. We desgn truthful, polynomal tme auctons to acheve socal welfare maxmzaton and/or the provder s proft maxmzaton wth good compettve ratos. Our mechansms consst of two man modules: () an onlne prmal-dual optmzaton framework for VM allocaton to maxmze the socal welfare wth server costs, and for revealng the payments through the dual varables to guarantee truthfulness; and () a randomzed reducton algorthm to convert the socal welfare maxmzng auctons to ones that provde a maxmal expected proft for the provder, wth compettve ratos comparable to those for socal welfare. We adopt a new applcaton of Fenchel dualty n our prmal-dual framework, whch provdes rcher structures for convex programs than the commonly used Lagrangan dualty, and our optmzaton framework s general and expressve enough to handle varous convex server cost functons. The effcacy of the onlne auctons s valdated through careful theoretcal analyss and trace-drven smulaton studes. Categores and Subject Descrpto C.4 [erformance of Systems]: Desgn studes; Modelng technques; I.. [Algorthms]: Analyss of algorthms General Terms Algorthms; Desgn; Economcs ermsson to make dgtal or hard copes of all or part of ths work for peonal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the ft page. Copyrghts for components of ths work owned by othe than the author(s) must be honored. Abstractng wth credt s permtted. To copy otherwse, or republsh, to post on serve or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Request permssons from ermssons@acm.org. SIGMETRICS 5, June 5-9, 05, ortland, OR, USA 05 ACM. ISBN 978--4503-3486-0/5/06...$5.00 DOI: http://dx.do.org/0.45/745844.745855. Keywords Cloud Computng; Aucton; Resource Allocaton; rcng; Onlne Algorthms; Truthful Mechansms. INTRODUCTION As a major model n cloud computng servces, Infrastructureas-a-Servce (IaaS) clouds are prolferatng n today s Internet. An IaaS cloud meets use realtme resource demands through vrtualzaton technologes, whch pack resources (e.g., CU, RAM, dsk) nto vrtual machnes (VMs). Major IaaS provde today typcally offer pre-confgured VM nstances of fxed types, wth the number of types ncreasng over the yea. For example, Amazon EC currently provdes 7 categores and 3 types of VMs [], a substantal growth from a few yea ago. A few recently emerged cloud platforms start to allow customzed VMs that bundle varous resources at user-specfed amounts [][3]. Although the granularty of resource provsonng keeps mprovng, fxed prcng polces are stll domnant n practce, chargng custome a fxed amount for each pre-confgured VM [] or each unt of resources [][3]. Despte ther apparent smplcty, fxed-prce polces nherently lack market aglty and effcency, jeopardzng both the provder s proft and custome utlty. Amazon EC s Spot Instance s a ft-step attempt to apply a market-based aucton mechansm on VM provsonng, but ts prces have been dscovered to be often not marketdrven [4], whch can lead to untruthful bddng [5]. Towards better market-based prcng, auctons have recently been desgned for cloud resource allocaton, for pre-confgured VMs of lmted types [6][7][8], or for customzed VMs wth user-specfed resource bundles [9][0][]. Most of the mechansms have focused on the one-shot or offlne settng, solvng for resource allocaton and payments assumng that the bds are gven all at once. Onlne VM auctons, where bdde come and go at wsh and allocatons and chargng decsons are to be made on the spot, have only been nvestgated n very lmted setups. Wang et al. [8] and Zhang et al. [] consder only one type of VMs avalable to all use n ther onlne auctons. Sh et al. [9] study auctons over multple rounds, whch are coupled together by an overall budget at each user, whle each acqured VM s stll used for a sngle round. The socal welfare and revenue optmzaton n [] s acheved based on a strong assumpton of allowng preempton of resources already occuped by a user, whch s arguable at best n practce. Ths work targets a more realstc and general setup n onlne VM aucton desgn wth the followng features. (I) User-specfed VM (future) start/end tme and tme-varyng resource bundle: Each
cloud user bds for a customzed bundle of resources to consttute her VM, whch can start executon at any future tme for any specfed duraton. Besdes, the resource composton of the VM can vary over ts duraton, accordng to the projected need of user workload. (II) Heterogeneous serve wth varous resource capactes and operatonal costs: The cloud provder dynamcally packs multple types of resources on heterogeneous physcal machnes (serve) nto the requested VMs. Varous server cost functons under dfferent server operatonal models are consdered n resource allocaton, whch have not been modeled n prevous cloud auctons. (III) Socal welfare maxmzaton as well as proft maxmzaton: We desgn auctons that maxmze the socal welfare on aggregate gan of the cloud provder and the use (system effcency), and auctons that maxmze the cloud provder s net proft (another realstc objectve), whle guaranteeng other desrable propertes ncludng truthful bddng, ndvdual ratonalty and computatonally effcency over the entre system runnng span. The desgn of onlne allocaton algorthms n our setup s ndeed challengng, when one ams to puue socal welfare or proft that closely approaches that n the optmal offlne soluton, computed usng complete nformaton n the system span. Even n the offlne settng, packng multple types of resources on heterogeneous serve nto customzed VMs of tme-varyng resource composton and dfferent duratons nvolves N-hard combnatoral optmzaton problems. In the onlne algorthm, the decson on packng or reservng resources (f the VM s to start at a future tme) for the requested VM should be made upon recept of each bd, wthout the assstance of any future nformaton. What s more, even when an onlne approxmate allocaton algorthm s n place, t can be dffcult to desgn a payment rule that works wth the allocaton algorthm to guarantee desred propertes such as truthfulness [3]. The classc VCG mechansm, essentally the only type of aucton that guarantees both truthfulness and economc effcency n the offlne settng [4], does not drectly work n the onlne case, snce t requres the computaton of exact optmal allocaton to guarantee truthfulness, whch cannot be calculated for the future requests. The challenge further escalates when our aucton model nvolves operatonal costs of serve n the computaton of socal welfare and proft. Most exstng aucton desgns gnore such (producton) costs of resources, but consder socal welfare as only the overall value of accepted bds and proft as the overall payment. Sgnfcant dffcultes are nvolved, preventng good results, when the costs of resources are deducted n calculatng socal welfare and proft, especally n the onlne settng: The allocaton problem wth server costs contans a mxture of packng and coverng constrants (packng VM requests wthn resource capactes, and coverng accepted requests by producng enough resources and payng server costs) such problems are known to be more challengng than problems wth only packng constrants such as the prevous models wthout server costs [5]. Further, we seek to consder more generc server cost functons that are convex nstead of lnear, and there were no approprate technques for handlng such non-lnear costs untl very recently (see Sec. for detals). Our Contrbutons. Ths paper leverages a set of latest, novel prmal-dual onlne optmzaton technques and randomzed reducton technques, to desgn a set of truthful, polynomal-tme onlne auctons for socal welfare maxmzaton or proft maxmzaton wth good compettve ratos. Our mechansms consst of two man modules: () an onlne prmal-dual optmzaton framework for VM allocaton to maxmze the socal welfare wth server costs, and for revealng the payments through the dual varables to guarantee truthfulness; and () a randomzed reducton algorthm to convert the socal welfare maxmzng auctons to ones that glean a maxmal expected proft for the provder, wth compettve ratos comparable to those for socal welfare. Ft, we model the socal welfare (proft) maxmzaton problem usng a prmal-dual optmzaton framework, and adopt a new applcaton of Fenchel dualty for the dual, whch provdes rcher structures for convex programs that gude the desgn and analyss of onlne auctons, than the commonly used Lagrangan dualty. Our optmzaton framework s general and expressve enough to handle varous convex server cost functons, e.g., cubc, lnear, or zero nfnty, representng dfferent server operaton models n real-world IaaS clouds. Second, we desgn effcent prmal-dual onlne auctons for socal welfare maxmzaton, whch extend the exstng onlne prmaldual resource allocaton framework to handle departures of resource requests, such that resources released by completed VM requests can be reused by later bds. Exstng onlne prmal-dual resource allocaton algorthms (e.g., [6][7]) do not handle resource re-use and tme-varyng resource demands n each request. To the best of our knowledge, the only onlne prmal-dual algorthms that address departure of resource requests are those for onlne schedulng (e.g., [8]), whch s structurally dfferent from our problem and the technques cannot be easly translated to our settng. Other hghlghts of the desgn nclude metculously desgned prcng functons for updatng the margnal prces per unt resource accordng to the current resource usage levels, whch play a key role n achevng truthfulness and good compettve rato. Thrd, we extend our socal welfare maxmzng auctons to proft maxmzng ones usng randomzed reducton, wth mnor losses n compettve ratos. To obtan good compettve ratos n terms of proft wth super-lnear server costs (Sec. 4), we ntroduce a new onlne prmal-dual analyss for proft. revous technques usually compare the proft of onlne auctons wth the optmal socal welfare, and do not easly generalze to our model wth server costs. In contrast, we compare the proft of our auctons wth the dual objectve of the socal welfare maxmzaton convex program. To our knowledge, our onlne prmal dual analyss for proft s novel n the lterature and may be of further nterest for other proft maxmzaton problems. Organzaton. We dscuss related work n Sec., and defne the problem model n Sec. 3. Sec. 4 presents our onlne aucton desgn under super-lnear server cost functons. Sec. 5 dscusses the onlne auctons under lnear server cost models. Smulatons are presented n Sec. 6. Sec. 7 concludes the paper.. RELATED WORK Allocatng pre-confgured types of VMs n an IaaS cloud to serve user jobs has been extensvely studed wth dfferent focuses. Beloglazov et al. [9] study energy-effcent allocaton algorthms for schedulng VMs to serve computng tasks. Joe-Wong et al. [0] seek to balance effcency and farness when allocatng VMs to use. Magulur et al. [] nvestgate stochastc models and algorthms for load balancng and VM allocaton to handle randomly arrvng workloads. None of them nvestgates onlne optmzed packng of customzed VMs at user-specfed resource amounts, whch s the focus of ths work. Aucton mechansms have been at the focal pont of recent lterature on VM prcng. Ln et al. [7] propose a second prce aucton for computng capacty allocaton and prcng. Zaman and Grosu [6] study on-demand VM allocaton through a truthful aucton, and show through experments that a hgher revenue can be acheved for the cloud provder. Ther models do not nclude talor-made VM assemblng. Wang et al. [5] model VM allocaton and prc-
ng as a mult-unt combnatoral aucton, apply the crtcal value method, and derve a mechansm that s truthful and collusonresstant. Zhang et al. [0] study customzed VM provsonng wthn one data center, and desgn a truthful aucton by applyng an L decomposton technque, achevng a.7-approxmaton n socal welfare n typcal scenaros. All these studes consder only one-round auctons wth all bds gven, neglectng the dynamcal nature of use demands. Wang et al. [8] model a dynamc aucton where bdde may request to occupy a VM for more than one decson nterval, and propose an onlne aucton mechansm. Wth smulatons, they show ther mechansm to be truthful and asymptotcally optmal n provder revenue when demand s suffcently hgh. However, the aucton model s over-smplfed and consde one type of VM only. In the VM aucton of Zhang et al. [], the bddng language and the user valuaton models capture the heterogeneous demands n a cloud market. However, only a sngle type of VM s consdered, sgnfcantly smplfyng the underlyng socal welfare maxmzaton. Sh et al. [9] nvestgate auctons over multple rounds whch are coupled together by the overall budget of each user, whle each acqured VM s stll used for only one round. We nvestgate a more practcal onlne setup, where each VM can be runnng for varous duratons nto the future, where t s sgnfcantly more challengng to guarantee truthfulness and effcency. Onlne VM allocaton s also studed n [], but a strong assumpton s made on allowng preempton of resources already occuped by a user. In addton, none of exstng cloud auctons consder server costs n the socal welfare or provder s proft, whch wll be ncluded n our model. The onlne prmal dual framework (see [6] for a survey) has been used to desgn onlne algorthms and auctons for varous problems, such as the sk rental problem, metrcal task system problem and ad auctons. The orgnal onlne prmal dual framework focuses on lnear programs, whch does not naturally model the convex cost functons consdered n ths work. Recently, there have been studes on extendng the onlne prmal dual framework to problems modeled by convex programs, such as onlne schedulng on speed-scalable machnes [8][], and onlne combnatoral auctons wth producton costs [7][3]. The former s structurally dfferent from our problem. The latter does not handle departures of resource requests, whle n practcal scenaros of VM allocatons, each VM only occupes the requested resources for a lmted perod of tme, and the resources can be released and reused afterwards. Ths work extends the prmal-dual framework to handle VM departures and resource recyclng. 3. ROBLEM MODEL 3. Cloud System and the Aucton We consder an IaaS cloud system consstng of S serve, offerng use R types of resources, as exemplfed by CU, RAM, dsk storage and bandwdth. We use [] to denote the set {,,...,}. Each server s [S] can provde a maxmum amount C of resource r [R]. Cloud use arrve over tme, and each requests one or multple talor-made VMs for workload executon, wth the amount of resources needed for each VM specfed. The IaaS cloud provder acts as the auctoneer and sells the VMs through auctons. A user submts a bd for each VM she requests upon her arrval. A total number of I bds are submtted n a large tme span,,...,t. Let B denote the th bd submtted at t (we allow multple bds to be submtted at the same tme and order smultaneous bds randomly). t s the start tme to run the VM f B s accepted and t + s the end tme, where t apple t <t +. Let d r(t) denote the amount of type-r resource requred by bd at tme t. By allowng d r(t) to resume dfferent values over t [t,t + ], we enable each VM to consume a dfferent amount of each type of resource over tme. For example, at dfferent stages of a MapReduce job, dfferent CU and bandwdth capactes are needed [4]. d r(t) s are specfed n the bd based on the projected resource need of the bdder s workload at dfferent tmes, e.g., accordng to prevous executon of smlar workloads. Dynamc scalng of resources occuped by a runnng VM s practcally feasble through hotplug technologes that adjust CU cycles, memory and dsks allocated to a runnng VM, as supported n varous vrtualzaton envronments ncludng en, VMWare and VrtualBox [5][6]. Let b be the overall wllngness-to-pay submtted n bd for the talor-made VM to run between t and t +, and v be the true valuaton of the respectve bdder. A bd can be expressed as (bddng language): B = {t,t +, {d r(t)} r[r],t[t,t + ],b}. () Upon recevng each bd, the cloud provder decdes whether to accept t, and on whch server to provson the requested VM f accepted. A bnary varable x s s set to f bd s accepted wth resources allocated on server s, and 0 otherwse. The provder also computes a payment ˆp for each wnnng bd. In practce, most cloud data cente keep ther serve on, whch reman n the low-power dle mode f no jobs are runnng, to avod tme-consumng bootng up f swtched completely off [7]. The decsons of server provsonng happen at a much larger tme scale than those for VM allocaton, e.g., Amazon EC adjusts ts server provsonng roughly once per month, accordng to dscussons wth Amazon employees. Therefore, we realstcally assume that all S serve are turned on n the span T under our nvestgaton. Each server consumes a basc amount of power wth no VM runnng, and the power usage ncreases wth the ncrease of resource occuped on the server. The operatonal cost of a server s manly due to the power cost, followng a smlar ncreasng trend wth power consumpton. We use f ( ) to denote the cost functon of server s on the amount of type-r resource used on the server, as ndcated by y (t). The cost functon s defned as follows: hy f (y (t)) = (t) +, y (t) [0,C ] () +, y (t) >C arameter 0 decdes the shape of the cost functon, accordng to dfferent operatonal models of the server. For example, Dynamc Voltage Frequency Scalng (DVFS) has been wdely supported n vrtualzaton platforms, whch adjusts the frequency or voltage of a CU on the fly (often n response to the workload) to conserve ts power consumpton [8]. When the CU voltage s elevated wth the utlzaton, the CU power usage rende a cubc ncrease wth the CU voltage [9], and hence we can approxmately use =n () where r denotes the CU. If DVFS s not enabled, measurements have shown that the server power consumpton ncreases roughly lnearly wth the utlzaton of CU, memory, dsk I/O and network I/O [30], and hence we set =0 n ths case. h ndcates the relatve weght of the cost due to each type of resource n the overall server cost. It has also been shown that power consumpton of memory, dsk I/O and network I/O are sgnfcantly lower than that of the CU, further ranked n a decreasng order among themselves [3], and the power usage due to dfferent resources s addtve [30], confrmng our addtve model of the costs due to dfferent resources. 3. Mechansm Desgn Goals We target the followng propertes n our aucton desgn. () Truthfulness: For any bdder, declarng her true valuaton of the
VM and true nformaton (e.g., bd arrval tme) n her bd always maxmzes her utlty, regardless of other use bds. () Computatonal effcency: olynomal-tme algorthms for resource allocaton and payment calculaton are needed for the aucton to run effcently n an onlne fashon. () Indvdual ratonalty: Each bdder obtans a non-negatve utlty by partcpatng n the aucton, and the cloud provder receves a non-negatve net proft. (v) Socal welfare maxmzaton or provder s proft maxmzaton: The cloud provder s proft equals the aggregate user payment mnus server costs,.e., ˆp x s f (y (t)). (3) [I] s[s] t[t ] s[s] r[r] Bd s utlty s v ˆp. The socal welfare over system span T s the sum of the provder s proft and the bdde aggregate utlty, [I] (v ˆp) s[s] xs (valuaton mnus payment of all wnnng bds), whch equals the aggregate valuaton of the wnnng bds mnus the server costs, f (y (t)), and [I] v s[s] [I] x s b x s s[s] t[t ] s[s] r[r] t[t ] s[s] r[r] f (y (t)) under truthful bddng. A cloud system operates at the maxmal effcency f socal welfare s maxmzed over the runnng span, beneftng both the cloud provder and use. An equally natural goal s to maxmze the provder s proft, whch we wll also puue wth effcent onlne aucton desgn. The offlne VM allocaton and wnner determnaton problem can be formulated as follows, supposng all I bds wthn system span T are known and truthful bddng s guaranteed. The objectve n (4) ndcates socal welfare maxmzaton, and can be easly changed to proft maxmzaton by replacng the socal welfare wth the provder s proft n (3). maxmze subject to: [I]: t appletapplet + [I] s[s] x s {0, },y (t) b x s t[t ] s[s] r[r] f (y (t)) (4) x s apple, 8 [I] (4a) s[s] d r (t)x s apple y (t), 8r [R],s [S],t [T ](4b) 0, 8r [R],s [S], [I],t [T ](4c) Here, constrant (4a) ndcates that each requested VM s provsoned on at most one server. (4b) sums up the amount of type-r resource needed by all accepted bds on server s at t (countng only bds whose VMs are runnng at t) nto y (t). Recall the defnton of the cost functon n (): by settng f (y (t)) = + when y (t) exceeds C (the overall capacty of type-r resource on server s), t s (mplctly) guaranteed that the allocaton of each type of resource on any server wll not go beyond the capacty lmt. The offlne problem n (4) s a convex mxed nteger program wth a concave objectve functon and lnear constrants. We relax the ntegralty constrants x s {0, } to x s 0 (constrant (4a) guarantees x s apple ), and apply Fenchel dualty [3] to the relaxed convex program. As compared to the well-known Lagrangan dualty defned genercally for convex and non-convex programs, Fenchel dualty s defned only for convex programs, and the derved Fenchel dual problems typcally present rcher structures that gude the desgn and analyss of prmal-dual onlne algorthms. Table : Notaton I # of bds T # of tme slots R # of resource types S # of serve t arrval tme of bd t (t + ) start (end) tme of bd t arrval tme of bd d r(t) demand of resource r at t of bd C capacty of resource r of server s x s serve bd on server s () or not (0), compettve ratos of algorthms D and D U r max [I] b / t[t,t + dr(t) ] L r mn [I] b / t[t,t + ] dr(t) y (t) amount of allocated resource r on s at t u utlty of bd p (t) margnal payment of type-r resource on s at t f (y (t)) cost functon of type-r resource on s f(p (t)) conjugate of f ( ) ˆp payment of bd n D and D Let u and p (t) be the dual varables assocated wth (4a) and (4b), respectvely. The Fenchel dual [3] of the relaxed convex program s as follows: mnmze u + f(p (t)) (5) subject to: u b [I] t[t,t + ] r[r] t[t ] r[r] s[s] d r (t)p (t), 8s [S], [I] (5a) p (t) 0, 8r [R],s [S],t [T ](5b) u 0, 8 [I] (5c) where f? (p (t)) s the conjugate of the cost functon f ( ), defned as f? (p(t)) = sup {p (t)y (t) f (y (t))} (6) y (t) 0 ROOSITION. The conjugate of f (y (t)) defned n () s 8 < f(p ( p(t) ) + + (t)) =, y(t) 0 apple C (h : ) C p (t) h (C ) +, y 0 (t) >C (7) p (t) where y(t) 0 =( ) h (+ ). See Appendx A for the proof. The offlne allocaton problem and ts Fenchel dual are establshed assumng complete knowledge about the system over ts entre lfespan. In practce, wth the arrval of bds, the varables and constrants emerge gradually. For example, on the arrval of bd, there s a set of new prmal varables x s for s [S] subject to s[s] xs apple. The cloud provder must decde mmedately whether to serve bd and on whch server, as well as the bdder s payment f served. In the followng, we desgn onlne prmal-dual allocaton algorthms and payment schemes based on (4) and (5). 4. ONLINE AUCTIONS FOR SOCIAL WEL- FARE AND ROFIT MAIMIZATION In ths secton, we focus on the case of superlnear server cost functons,.e., h > 0 and > 0 n (), and desgn onlne auctons for socal welfare maxmzaton (Sec. 4.) and proft maxmzaton (Sec. 4.). We wll dscuss the case of lnear server cost
functons (wth zero server cost as a specal case) n Sec. 5. 4. Onlne Aucton for Socal Welfare Maxmzaton 4.. Aucton Desgn Decdng whether to serve a new bd and on whch server s equvalent to choosng a feasble assgnment for the new prmal varables x s. If the cloud provder decdes to serve bd on some server s, then let x s =, and ncrease the amount of allocated resources y (t) by d r(t) on server s for all resources r [R] and for all tme slots t [t,t + ]. As a result, the total server cost of s, t[t ] r[r] f (y (t)), ncreases accordngly. Otherwse, x s wll be zero for all serve s [S]. VM Allocaton. The queston s how to decde whether to serve bd and on whch server. For ths we wll look nto the dual program and resort to the KKT condtons [33]. Wth bd, there s a new dual varable u 0 subject to constrants (5a), that s, u b t[t,t r[r] + dr(t)p(t) for all s [S]. The ] KKT condtons ndcate that n the offlne prmal and dual solutons to (4) and (5), x s must be zero unless constrant (5a) s tght for server s. Thus, we let u be the maxmal of 0 and the rght hand sde (RHS) of constrants (5a), that s, u =max 0, max s[s] b t[t,t + ] r[r] d r(t)p (t). (8) Accordngly, we adopt the followng method to decde whether to accept a bd and on whch server: the cloud provder does not serve bd,.e., x s =0for all s [S], f no server s acheves a nonnegatve value on the RHS of (5a) (.e., u =0), and otherwse serves bd on the server s that maxmzes the RHS (.e., x s =, and x s =0, 8s 6= s ). The ratonale s as follows: If we nterpret p (t) as the margnal prce (a.k.a. payment) per unt of type-r resource on server s at tme t, then the second term on the RHS of (5a) becomes the total payment that bd should pay for the requested resources,.e., ˆp = t[t,t r[r] + dr(t)p(t). (9) ] So the RHS of (5a) for a gven s s the utlty of bd f t would be served on server s (valuaton mnus payment, assumng truthful bddng). Therefore, the above method effectvely assgns bd to the server that maxmzes ts utlty, and u s bd s utlty. In ths way, we target utlty maxmzaton for each bd, whch leads to truthfulness and socal welfare maxmzaton. ayment Desgn. Furthermore, the cloud provder updates the margnal prces p (t) as the amounts of allocated resources y (t) change, after calculatng payment ˆp of accepted bd usng (9), to ensure that: () the gan n socal welfare when a bd s served outweghs the loss n total server cost, and () the cloud does not allocate all resources to low value bds that come early at the rsk of havng no capacty left for hgh value bds n the future. Indeed, desgnng the onlne prcng rules s the key to obtan a good compettve rato n socal welfare, as compared to the offlne optmum. Agan, our method of prcng s drven by the structure of the offlne prmal and dual solutons. Let ÿ (t) be total demand of resource r on server s at tme t, and p (t) be the respectve margnal prces n the offlne prmal and dual solutons. If ÿ (t) s smaller than the capacty C, the margnal prce p (t) shall be equal to the margnal cost per unt of the resource, f 0 (ÿ (t)); f the demand meets ts capacty, the margnal prce shall serve as a threshold to flter out the low value bds, such that the demand subject to the prces s equal to the capacty. In ths onlne settng, however, the cloud provder can see only Algorthm : rmal-dual Onlne Aucton D Input: S, R, C, h, Output: x, p DEFINE f (y (t)) accordng to (), 8s [S],r [R]; DEFINE p (y (t)) accordng to (0), 8s [S],r [R]; 3 INITIALIZE x s =0,y (t) =0,u =0,p (t) =0, 8 [I],s [S],r [R],t [T ]; 4 Upon the arrval of the th bd 5 Get bddng language B accordng to (); 6 (x, ˆp, p, y) =CORE(S, R, B, p, y, p(y)); 7 f 9s [S],x s =then 8 Accept the th bd and allocate resources on server s to provson the VM requested n bd ; 9 Charge the th bd at ˆp ; 0 else Reject the th bd; end the current demands of resources, not the fnal demands. Therefore, t wll predct the fnal demands based on the current ones and set prces accordngly. Our approach s predctng the fnal demand of a resource r on server s to be tmes ts current demand f the predcted fnal demand does not exceed the capacty, for a fxed parameter > to be determned later, and set the margnal prce to be p (t) =f( 0 y (t)). If the above predcted fnal demand exceeds the capacty, the cloud provder needs to predct the fnal threshold prce (subject to whch the demand equals the capacty) and use t as the margnal prce. For ths we use the state-of-the-art technque n onlne resource allocaton (e.g., [34]), and let the margnal prce ncrease exponentally n the current demand,.e., p (t) =f(c 0 )exp( (y (t) C / )), where s a parameter to be determned later, f the predcted fnal demand exceeds the capacty. In summary, the margnal prce for each resource on each server at each tme, p (t), s defned to be a functon on y (t) (for all r [R], s [S], and t [T ]): ( f 0 ( y (t)), y (t) apple C p (y (t)) = f 0 (C )e (y(t) C ), y (t) > C (0) We wll show n Sec. 4.. that such a prce functon leads to nce propertes, guaranteeng compettveness of the onlne aucton. Aucton Mechansm. Drected by the dscussons above, we desgn the onlne aucton algorthm D, as gven n Alg., wth the one-round algorthm CORE gven n Alg.. Upon arrval of each bd,we update prmal varables by settng x s of the selected server to serve bd to (lne 7), and ncrease the utlzaton y (t) for dfferent resources on ths server for future tme slots t [t,t + ], accordng to the demand of bd (lne 8). We update the dual margnal prce varables p (t) s accordng to (0) (lne 0). We note that recyclng of resources s mplctly handled by ncreasng y (t) s wth bd s resource demand, only for tme slots wthn specfed duraton [t,t + ]. 4.. Analyss () Truthfulness, Indvdual Ratonalty, and olynomal Tme THEOREM. The onlne aucton D n Alg. s truthful and ndvdually ratonal, and processes each bd n O(RST ) tme. The detaled proof can be found n Appendx B. () Compettveness n Socal Welfare
Algorthm : One-Round Aucton Algorthm CORE Input: S, R, B, p, y, p(y) Output: x, ˆp, p, y Update the dual varable u (utlty) accordng to (8); f u > 0 then 3 Defne s = argmax s[s] b t[t,t + ] r[r] dr(t)p(t) ; 4 Calculate the payment of the th bd: 5 ˆp = t[t,t + ] r[r] dr(t)p (t).; 6 Update the prmal varables: 7 x s =, and x s =0for all s 6= s ; 8 y (t) =y (t)+d r(t) 8r [R],t [t,t + ]; 9 Update the dual margnal prce varables: 0 p (t) =p (y (t)) 8r [R],t [t,t + ]; else x s =0for all s [S]; 3 end We next analyze the compettve rato of our onlne aucton,.e., the wot-case upper-bound rato of the socal welfare acheved by the offlne soluton of (4) to the overall socal welfare acheved by our onlne aucton at the end of T. We start by ntroducng an onlne prmal-dual analyss framework, whch bounds the rato accordng to a bound between the ncrease of prmal objectve value and the ncrease of dual objectve value at each step of the onlne algorthm. Let 0 =0and D 0 be the ntal values of the prmal and dual objectves. Let and D denote the prmal and dual objectve values after handlng bd. y (t) denotes the amount of allocated type-r resource on server s after handlng bd, and p (t) denotes the correspondng margnal prce. Note that I and D I are the prmal and dual objectve values at the end of T. Let OT be the optmal prmal objectve value of (4),.e., the offlne optmal socal welfare. LEMMA. If () there s a constant such that the ncremental ncrease of the prmal and dual objectve values dffer by at most an factor,.e., (D D ), for every step, and () the ntal dual objectve value D 0 s at most OT, then the algorthm s compettve. ROOF. Summng up the nequalty of each step, we have I = ( ) (D D )= (DI D 0 ). Here, we use the fact that 0 =0. By weak dualty [33], D I OT. Further by our assumpton that D 0 apple OT, we have OT. So the algorthm s compettve. I In fact, the ntal dual value of our algorthm D s D 0 =0. Accordng to the above proof, we can show that our algorthm s (nstead of ) compettve, f we can fnd the smallest such that (D D ) for all steps, snce t acheves I = (DI D 0 )= DI OT. The key to dentfy ths s to show that our constructed margnal prcng functon p (t) n (0) satsfes an Allocaton-ayment Relatonshp contngent on ths, whch s suffcent to guarantee the nequalty n Lemma, based on Theorem. DEFINITION. The Allocaton-ayment Relatonshp for a gven parameter s p (t) y(t) y (t) f (y(t)) f (y (t)) f (p? (t)) f (p? (t)), () for all [I], r [R], s [S], and t [t,t + ]. The Allocaton-ayment Relatonshp shows that the dfference between payment for type-r resource on server s (accordng to the old prce before handlng ) and the ncremental cost of server s due to bd s addtonal use of resource r, s no smaller than of the value ncrease of the conjugate f? due to the adjustment of the margnal prce. Snce the payment of bd accordng to the adjusted prce s no smaller than p (t)d r(t), t guarantees that the payment from a bd f served can cover the loss n the server cost to a guaranteed extent. Concretely, the followng lemma shows that the net proft s lower bounded by a constant fracton of the ncrease n the dual objectve due to the ncrease of dual prces p (t). LEMMA. If the Allocaton-ayment Relatonshp holds for, then for every accepted bd and the correspondng server s ˆp t[t,t + ] r[r] f (y (t)) f (y (t)) (D D u ) ROOF. Recall that ˆp = t[t,t + ] r[r] dr(t)p (t) = t[t,t r[r] + y(t) y ] (t) p (t) and that D D = u + t[t,t + ] r[r] f? (p (t)) f? (p (t)). The lemma follows by summng the Allocaton-ayment Relatonshp () over all r [R] and t [t,t + ]. THEOREM. If the Allocaton-ayment Relatonshp holds for, then (D D ) for all [I]. ROOF. If bd s rejected, then = D D =0. Next, assume bd s accepted and let s be the server to whch bd s allocated (x s =). The change of prmal objectve s = b t[t,t + ] r[r] f (y (t)) f (y (t)) Note that b = u +ˆp. By Lemma, we get that u + (D D u ) By u 0 and, we have (D D ). We next fnd the smallest for whch our margnal prce functons satsfy the respectve Allocaton-ayment Relatonshp. Note that each nequalty () n Defnton nvolves only quanttes about the same resource r and the same server s. Therefore, we wll dentfy a separate rato for each par of r and s, such that the margnal prcng functons on r and s satsfy the Dfferental Allocaton-ayment Relatonshp decded by ths (Defnton ). We can then take as the maxmum of all s,.e., =max r[r],s[s], to satsfy () for all r and s. Wthout loss of generalty, we assume n the followng dscussons that the demands d r(t) are much smaller compared to the server capacty C. Then we can derve a dfferental veon of the Allocaton-ayment Relaton () based on d r(t) =dy (t) for all s [S],t [t,t + ], as follows: DEFINITION. The Dfferental Allocaton-ayment Relatonshp for a gven parameter s p (t)dy (t) f 0 (y (t))dy (t) f? 0 (p (t))dp (t), 8 [I],r [R],s [S],t [t,t + ]. () LEMMA 3. The margnal payment functon defned n (0) satsfes the Dfferental Allocaton-ayment Relatonshp for =max 4( + ), (+ ) ln( U r h (+ )C,
wth paramete where =max, ( + ), =max C, C ( ) ln( U r h (+ )C U r =max [I] b t[t,t+ ] d r(t) s the maxmum value per unt of resource r per unt of tme. We wll need the followng lemma, whch states that the margnal payment s greater than the margnal cost by at least a + factor. LEMMA 4. For y (t) [0,C ] and the correspondng p (t), p (t) ( + )f 0 y (t) = h ( + ) y (t), for the paramete n Lemma 3. The proofs of Lemma 4 and Lemma 3 are gven n Appendces C and D. We next obtan the compettve rato of onlne aucton D. THEOREM 3. The onlne aucton D n Alg. s -compettve n socal welfare, for =max r[r],s[s] wth the paramete gven n Lemma 3. ROOF. By Lemma 3, the margnal prces used by onlne aucton D satsfes the Dfferental Allocaton-ayment Relatonshp defned n () for. By the assumpton that each dy = y (t) y (t) =d r(t) s very small compared to the capacty (and that dy (t) =y(t) y (t) =0for s 6= s ), we get that f y(t) f y (t) = f 0 y (t) y(t) y (t) f? p (t) f? p (t) = f?0 p (t) p (t) p (t). So () holds mples that the Allocaton-ayment Relatonshp () also holds for. Then, by Lemma, Theorem, the theorem follows. 4. Onlne Aucton for roft Maxmzaton Next, we present an onlne aucton for proft maxmzaton based on the socal welfare maxmzng onlne aucton D, nspred by a randomzed reducton technque [35]. Whle the dea of ncreasng the payment by a randomly chosen power-of- factor s smlar to [35], the analyss of [35] does not extend to our settng wth server costs and our analyss s fundamentally dfferent. In partcular, we wll use an onlne prmal-dual analyss, comparng the expected proft of our onlne aucton to the dual objectve value of the socal welfare optmzaton problem. To the best of our knowledge, usng the onlne prmal dual framework to analyze proft s new n the lterature, and the only known technque for analyzng proft wth resource costs s the work by Blum et al. [3]. However, the compettve rato acheved by ther technque grows logarthmcally n the number of bds, whch s undesrable as we are nterested n systems nvolvng a large number of bds. In contrast, our compettve rato s ndependent on the number of bds. Further, the technque of [3] ncu an addtve loss n proft (other than the multplcatve compettve rato) whle our technque does not. 4.. Aucton Desgn We ft ntroduce a few paramete: mn s the mnmum of ; L r and U r are the lower and upper bounds of a user s value, Algorthm 3: Randomzed Onlne Aucton RD Input: S, R, C, h,, L r, U r Output: x, p RU DEFINE =max r r[r] L r ; INITIALIZE x s =0,y (t) =0,u =0,p (t) =0, 8 [I],s [S],r [R],t [T ]; 3 Upon the arrval of the th bd 4 Get bddng language B accordng to (); 5 (x, ˆp, p, y) =CORE(S, R, B, p, y, p(y)); 6 f 9s [S],x s =then 7 Update p =max r[r] t[t,t + Lr dr(t), ] R ˆp 8 Generate a random number such that: wth prob. = j wth prob. for j =,...,log log (3) 9 f b p then 0 Accept the th bd, allocate the VM on server s, and charge p ; else Reject the th bd; 3 end 4 else 5 Reject the th bd; 6 end per unt of resource per unt of tme, respectvely. L r =mn [I] b t[t,t + ] d r(t) mn = mn s[s],r[r] (assumng truthful bddng such that v = b ). b U r =max [I] t[t,t + ] d r(t) (4) The onlne aucton RD s presented n Alg. 3. The dea s to use D to ft obtan a tentatve allocaton and payment for each bd, and then re-examne each tentatvely accepted bd wth a boosted payment to mprove proft. If the bddng prce of the bd s larger than the respectve boosted payment, t wll be accepted; otherwse t wll be rejected. Here, D serves as a pre-screenng step that flte out low-value bds, gvng us a set of bds whose total value s comparable to the offlne optmal socal welfare (accordng to the compettve analyss of D ). The problem s that the payment chosen by D for, say, the th bd, could be much smaller than ts true value b, leavng us wth lttle proft. To resolve ths problem, RD ncreases the tentatve payment n two steps to guarantee that t s close to the true value v wth non-trval probablty. Ft note that for any resource r, by the defnton of L r, the true value (bd prce) of the th bd s at least b t[t,t + ] dr(t)lr. Snce the above holds for any resource, we further get that Lr b r[r] dr(t) ] R t[t,t + In lght of ths observaton, RD rases the tentatve payment to p =max Lr r[r] dr(t), ] R ˆp t[t,t +
The above payment can stll be very far from the true value v. Hence, RD further multply p by a randomly chosen power of, denoted as, whch s drawn from a carefully chosen dstrbuton (.e., (3)). By dong so, the fnal payment s wthn a factor of (and lower than) the true value v wth non-trval probablty. We note that Alg. 3 requres U r and L r as nput, whose exact values are not known before all bds have arrved. Instead, we adopt estmated values of the upper and lower bounds as nput to our onlne algorthm, e.g., based on past experence. We wll show n the smulatons good performance of the aucton even f the estmaton s qute dfferent from the actual value. 4.. Analyss () Truthfulness, Indvdual Ratonalty, and olynomal Tme THEOREM 4. The randomzed onlne aucton RD n Alg. 3 s truthful and ndvdually ratonal, and processes each bd n O(RST ) tme. The proof s gven n Appendx E. () Compettveness n roft THEOREM 5. The randomzed onlne aucton RD n Alg. 3 s O( +log )-compettve n expected proft, where s gven RU n Theorem 3, =max r r[r] L r wth U r and L r defned n (4). ROOF. Let R denote the expected proft that RD generates from the ft bds (R 0 =0). We wll use an analyss smlar to the onlne prmal dual approach, and show that for any [I], O +log (R R ) D D (5) where D s dual objectve of prmal dual aucton D after handlng the ft bds. The theorem follows because the expected proft of RD at the end of the nstance satsfes O +log R I D I D 0 OT, where the last nequalty holds because D I OT (weak dualty) and D 0 =0(defnton of D ). It remans to prove (5). If bd s not accepted by D, t s not accepted by RD ether. Therefore, both sdes of (5) are zero. Next, suppose bd s accepted by D. Let c = t[t,t + ] r[r] f 0 y (t) dy (t) (6) denote the ncrease of server costs f bd s accepted. Then, R R log = r[ =] p c + = p c + log j= r[ = j ] b j p j p c log j= b j p j p c (7) where b j p equals f b j p, and equals 0 otherwse. By Lemma and Lemma 3 (also recall the defnton of c n (6)), we have p c ˆp c (D D u ) (8) So the ft term of (7) alone s almost suffcent for showng (5) modulo the u term. The rest of the proof s dvded nto two cases dependng on whether b < p (only the ft term n (7) s non-zero, but b and, thus, u are small), or not (b and, thus, u are large, but we get postve contrbuton from the second term of (7)). Case : b < p. Note that p ˆp = r[r] t[t,t + ] p (t)dy (t). By Lemma 4, p s at least r[r] t[t,t + ](+ )f 0 y (t) dy (t) (+ mn)c Note that b j p =0for all j. We get that R R = p c 6 (ˆp c)+ p c ( p ˆp) 3 6 (ˆp c)+ mn p ( p 3(+ mn ( + ) mn)c) 6 (ˆp c)+ mn u (u 6(+ mn apple b < p) ) 6 (D D u )+ mn u 6(+ mn (by (8)) ) By defnton of, +. Recall =max r[r],s[s]. So +. uttng together, (5) follows because mn R R 6 (D D u )+ mn 6 u 6 (D D ) Case : b p. On one hand, by our choce of p, t s at least Lr p r[r] dr(t). ] R t[t,t + On the other hand, by the defnton of U r, the true value s at most b apple r[r] t[t,t + ] dr(t)ur So b p apple RUr L r apple. Snce s a randomly chosen power of between and, wth probablty, satsfes b apple p apple log b and, thus, RD accepts bd and obtans proft at least b. Therefore, R R ( p c)+ b (recall (7)). By log (8), the ft term s at least (D D u ). The second term s at least u 4log because bd s utlty s at most ts value. uttng together, the expected proft from bd s at least R R (D D u )+ 4log (D D u +4 log )+ = +4 log (D D ) So the theorem holds n case as well. () Compettveness n Socal Welfare u +4 log THEOREM 6. Randomzed onlne aucton RD n Alg. 3 s -compettve n expected socal welfare, where s gven n Theorem 3. ROOF. Consder any bd that s accepted by D. By the defnton f RD, t wll tentatvely accept the bd and then reexamne t wth randomly chosen boosted payment. Wth probablty, =. Further note that when =, b p and, thus, RD wll accept bd. In sum, for every bd that s accepted by D,RD would accepted t wth probablty at least. Therefore, the expected socal welfare of RD s at least a half of the socal welfare of D. 5. ETENSION TO LINEAR SERVER COSTS In ths secton, we extend the aucton desgn n Sec. 4 to the settng of lnear server cost functons,.e., =0n () such that hy f (y (t)) = (t), y (t) [0,C ] (9) +, y (t) >C whch says that the margnal cost per unt of resource usage s a constant (h ) wthn the capacty constrant. u
Algorthm 4: rmal-dual Onlne Aucton D Input: S, R, C, h, Output: x, p DEFINE f (y (t)) accordng to (9); DEFINE p (y (t)) accordng to (); 3 INITIALIZE x s =0,y (t) =0,u =0,p (t) = L r h + h, 8 [I],s [S],r [R],t [T ]; RS 4 Upon the arrval of the th bd 5 Get bddng lauguage B accordng to (); 6 (x, ˆp, p, y) =CORE(S, R, B, p, y, p(y)); 7 f 9s [S],x s =then 8 Accept the th bd on server s, and charge t at ˆp ; 9 else 0 Reject the th bd; end ROOSITION. The conjugate of f (y (t)) defned n (9) s f? (p (t)) = 0, p(t) apple h (p (t) h )C, p (t) >h (0) The proof can be found n Appendx F. 5. Socal Welfare Maxmzaton We adapt onlne aucton D to onlne aucton D for lnear cost functons, as gven n Alg. 4. Among the nput, >0 s a parameter such that T s the mnmum resource occupaton tme span among all bds (whch can be an estmated lower bound). We defne a new margnal payment functon: p (y (t)) = Lr h RS RS(U r h ) L r h y(t) C + h, () where L r and U r are defned n (4). For any bd such that b / t[t,t + d r(t) apple h, ts value s smaller than the server cost needed to serve t and, thus, wll be rejected. So we can assume wthout loss of generalty that L r >h. The margnal prce p (t) s a functon of the amount of currently allocated resource y (t). p (t) s hgher than margnal operatonal cost h to guarantee non-negatve utlty of the provder. The ntal margnal prce s low enough such that any bd (subject to L r whch lower bounds a bd s value per unt of resource per unt of tme) wll be accepted. Then, the margnal prce p (t) ncreases as y (t) ncreases to ensure that a server wll not allocate all capacty of a resource to low value bds. Fnally, the margnal prce s hgh enough (larger than U r upper bound of a bd s value per unt of resource per unt of tme) when y (t) >C to make sure a server wll not allocate more resources than ts capacty. D also uses CORE as a sub-routne. The only dfference between D and D are the margnal prce functons and ntal values of the margnal prces, due to dfferent server cost functons. THEOREM 7. The onlne aucton D n Alg. 4 s truthful and ndvdually ratonal, and runs n polynomal tme. The proof s smlar to the proof of Theorem and thus omtted. THEOREM 8. The onlne aucton D n Alg. 4 s -compettve n socal welfare wth = RS(Ur h) max ln( r[r],s[r] L r h ), () assumng that the offlne optmal socal welfare s at least OT RS r[r] s[s] T (L r h )C. Before we get to the proof of the theorem, let us ft explan how to nterpret the assumed lower bound on the offlne optmal socal welfare. Recall that T s the mnmum tme span that a bd occupes the requested resources for. L r h s the mnmum socal welfare generated by a bd that demands resource r and s allocated to server s, per unt of resource r and per unt of tme. Thus, T L r h C s the mnmal socal welfare generated by all bds that demand resource r and are allocated to server s f the entre capacty of resource r on server s s occuped for each tme slot. So the assumpton n the above theorem s essentally sayng that n the offlne soluton, there are enough workloads to exhaust at least one resource on one server at a tme slot, whch s easly satsfed n real-world cloud systems. RS(Ur h) ROOF. Let =ln L r h. We wll show that the margnal payment functon defned n () satsfes the Dfferental Allocaton-ayment Relatonshp for all resource r, server s, and tme t wth parameter,.e., p (t)dy (t) f 0 (y (t))dy (t) f (p?0 (t))dp (t) and that the ntal value of the dual objectve s at most OT. Then, gven = max r[r],s[r], the theorem wll follow from Lemma and Theorem. Accordng to (9) and (0), h, f(y 0 0 apple y (t)) = (t) apple C +, y (t) >C f (p?0 0, 0 apple p(t) <h (t)) = C p (t) h By (), we get p (t) > U r when the demand exceeds the capacty,.e., y (t) > C. So by our choce of p (t), the cloud provder wll never allocate more resources than ts capacty,.e., y (t) apple C. In addton, the ntal margnal prce p (0) = Lr h + h >h, and the margnal prces are nondecreasng. Therefore n the rest of the dscusson, we may as- RS sume that f(y 0 (t)) = h and f (p?0 (t)) = C. uttng them nto the above dfferental nequalty, t becomes p (t) C h dy (t) dp (t), whch holds wth equalty by our RS(Ur h) choce of p (t) and that =ln L r h. Fnally, accordng to (5) and (0), D 0 = t[t ] s[s] r[r] C p(0) h = Lr h t[t ] s[s] r[r] C RS = RS r[r] s[s] T (L r h )C apple OT. The last nequalty s based on the assumpton n the theorem. 5. roft Maxmzaton Next, we present an extenson of D that acheves a compettve rato comparable to n provder s proft. Alg. 5 gves our randomzed onlne aucton for proft maxmzaton. Smlar to how RD extends D,RD ft use D as a black box to obtan a tentatve allocaton x and payment ˆp for each bd. Then t rases the payment by a factor of and accepts the bd only when ts value s hgher than the new payment. The dfference les n the dfferent values of, whch decdes the dstrbuton from whch s sampled. THEOREM 9. The onlne aucton RD n Alg. 5 s truthful and ndvdually ratonal, and runs n polynomal tme.
Algorthm 5: Randomzed Onlne Aucton RD Input: S, R, C, h,, L r, U r Output: x, p DEFINE f (y (t)) accordng to (9); DEFINE p (y (t)) accordng to (); RS(U 3 DEFINE =max r h ) r[r],s[s] L r h ; 4 INITIALIZE x s =0,y (t) =0,u =0,p (t) = L r h + h, 8 [I],s [S],r [R],t [T ]; RS 5 Upon the arrval of the th bd 6 Get bddng language B accordng to (); 7 (x, ˆp, p, y) =CORE(S, R, B, p, y, p(y)); 8 f 9s [S],x s =then 9 Generate a random number such that: wth prob. = j wth prob. for j =,...,log log (3) f b ˆp then 0 Accept the th bd on server s, and charge ˆp ; else Reject the th bd; 3 end 4 else 5 Reject the th bd; 6 end The proof s smlar to the proof of Theorem 4 and thus omtted. The proft guarantee wll be presented n Theorem 0. We ft show a few techncal lemmas that are needed for the proof of Theorem 0 (proofs of lemmas gven n appendces G I). Let A? be the set of bds accepted by socal welfare maxmzng onlne aucton D, and O? be the set of bds accepted by the offlne omnscent optmal algorthm for socal welfare maxmzaton. c s the cost occu by an accepted bd n (6). LEMMA 5. The optmal proft of any truthful and ndvdually ratonal aucton (onlne or offlne) n our system s upper bounded by A (b? c )+ O? :b <ˆp (b c ). (4) LEMMA 6. The ft term of (4) s upper bounded by 4log tmes the expected proft of onlne aucton RD. LEMMA 7. The second term of (4) can be upper bounded by tmes the proft of onlne aucton D,.e., O? :b <ˆp (b c ) apple A? (ˆp c ) (5) where s defned n (). LEMMA 8. The expected proft of RD s at least a half of the proft of D. ROOF. It follows from that for every bd,rd uses the same allocaton and payment as n D wth probablty. THEOREM 0. The randomzed onlne aucton RD n Alg. 5 s + -compettve n terms of expected proft. 4 ln ROOF. Combnng Lemmas 5, 6, 7 and 8, the optmal proft s upper bounded by 4log + tmes the expected proft of RS(U RD. By the choce of =max r h ) r[r],s[s] L r h and the defnton of n (), we get that log = ln = ln. So the theorem follows. ln We also gve the compettveness n socal welfare acheved by RD n the followng theorem, wth proof gven n Appendx J. THEOREM. The randomzed onlne aucton RD n Alg. 5 s 4 -compettve n terms of expected socal welfare. Fnally, we note that as a specal case, the onlne auctons n ths secton also handle the case wth no server costs wthn the capacty, whch s equvalent to havng h =0, and the server cost functons as the followng zero nfnty functons: 0, f (y (t)) = y(t) [0,C ] +, y (t) >C (6) All the propertes that we have shown for lnear cost functons apply to zero nfnty costs, wth proofs omtted due to duplcty. 6. ERFORMANCE EVALUATION We evaluate our auctons usng trace-drven smulatons, explotng Google cluster-usage data [36], whch contans nformaton ncludng resource demands (CU, RAM, Dsk) for each job submtted to the Google cluster, job arrval tmes and duratons. We translate each job nto a VM bd, requestng R =3types of resources at the demands extracted from the traces (note demand d r(t) here s not much smaller than C, though our theoretcal analyss assumed so). Each tme slot s 0 seconds [37], and a bd arrves every [, 0] tme slot(s) (we set ths arrval rate relatvely low to test our algorthm n extreme scenaros, snce the more bds arrve concurrently, the closer our onlne algorthm s to the offlne one). The duraton of each VM s between 0 and 3600 tme slots. We set the bddng prce of each bd by multplyng the overall resource demands n the bd by unt prces randomly pcked wthn dfferent ranges, accordng to the upper and lower bounds of use value per unt of resource per unt of tme, U r and L r, whch wll be vared n dfferent experments. We wll vary the span between bd arrval tme and the VM start tme. We smulate serve wth heterogenous resource capactes (C ) followng the dstrbuton of server confguratons summarzed from the Google data as follows (CU and Memory unts are normalzed so that the maxmum capacty s ): # of machnes 673 3863 00 795 6 (percentage) (53%) (30%) (8%) (6%) (3%) CU 0.50 0.50 0.50.00 0.5 Memory 0.50 0.5 0.750.00 0.5 Snce the Google data does not provde dsk confguratons of serve, we set the dsk storage capacty of our serve randomly wthn [30, 800](GB). The total capacty of each type of resource to provson, and hence the number of serve to smulate, s roughly accordng to the total amount of demand from all bds multplyng a random number n [0.4, 0.8]. The total number of tme slots T s set to 5I +3600, snce on average a bd arrves n 5 tme slots and an extra 3600 tme slots are reserved to serve long-runnng bds that come near the end of the batch. h s set wthn [0.4, 0.6] for CU (dfferent for dfferent serve s), and wthn [0.005, 0.0] for RAM and dsk, roughly followng the percentage measured n [3]. For superlnear cost functons, we set wthn [.7,.] for CU and wthn [0.5, ] for RAM and dsk [9][3]. By default, =500, U r =50, L r =. We compare our algorthms wth the offlne optmum, as well as two exstng schemes Twce-the-Cost (TC) and Twce-the-Index (TI) [3] (we dentfed a lack of comparable approaches from the VM aucton lterature). TC and TI share smlar basc deas wth our onlne auctons, but adopt dfferent margnal prcng functons: For TC, p (y (t)) = f(y 0 (t)),.e., the current margnal payment s twce of the current margnal cost; for TI, p (y (t)) =
Compettve Rato 4 3 0 50 00 00 400 500 600 700 800 900 000 Number of Bds Compettve Rato 6 4 0 0 50 00 50 500 600 700 800 900 000 Number of Bds Compettve Rato 7 6 5 4 3 0% 60% 00% 40% 80% 500 600 700 800 900 000 Number of Bds Fgure : Compettve rato of D n socal welfare among dfferent Compettve Rato 8 6 4 0% 60% 00% 40% 80% 500 600 700 800 900 000 Number of Bds Fgure 4: Compettve rato of R D n proft over dfferent percentages of estmaton over real L r Fgure : Compettve rato of R D n proft among dfferent U r Socal Welfare 6 x 08 4 0 D TI TC 500 000 500 000 500 3000 3500 4000 Number of Bds Fgure 5: Comparson among D, TC and TI Fgure 3: Compettve rato of R D n proft over dfferent percentages of estmaton over real U r roft 8 x 07 6 4 0 RD TI TC 500 000 500 000 500 3000 3500 4000 Number of Bds Fgure 6: Comparson among R D, TC, TI Socal Welfare /roft(log) 0 5 0 5 0 W(D) W(TC) (RD) (TC) 500 000 500 000 500 3000 Number of Bds Fgure 7: Comparson between D and TC for socal welfare, and between R D and TC for proft f 0 (y (t)),.e., the current margnal payment s the margnal cost on twce of the current resource usage. 6. Comparson wth Offlne Optmum We ft study the compettve ratos acheved by our onlne auctons, computed by dvdng the offlne optmal socal welfare obtaned by solvng (4) exactly usng Gurob Optmzer, by the socal welfare or proft acheved by the respectve onlne algorthm. Due to the hgh tme complexty of solvng the offlne convex program wth a large number of varables, we set the largest number of bds to be 000 n ths set of experments. Fg. llustrates that a smaller compettve rato comes wth a larger, whch s the average number of tme slots between the bd arrval tme to ts specfed VM start tme over all the bds. Consderng the onlne settng, ths result s nterestng but qute reasonable: a group of bds apply for a future occupaton duraton; the earler they arrve before ther actual resource occupaton starts, the more future nformaton the system can learn, and better allocaton and payment decsons can be made. Fg. shows that the compettve rato of expected proft wth RD becomes larger wth the ncrease of U r. Accordng to Theorem 5, a larger U r not only negatvely nfluences the compettve rato of RD, but also determnes the dstrbuton of that further ncreases the rato. In Fg. 3 and 4, we use estmated values of U r or L r as nput to RD, whch are at dfferent percentages as dvded by the real U r or L r. We see that both overestmaton and underestmaton have mnor nfluence n the performance, as compared to that acheved by the real U r or L r (the case of 00% n the fgures), and underestmaton of U r and overestmaton of L r are more desrable. In addton, all fgures show that the compettve rato becomes smaller wth the ncrease of the number of bds. Accordng to our setup, when the number of bds ncreases, the total demands of resources ncrease, and more serve are provsoned. In our onlne auctons, we always choose a cheapest server for each comng bd n (8), and hence the soluton space becomes larger when the number of serve ncreases, leadng to better compettve ratos. The results for D and RD are smlar to the cases of D and RD, and we omt them here due to space constrants. 6. Comparson wth Exstng Schemes We next compare the socal welfare and proft acheved by our auctons wth TC and TI at larger scales of the system. Fg. 5 shows that the socal welfare acheved by our D outperforms those by TC and TI. Especally, we observe through our experments that the margnal payment functon n TC can not flter out low value bds when the used resource of a server approaches ts full capacty, and wth TI, maxmally only half of the capacty on a server can be allocated due to the + part of the cost functon, both leadng to lower socal welfare. Fg. 6 further reveals the hgher proft acheved by our RD, as compared to TC and TI. Due to space lmt, we compare D and RD wth TC n terms of socal welfare (W ( )) and proft ( ( )), respectvely, n the same Fg. 7 (note the values are n log scale). Our algorthms outperform TC n both cases. 7. CONCLUDING REMARKS Ths work desgns truthful and effcent onlne VM auctons where cloud use bd for resources nto the future for talor-made VMs wth dfferent runnng duratons, targetng socal welfare maxmzaton and cloud provder s proft maxmzaton. We consder server costs n our aucton model, and handle the resultng sgnfcantly more challengng mechansm desgn by leveragng novel prmaldual onlne optmzaton and randomzed reducton technques. Our prmal-dual framework adopts a new applcaton of Fenchel dualty and handles varous convex server cost functons. It further allows request departures and resource recyclng whle guaranteeng good compettve ratos, whch exstng onlne prmal-dual resource allocaton frameworks do not handle. For proft maxmzaton, we ntroduce a new onlne prmal-dual analyss to obtan good compettve ratos wth super-lnear server costs, whch s new n the lterature Trace drven smulatons valdate our theoretcal analyss and show good performance of our mechansms as compared to the offlne soluton and exstng mechansms on smlar frameworks.
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AENDI A. ROOF OF ROOSITION ROOF. f (p? (t)) = ( p sup (t)y (t) h y (t) +, y (t) [0,C ] y 0 p (t)y (t), y (t) >C Observe that =. Then we only need to obtan the conjugate of f ( ) from. Let g (y (t)) = p (t)y (t) h y (t) +. Dfferentate g (t) wth respect to y (t) and set (p (t) y (t) h y (t) + ) 0 =0. (7) We obtan the soluton of the equaton as y(t) 0 p =( (t) ) h (+ ). Note that the doman of g (t) s y (t) [0,C ]. Hence the supremum happens at y(t) 0 only f y(t) 0 les n [0,C ]. Otherwse when y(t) 0 >C, (p (t) y (t) h y (t) + ) 0 > 0, such that g (y (t)) monotoncally ncreases wth y n [0,C ] and the supremum happens at C. Mergng the two cases, we derve the conjugate of the cost functon: 8 >< f(p ( p(t) ) + + (t)) =, y(t) 0 apple C h >: + C p (t) h C, y 0 (t) >C (8) B. ROOF OF THEOREM ROOF. (Truthfulness n bddng prce) The margnal prces that the cloud provder presents to bd depend only on the demands of resources before the arrval of bd and the demand of bd, thus, are ndependent on bd s bddng prce. Further, the cloud provder always assgns bds to serve to maxmze each bd s utlty gven the current margnal prces. So t falls nto the famly of sequental posted prce mechansms (e.g., [38]) and, thus, a bdder cannot mprove ts utlty by lyng about ts bddng prce. (Truthfulness n arrval tme) Snce the margnal prces are nondecreasng n the amount of allocated resources, whch s nondecreasng over the resource allocaton tme,.e., let y (t,t) denote the amount of resource r on s n t whch has been allocated by t (t apple t) and y (t,t) s non-decreasng over t. Hence a bdder cannot decrease the total prce of the resource that t requests by delayng ts arrval. Note that the arrval tme of a bd s the ft tme the bdder s aware of her demands so the arrval tme can not be earler. (Truthfulness n resource occupaton tmes) Droppng part of the true resource occupaton duraton n the request would rsk falng to complete the job. So no bdder would do that. On the other hand, the margnal prces are non-negatve accordng to (0). So requestng a supeet of the true resource occupaton duraton ncreases a bdder s payment and decreases her utlty. (Indvdually ratonal) Accordng to (8), the utlty of a bdder s always non-negatve. The proft of the provder s also non-negatve based on (0) whch mples p (y (t)) >f(y 0 (t)). (olynomal runnng tme) We assume that the algorthm can compute the dfferentals of margnal server cost functons,.e., f 0 n constant tme. (Otherwse, the runnng tme wll be O(RST ) tmes the tme complexty of computng f.) 0 To process a bd, the algorthm ft sums up the margnal prces for all requested resources over the occupaton duraton for each server s [S] to compute the payment that bd should pay f t would be served on server s. Ths step runs n O(RST ) tme. Then, the algorthm computes u and decdes the allocaton and payment of bd by checkng the utlty of the bd f t would be served on each server s, whch can be done n O(S) tme. Fnally, the algorthm updates the amount of allocated resources y (t) and the respectve margnal prces p (t), whch can be done n O(RST ) tme. C. ROOF OF LEMMA 4 ROOF. It s easy to verfy that the nequalty n the lemma holds wth equalty when y (t) apple C accordng to (0) and the choce C of n Lemma 3. Next, assume that y (t). In ths case, p (t) =h ( + )C e (y(t) C ). So the nequalty (+ )e C s equvalent to e y(t). Recall that y (t) C ( + ) /. It s easy to verfy that the above nequalty holds wth equalty for y (t) = C e y(t) y (t) e y(t) ( y (t). Next, t suffces to show that s non-decreasng as y(t) ncreases. Its dervatve s y (t) +. By our choce of C and the C assumpton that y (t), y(t) 0 and the above dervatve s non-negatve. So the lemma follows. ) D. ROOF OF LEMMA 3 ROOF. Ft, we explctly wrte down the dfferentals of the server cost functons (n ()) and ther convex conjugates (n (6)): f(y 0 h (t)) = ( + )y (t), y (t) [0,C ] +, y (t) >C ( f (p?0 p ( (t) ), p (t)) = (+ )h (t) apple h ( + )C C, p (t) >h ( + )C By ln( U r C ( ) h (+ )C and the defnton of margnal payment n (0), we get that when y (t) =C, p (t) =h ( + )C e (C C ) U r. By our choce of p (t), the cloud provder wll never allocate more resources than ts capacty,.e., y (t) apple C. Therefore, we may assume n the rest of the proof that y (t) apple C and, thus, f(y 0 (t)) = h ( + )y (t). Next, accordng to the pece-wse defnton of f,?0 the proof s dvded nto two cases. Case : y (t) apple C. In ths case, the Dfferental Allocaton-ayment Relatonshp s p (t) h ( + )y (t) dy (t) By the margnal payment defnton n (0), p(r) h (+ ) dp(t) (9) p (t) =f 0 ( y (t)) = h ( + ) y (t). (30) uttng them together, (9) s equvalent to (cancellng common terms of both sdes) ( +. If, ) then =max, ( + ) =and + = 4 apple 4 4. So we get that + = apple 4 <
. If <, then =(+ ), and + C = ( + ) apple e( + ) <. Case : apple y(t) apple C. In ths case, the Dfferental Allocaton-ayment Relatonshp s p (t) h ( + )y (t) dy (t) Recall that when y (t) p (t) =h ( + C dp (t) (3) C, the margnal payment s )C e (y(t) C ) (3) By Lemma 4, to show (3), t suffces to show + p (t)dy (t) C dp (t). On the other hand, by the defnton of the margnal payment n (3), we have dp (t) = p (t)dy (t). So t remans to show that apple C (+ ). By our choce of paramete, ether = or = C apple e C ( apple e) = C (+ ) e( + ) apple C (+ ), ln( U r C ( ) h (+ )C apple U C ln( r h (+ )C (+ = ) C (+ ) apple C. (+ ) So the lemma holds n case. ln( U r h (+ )C E. ROOF OF THEOREM 4 ( ) ROOF. RD also posts a take-t-or-leave-t prce for each bd and serves the bd f and only f ts bddng prce s larger than the posted prce, whle changng the payment from ˆp n D to p, where and p are ndependent of b s bddng prce. So t s stll a sequentally posted prcng mechansm and, thus, truthful n terms of bddng prce. The proof of truthfulness n bd arrval tme and duraton tmes, ndvdual ratonalty, and tme complexty of bd processng follows smlarly to the proof of Theorem, and s hence omtted. F. ROOF OF ROOSITION ROOF. The conjugate of f (y (t)) s defned n (6). By the defnton of f, p (t)y (t) f (y (t)) s equal to (p (t) h )y (t) f y (t) apple C, and s f y (t) > C. If p (t) h < 0, then the rght hand sde of (6) s maxmzed when y (t) =0wth maxmum value 0; f p (t) h 0, then t s maxmzed when y (t) = C wth maxmum value (p (t) h )C. G. ROOF OF LEMMA 5 ROOF. The proft of any truthful and ndvdually ratonal aucton s upper bounded by ts socal welfare and, thus, by the optmal socal welfare,.e., O? (b c )= O? :b ˆp (b c )+ O? :b <ˆp (b c ) apple :b ˆp (b c )+ O? :b <ˆp (b c ) = A? (b c )+ O? :b <ˆp (b c ) H. ROOF OF LEMMA 6 ROOF. Due to the lnear cost functon n ths case, the ncurred cost by an accepted bd s c = t[t,t r[r] + ] dr(t)h, where s s the server that serves the th bd. By the defnton of D, for any tentatvely accepted bd A?, the tentatve proft from bd s at least ˆp c = r[r] t[t,t + d r(t)p (t) d r(t)h ] r[r] dr(t) p(0) h(t) ] t[t,t + Snce p (0) = Lr h + h, we further have RS Lr ˆp c r[r] dr(t) h ] t[t,t + RS On the other hand, the maxmum proft from bd s at most b c apple r[r] t[t,t + ] dr(t)(ur h). We have that b c ˆp c apple RS(U r h ) L r h =. By the defnton of n (3), s a randomly chosen power of between and and, thus, satsfes that (b c ) apple (ˆp c ) apple (b c ) wth probablty at least. In ths log case, RD wll accept the th bd and generate proft (ˆp c ) (b c) from the th bd. Therefore, the expected proft generated by RD from every bd A? (that s, tentatvely accepted by D ) s at least (b c). The expected proft of RD s at least log 4log A? (b c ). I. ROOF OF LEMMA 7 ROOF. Consder a vrtual nstance of our onlne problem where the th bd n the vrtual nstance has the same demands d r(t) over [t,t + ] as n the orgnal nstance, but has bddng prces mn{b, ˆp } nstead of b. Then, the onlne aucton D would accept the same set of bds n the vrtual nstance as n the orgnal nstance. The socal welfare of D n the vrtual nstance s therefore A (ˆp? c ). Further note that choosng all O? such that b < ˆp s a feasble soluton n the vrtual nstance and gets socal welfare O? :b <ˆp (b c ). The lemma now follows from the compettve rato of onlne algorthm D. J. ROOF OF THEOREM ROOF. The set of accepted bds of the onlne aucton RD s a subset of that of D. Let B? denote the set of accepted bds of RD. Recall A? denotes the set of accepted bds of D. Defne a random varable b 0 as follows: b 0 b f b = ˆp (33) 0 f b < ˆp The expected socal welfare of RD s: E B? (b c ) = E A? (b 0 c ). (34) Note that A? = { : b ˆp }. By lnearty of expectaton, (34) = E :b ˆp (b 0 c ) = :b ˆp E b 0 c = :b ˆp (b c ) r[b ˆp ] :b ˆp (b c ) r[b ˆp ] r[ =] A? (b c ) Therefore, the expected socal welfare of RD s at least half of that of D. The theorem then follows from Theorem 8.