Pricing Cloud Bandwidth Reservations under Demand Uncertainty

Transcription

1 Prcng Cloud Bandwdth Reservatons under Demand Uncertanty D Nu, Chen Feng, Baochun L Department of Electrcal and Computer Engneerng Unversty of Toronto {dnu, cfeng, bl}@eecg.toronto.edu ABSTRACT In a publc cloud, bandwdth s tradtonally prced n a pay-asyou-go model. Reflectng the recent trend of augmentng cloud computng wth bandwdth guarantees, we consder a novel model of cloud bandwdth allocaton and prcng when explct bandwdth reservaton s enabled. We argue that a tenant s utlty depends not only on ts bandwdth usage, but more mportantly on the porton of ts demand that s satsfed wth a performance guarantee. Our objectve s to determne the optmal polcy for prcng cloud bandwdth reservatons, n order to maxmze socal welfare,.e., the sum of the expected profts that can be made by all tenants and the cloud provder, even wth the presence of demand uncertanty. The problem turns out to be a large-scale network optmzaton problem wth a coupled objectve functon. We propose two new dstrbuted solutons based on chaotc equaton updates and cuttng-plane methods that prove to be more effcent than exstng solutons based on consstency prcng and subgradent methods. In addton, we address the practcal challenge of forecastng demand statstcs, requred by our optmzaton problem as nput. We propose a factor model for near-future demand predcton, and test t on a real-world vdeo workload dataset. All ncluded, we have desgned a fully computerzed tradng envronment for cloud bandwdth reservatons, whch operates effectvely at a fne granularty of as small as ten mnutes n our trace-drven smulatons. Categores and Subject Descrptors K.6.2 [Installaton Management]: Prcng and resource allocaton; Performance and usage measurement; G.3 [Probablty and Statstcs]: Tme seres analyss General Terms Algorthms, Economcs, Measurement, Performance Keywords Cloud Computng, Bandwdth Prcng, Dstrbuted Optmzaton, Predcton, Tme Seres Permsson to make dgtal or hard copes of all or part of ths work for personal 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 frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. SIGMETRICS 12, June 11 15, 212, London, England, UK. Copyrght 212 ACM /12/6...$ INTRODUCTION Cloud computng delvers Infrastructure as a Servce (IaaS that ntegrates computaton, storage and network resources n a vrtualzed envronment. It represents a new busness model where applcatons as tenants of the cloud can dynamcally reserve nstances on demand. However, a major rsk to these tenants usng cloud servces s that unlke CPU and memory, bandwdth s not guaranteed n current-generaton cloud platforms (e.g., Amazon EC2, leadng to unpredctable network performance [6, 2]. A lack of bandwdth guarantee mpedes cloud adopton by applcatons that requre such guarantees, such as transacton processng web applcatons [14] and vdeo-on-demand (VoD applcatons [4]. The utlty of tenants runnng these applcatons depends not only on the bandwdth usage, but more mportantly on how many of ther end-user requests are served wth guaranteed performance. Wth an ever-ncreasng demand for performance predctablty, a recent trend n networkng research s to augment cloud computng to explctly account for network resources. In fact, datacenter engneerng technques have been developed to expand the tenantcloud nterface to allow bandwdth reservaton for traffc flowng from a vrtual machne (VM n the cloud to the Internet [7, 12]. We envson that n future cloud platforms, bandwdth reservaton wll be a value-added feature that attracts tenants who seek bandwdth guarantees. Unfortunately, even wth cloud bandwdth reservaton enabled, due to demand uncertanty, t s stll dffcult for a tenant to predct how much bandwdth t needs at a partcular tme. The usual approach of over-provsonng ncurs hgh costs to tenants and does not really provde quanttatve servce guarantees. To promote guaranteed servces, we beleve that a new cloud servce model should be ntroduced, n whch a tenant smply needs to specfy a percentage of ts (bandwdth demand to be served wth guaranteed performance, whch we call the guaranteed porton, whle the rest of ts demand wll be served wth best effort. It s then the cloud provder s responsblty to satsfy the guaranteed porton of the tenant wth a hgh probablty. Snce the cloud provder has vast hstorcal workload data, t can leverage statstcal learnng to predct tenant demands and make actual bandwdth reservatons for the tenants. In ths paper, we study how to prce the above guaranteed servce. It s worth notng that usage-based prcng (pay-as-you-go s not sutable for prcng bandwdth guarantees. For example, t s more costly to guarantee the performance of a tenant wth bursty demand than a tenant wth constant demand, even f they have ncurred the same usage (number of bytes transferred. As a result, on top of the usage fee, the cloud should charge each tenant an extra reservaton fee, dependng on ts unque demand statstcs. Our objectve s to farly set such reservaton fees, wth the followng

2 challenges. Frst of all, the cloud provder usually multplexes tenant demands to save the servce cost. Due to resource sharng, the absolute amount of bandwdth reserved for each tenant s unknown. It s a challengng queston to fnd out each tenant s far share n the aggregate servce cost. Second, a prcng polcy, when mposed to the market, may affect tenants demand; such demand change n turn affects prcng decsons, leadng to potentally unstable teratons. To overcome these dffcultes, we defne the reservaton fee of each tenant as a functon of ts specfed guaranteed porton nstead of the absolute amount of bandwdth reserved. We also express each tenant s utlty as a functon of ts guaranteed porton, whch essentally measures the Qualty of Servce (QoS at the tenant. Under ths new model of prcng and utlty, each tenant wll choose a guaranteed porton to maxmze ts surplus, whch s ts utlty mnus prce. Note that n realty, a tenant may choose a guaranteed porton close to 1 nstead of beng 1 out of cost concerns, whle havng the remanng demand served wth best effort. We study a cloud provder whose objectve s to maxmze the socal welfare of the system,.e., the total expected tenant utlty under demand uncertanty mnus the aggregate servce cost. Although the cloud cannot know the exact form of utlty at each tenant, t can affect each tenant s choce of guaranteed porton through prcng, and thus control the socal welfare acheved. To handle the coupled cost functon (due to multplexng, we propose a novel algorthm based on chaotc equaton updates, for whch we provde a suffcent convergence condton. We further propose a dstrbuted verson of the cuttng-plane method wth guaranteed convergence. These methods are step-sze-free and proved to be more effcent than tradtonal subgradent methods n smulatons. In addton, we gve explct solutons to optmal prcng under certan specal cases and pont out the dependence of reservaton prcng on demand statstcs such as burstness and covarances. Snce a man duty of the cloud provder s to reserve bandwdth for the tenants, demand forecast consttutes an mportant part n the reservaton-based servce. Toward ths end, we propose a factor model to predct the expectatons as well as covarances of tenant demands n the near future, based on prncpal component analyss (PCA. Fnally, we evaluate the proposed algorthms on the workload traces of a real-world VoD system called UUSee [3]. We conduct trace-drven smulatons of bandwdth reservaton and algorthmc prcng based on demand predcton. The system s shown to operate effectvely at a fne granularty (of as small as 1 mnutes. The remander of the paper s organzed as follows. We revew related work n Sec. 2, and present our system model n Sec. 3. We formulate the problem of socal welfare maxmzaton n Sec. 4, where we outlne the condton for optmal prcng and dscuss ts economc mplcatons. To solve the optmal prcng problem dstrbutvely, n Sec. 5, we propose two algorthms: chaotc prce update and the dstrbuted cuttng-plane method, and study ther convergence performance. In Sec. 6, we present our statstcal methods for demand forecast. We conduct trace-drven smulatons n Sec. 7, and conclude the paper n Sec BACKGROUND AND RELATED WORK Cloud computng, e.g., Amazon EC2, s usually offered wth usage-based prcng (pay-as-you-go [6, 11]. Dfferent from payas-you-go, resource reservaton nvolves payng a negotated cost to have the resource over a tme perod, whether or not the resource s used. Although sutable for delay-nsenstve applcatons, pay-asyou-go s nsuffcent as a busness model for bandwdth-ntensve and qualty-strngent applcatons lke VoD, snce no performance guarantees are provded n general. The good news s that cloud bandwdth reservaton s becomng techncally feasble. There have been proposals on datacenter traffc engneerng to offer elastc bandwdth guarantees for egress traffc from vrtual machnes (VMs [12]. The dea of vrtual networks has also been proposed to connect the VMs of the same tenant n a vrtual network wth bandwdth guarantees [7,12]. Further, explct rate control has been proposed to apporton bandwdth accordng to flow deadlnes [22]. Such research progress has made the cloud more attractve to bandwdth-ntensve applcatons such as vdeoon-demand and MapReduce computatons that rely on the network to transfer large amounts of data at hgh rates [25]. Netflx, as a major VoD provder, moved ts data store and vdeo encodng and streamng servers to Amazon AWS [2] n 21 [4]. To support guaranteed cloud servces, we need new polces to prce not only the bandwdth usage but also bandwdth reservatons. Our prcng model s partally nspred by prcng electrc power consumpton and capacty reservaton under demand uncertanty [19]. However, due to the computng capablty and abundant workload data n the cloud, our bandwdth reservaton prcng theory s essentally a dstrbuted optmzaton problem based on statstcal learnng. Amazon Cluster Compute [1] allows tenants to reserve, at a hgh cost, a dedcated 1 Gbps network wth no multplexng. Instead of provsonng a fxed amount of capacty, we beleve that tenants should be allowed to specfy a guaranteed porton of demand, as a way to control QoS level, whle cloud provders should dynamcally vary bandwdth reservatons based on demand predctons. Our approach has the unque advantage that tenants are exempted from demand estmaton, for whch they do not have expertse. In contrast, the cloud can easly access tenant demand hstory from onlne montorng, and s computatonally capable of accurate demand forecast. Snce prcng guaranteed portons crtcally depends on accurate estmates of demand statstcs, we target applcatons wth predctable demands, such as vdeo access. As measurements show that vdeo workload demonstrates regular durnal perodcty [5,17, 23,24], varous technques have recently been proposed to forecast large-scale VoD traffc. Seasonal ARIMA models have been ntroduced n [16, 17] to predct non-statonary demand evoluton at a fne granularty. Prncpal component analyss (PCA has been proposed n [13] to extract vdeo demand evoluton patterns over longer perods (of weeks or months and forecast coarse-graned daly populatons. We combne the strengths of both approaches by fndng the common factors drvng the demand evoluton of all tenants usng PCA at a fne granularty. We then make predctons for ndvdual tenants as regressons from factor forecasts obtaned from seasonal ARIMA models. Unlke [13], our approach makes short-term predctons wth a lead tme of 1 mnutes, enablng autoscalng of resource allocaton. Our optmal prcng algorthms are related to network utlty maxmzaton (NUM, whch has been extensvely studed n the past, wth varous dstrbuted algorthms proposed. See [1, 18] for thorough surveys. Most of these algorthms assume no couplng n the objectve functon, and thus cannot be appled to our problem wth a coupled cost term. One exstng approach to handle coupled objectves s called consstency prcng [1, 21], whch s based on dual decomposton and subgradent methods. However, subgradent methods suffer from the curse of step szes, n that small steps ncur bg delays (many rounds of message exchanges between the cloud and tenants, whle bg steps yeld bg optmalty gaps. Varyng step szes strategcally s dffcult n realty. In ths paper, we propose two step-sze-free algorthms: 1 chaotc prce update, 2 the cuttng-plane method. The frst one s based on teratve equaton updates nstead of decomposton and acheves rapd conver-

3 gence under certan condtons. The second s a search algorthm wth a guaranteed convergence speed. 3. A NEW TENANT-CLOUD AGREEMENT Our system model s a generalzaton of the operaton mode of the current cloud. Current cloud provders charge tenants a usage fee based on the number of bytes transferred n the past hour, and do not provde bandwdth guarantees. We extend ths model to allow tenants to make reservatons for bandwdth guarantees explctly. The system operates on a short-term bass, e.g., based on hours or tens of mnutes. At the begnnng of each short perod, each tenant specfes a guaranteed porton to guard aganst performance rsks. The cloud decdes the actual bandwdth reservaton for tenants through demand estmaton based on workload analyss, and charges both a usage and reservaton fee. We now descrbe our system model n detal. We consder one such short perod, where N bandwdth-senstve tenants are present. Suppose that n ths perod, tenant s bandwdth demand s a random varable D (Mbps wth mean µ = E[D ] and varanceσ 2 = Var[D ]. We assume the cloud can predct µ and σ based on demand hstory and share them to tenant before the perod starts. In our proposed market, the key commodty traded s a noton called the guaranteed porton nstead of the absolute amount of bandwdth. Specfcally, the tenants and cloud wll comply to the followng servce agreement S (w,ǫ,r : Before the perod starts, each tenant specfes a guaranteed portonw [,1]; The cloud guarantees w fracton of demand D wth a hgh probablty 1 ǫ; outage s allowed to happen wth a small probablty ǫ, durng whch the bandwdth allocated to tenant s lmted to R. The parameters ǫ and R are a part of a servce level agreement (SLA advertsed by the cloud provder. We ntroduce the rsk factor ǫ because for random demand, regardless of how much bandwdth s allocated, there exsts a small rsk of resource shortage. Let q denote the actual bandwdth usage (realzed data rate n ths perod. Under a guaranteed porton w, servce S (w,ǫ,r s supposed to lead to the followng actual usage of tenant : { wd q (w =, w.p. 1 ǫ, (1 mn{w D,R }, w.p. ǫ,.e., wth probablty 1 ǫ the actual usage q s a realzaton of w fracton of ts demand D, whle durng outage (whch happens wth a small probablty ǫ, the actual usage q s ratoned byr. Clearly, tenant wll choose w based on both the utlty U and the prce of guaranteeng w porton of ts demand D. Unlke most pror work on network utlty maxmzaton [1] that assumes the utlty U depends on a sngle varable such as rate, we model utlty U (q,d of tenant as a functon of both actual usage q and the demand D. For example, a vdeo content provder (or a VoD company may have a lnear utlty gan (or revenue α q from usage q and a convexly ncreasng utlty loss e A (D q for the dened requests D q, wth α,a beng tenant-specfc parameters: U ( q(w,d = αq (w e A (D q (w, (2 where the utlty loss term can model the reputaton degradaton and potental revenue loss due to unfulflled demand 1. We assume U s concave and monotoncally ncreasng nq. The prce for tenantto use servces (w,ǫ,r s dvded nto two parts: a usage fee and a reservaton fee. As most current cloud provders do, we assume unform prcng for usage: each tenant pays $p for every unt bandwdth consumed. As a key departure from current clouds, we ntroduce a reservaton fee, whch s a functon of the guaranteed porton nstead of the absolute amount of bandwdth reservaton: each tenants charged a prce of $k w for havngw porton of ts demand guaranteed. We prce the guaranteed porton w rather than the absolute bandwdth, because tenants usually have no dea about how much bandwdth they need. Instead, they can ntutvely know how much percentage of guarantee s desred. Ths new busness model frees each tenant from the computatonal burden of demand predcton: t smply submts ts desred guaranteed porton w, whle the cloud provder computes the actual bandwdth reservaton as well as decdes the reservaton fee k w for each tenant. We defne the surplus of tenant as ts utlty mnus ts prce: S (w,p,k := U ( q(w,d pq k w. (3 Gven prces p and k, a ratonal tenant wll choose a w to maxmze ts surplus S. Tenant wll not always choose w = 1 because when ts demand s bursty, the prce to guarantee 1% of D may be hgh. In ths case, tenant wll choose a w close to 1 nstead of beng exactly 1, whle the rest of ts demand(1 w D wll be served wth best efforts. Based on tenant-specfed guaranteed portons w 1,...,w N, the cloud should guarantee the demands w 1D 1,...,w ND N for servce. Denote w := [w 1,...,w N] T. To realze the above servce guarantees, the cloud provder needs to reserve a total bandwdth capacty of K(w. Dependng on the technology used, the value of K could vary sgnfcantly from one case to another. For example, a smple non-multplexng technology s to reserver capacty for each tenant ndvdually such that demand w D s satsfed wth hgh probablty,.e., Pr(w D > R < ǫ, (4 and correspondngly, reserve capacty K = R n total. In contrast, a multplexng technology wll reserve capactyk for the tenants altogether such that the aggregate demand s satsfed wth hgh probablty,.e., Pr( w D > K < ǫ, (5 and durng outage (when wd > K, the usage q of tenant s ratoned to R wth R = K. In both cases, K s an mplct functon ofwdefned by the probablstc constrants (4 and (5, respectvely. To determne K(w, the cloud provder must estmate the future demand statstcs of all the tenants, and convert tenant-specfed guaranteed portons w nto the actual total bandwdth reservatonk. Smlarly, the cloud provder has two knds of servce costs: usage and reservaton costs. We assume ter-1 ISPs charge the cloud provder $b for every unt bandwdth actually used. Furthermore, reservng bandwdth capacty K wll ncur a reservaton cost of $c(k. Due to multplexng gan, to guarantee a smlar servce level, multplexng wll ncur a lower K and thus a lower reservaton cost c(k than wthout multplexng. 1 Even though the cloud provder may stll be able to fulflld q n a best-effort fashon, the tenant wll have no knowledge f ths s the case, and wll not be able to factor t nto ts expected utlty.

4 We defne the cloud proft Π as the dfference between ts total revenue and total cost,.e., Π(w := ( ( pq(w +k w c K(w bq (w. (6 4. PRICING TOWARDS SOCIAL WELFARE MAXIMIZATION We study a cloud provder as a socal planner whose objectve s to maxmze socal welfare W(w, whch s defned as the total tenant utlty mnus the total servce cost: W(w:= U c ( K(w = Π(w+ bq (w S (w,p,k (7 Under random demands, the cloud ams to decde a set of optmal guaranteed portons w = [w 1,...,w N] T for the tenants to maxmze the expected socal welfare by solvng max we[w(w] s.t. w 1. To solve (8, we frst derve the expected socal welfare n a smple approxmated form. Note that E [ U ] s bounded as follows: E[U ] (1 ǫe[u (w D,D ]+ǫe[u (,D ], E[U ] (1 ǫe[u (w D,D ]+ǫe[u (R,D ]. When the rsk factor ǫ s small, we have (8 E [ U (q,d ] (1 ǫe [ U (w D,D ]. (9 To smplfy notatons, we defne U (w := E [ U (w D,D ], (1 whch turns out to be monotoncally ncreasng and concave n w under very mld techncal condtons. Smlarly, the expected usage of tenants E [ q (w ] (1 ǫe [ w D ] = (1 ǫwµ. (11 Therefore, the expected surplus of tenant s E [ S (w,p,k ] =E [ U ] pe [ q ] kw =(1 ǫ ( U (w pw µ kw, (12 and the expected proft of the cloud provder s E[Π(w] = (p b(1 ǫ w µ + k w c ( K(w. (13 Substtutng the above nto (7 gves the expected socal welfare as E[W(w] = (1 ǫ ( ( U(w bw µ c K(w. ( An Equvalent Prcng Problem In realty, although the cloud provder has full knowledge about ts servce cost c(k(w, t does not know the utlty functon of each tenant. In other words, maxmzng E[W(w] n terms of w requres the cloud to know the utlty U of each tenant and s nfeasble. We now convert problem (8 nto an equvalent prcng problem. Note that the expected socal welfare s also the sum of the expected cloud proft and the total expected tenant surplus,.e., E[W(w] = E[Π(w]+ E[S (w,p,k ]. (15 w (p,k = argmax w E [ S (w,p,k ] Tenant w (p,k Tenant Cloud Provder p,k Tenant N Update p,k to ncrease socal welfare Fgure 1: Iteratve updates of prces and guaranteed portons. Furthermore, when charged wth prces p,k and facng a random demand D, a ratonal tenant wll choose a guaranteed porton w to maxmze ts expected surplus,.e., w = argmax w E [ S (w,p,k ], (16 whch defnes an mplct functon w (p,k of the prces. The cloud can affect guaranteed porton choces w = [ w 1,..., w N] T va approprate choces of prces p, k 1,...,k N, and control the correspondng expected socal welfare E[W( w]. Therefore, the socal welfare maxmzaton problem (8 s converted nto an equvalent optmal prcng problem: max p,k 1,...,k N E[W( w] = E[Π( w]+ E[S ( w,p,k ], (17 whch, by combnng (14 and (17, can be rewrtten as max (1 ǫ ( ( U( w b w µ c K( w, (18 p,k 1,...,k N where w = w (p,k s determned dstrbutvely by each tenant va surplus maxmzaton (16. Such a dstrbuted optmzaton s llustrated n Fg. 1. Now the cloud provder does not need to know U : t smply charges tenant the usage prce p and reservaton prce k, and expect a w(p,k chosen by tenant. We denote the optmal prces that solve problem (18 asp,k 1,..., k N. The optmal prcng problem (18 s equvalent to the orgnal problem (8, because by adjustng p and k, w (p,k can take any value n [, 1]. In other words, the guaranteed porton w (p,k chosen by tenant under optmal prcng p,k s exactly the guaranteed portonw that maxmzes the expected socal welfare,.e., we have w = w (p,k. (19 Therefore, once a set of optmal prces s obtaned, we essentally have found a decentralzed soluton to expected socal welfare maxmzaton (8, whch was orgnally mpossble to solve. Nonetheless, the optmal prcng problem (18 s not easy to solve ether. At a frst glance, (18 can be understood as a network utlty maxmzaton (NUM problem [1] that may be solved va decomposton among the tenants. A closer look at (18 suggests that the term c(k( w n the objectve functon may be coupled among all w s, so that (18 cannot be decomposed nto a set of subproblems, each solved at a tenant dstrbutvely. Couplng happens n the cost term when the cloud multplexes tenant demands and books a capacty K(w for the aggregated demand. As shown n Fg. 1, the key to the soluton s that the cloud provder must be able to update p and k towards the drecton that ncreases E[W( w]. And a good prce update algorthm should requre fewer rounds of message-passng between the cloud and tenants before reachng optmalty. Before presentng the dstrbuted solutons to (18 n Sec. 5, let us frst provde a number of nsghts on how to make prcng pol-

5 ces, by checkng the KKT condtons [8] that the optmal prces p,k 1,...,k N must satsfy. Proposton 1. The optmal prcesp,k1,...,k N must satsfy (1 ǫ(p bµ +k c ( K K( w w =,, (2 w= w where w = [ w 1,..., w N] T wth w = w (p,k gven by (16. Proof: Please refer to our techncal report [15] for the proof. An nspecton of (2 reveals that one set of optmal prces s { p = b, k = c ( K(w (21 / w w= w,. Although (21 s not the only set of optmal prces, our fndng comples wth the economc ntuton that a welfare-maxmzng cloud provder should charge margnal cost for both traffc usage and guaranteed reservaton. In (21, we can also observe that k depends on w, whch n turn depends on p,k 1,...,k N. Due to such couplng, (21 s not yet a closed-form soluton for the reservaton prce k. 4.2 No Multplexng vs. Multplexng All To draw nsghts, we take a look at two specal servce technologes that may be adopted by the cloud: non-multplexng and multplexng across all the tenants. For smplcty, we assume a lnear reservaton cost (whch wll be relaxed later: c(k = βk. When multplexng s not used, we can derve the optmal prces n a closed form from (21. Wthout multplexng, recall that the capacty R s reserved for each tenant ndvdually, such that Pr(w D > R < ǫ and K = R. When D s a Gaussan random varable (ths assumpton wll be verfed n Sec. 6, t s easy to check that R (w = ( µ +θ(ǫσ w, (22 where θ(ǫ = F 1 (1 ǫ s a constant, wth F( beng the CDF of normal dstrbutonn(,1. Snce the cost functon s naturally decoupled among tenants, accordng to (21, the optmal prces are mmedately gven n a closed form by p = b and k = β ( µ +θ(ǫσ,. (23 When multplexng s used, however, optmal prces have no explct solutons. Recall that wth multplexng, a capacty K s reserved to accommodate all the tenants together, such that the aggregate (nstead of ndvdual demand s satsfed wth hgh probablty: Pr( wd > K < ǫ. Snce the random demandsd 1,...,D N of dfferent tenants may be correlated, we denote ρ j the correlaton coeffcent of D and D j, wth ρ 1. For convenence, let µ = [µ 1,...,µ N] T and Σ = [σ j] be the N N symmetrc demand covarance matrx, wthσ = σ 2 andσ j = ρ jσ σ j for j. Under Gaussan demands, K can be wrtten as K(w=E [ ] wd +θ(ǫ Var [ ] wd =µ T w+θ(ǫ w T Σw. (24 Substtutng the above K(w nto (21 gves p = b and ( k = β µ +θ(ǫ w T Σw,, (25 w w=w where w = w (p,k. Clearly, wth multplexng, k s not gven n a closed form yet, due to the coupled cost functon. We note that whether wth or wthout multplexng, K(w s a convex functon and so s c(k(w. In fact, we can relax the lnear cost assumptonc(k = βk, as long asc ( K(w s strctly convex and monotoncally ncreasng for each w [,1]. There s an nterestng connecton between the non-multplexng and multplexng cases: the optmal soluton for non-multplexng can be used to bound the optmal soluton for the multplexng case. Specfcally, the optmal prces {k } of non-multplexng upperbound{k } of the multplexng case, whereas the optmal portons {w } of non-multplexng lower-bound {w } of the multplexng case. Ths s ntutve because multplexng leads to a reduced cost c(k, stmulatng tenants to ncrease ther choces of the guaranteed porton. The proof of ths connecton nvolves the use of Cauchy-Schwarz nequalty and s omtted due to space lmts. We wll use ths connecton n our dstrbuted algorthms. 4.3 Economc Implcatons Condton (2 has several economc mplcatons, whch apply to a general cost functon, although we may use the non-multplexng case for explanaton due to ts smplcty. Frst, merely adoptng a usage prce p cannot maxmze socal welfare: when k = for all, there s no p that can satsfy (2. In other words, a postve reservaton fee k > s necessary to acheve welfare optmalty, snce n the presence of demand uncertanty, only k can ncorporate a rsk factor (e.g., σ n (23 nto prcng. Ths reveals that current cloud bandwdth prcng schemes are neffcent n terms of provdng servce guarantee aganst demand fluctuaton. On the other hand, a usage prce p s not necessary: even f p =, the expected welfare s maxmzed as long as k satsfes (2. In ths case, the reservaton fee can be rased to compensate the loss from no usage fee. Furthermore, heterogeneous reservaton prces k 1,...,k N are necessary to acheve optmalty, each k dependng on the statstcal characterstcs of tenant s demand D. Ths conforms to the ntuton that tenants have dfferent degrees of demand volatlty, ncurrng dfferent costs for servce guarantees. For example, wthout multplexng, k depends on σ n (23: the more bursty a tenant s demandd, the more capacty that must be reserved to guard aganst fluctuaton, and thus the hgher the prce. In contrast, n terms of usage prcng, t s effcent enough to charge a homogeneous prcepfor every unt bandwdth consumed. 5. DISTRIBUTED SOLUTIONS As has been noted, the man challenge to solvng the optmal prcng problem (18 s that the reservaton cost c ( K(w s coupled among all the tenants and s not decomposable n general. One exstng approach to handle coupled objectve functons s to fnd the dual problem of (18 and to decompose the dual among all the tenants and the cloud provder by ntroducng auxlary varables. Such an approach s called consstency prcng [1, 21]. Subgradent methods are among the most popular technques to update the prces towards the optmalty of dual problems. However, they suffer from the curse of step szes. For the fnal output to be close to the optmalty, subgradent methods choose small step szes to updatek, leadng to slow convergence and many teratons of message-passng between the cloud and tenants. In ths paper, we propose two novel step-sze-free algorthms for prce updates that can quckly converge to the optmalty of (18. The frst algorthm, called Chaotc Prce Update, does not rely on decomposton at all: nstead, t resorts to teratve equaton updates based on the KKT condtons (2. The second algorthm, called the Cuttng-Plane Method, reles on dual decomposton but does not update k usng step szes: t s essentally a search algorthm

6 h(w ν(w h(w ν(w h(w ν(w k ( Intal Value 1 k (2 k ( Intal Value 2 k k k k (3 k (1 k (1 w (1 w (3 w w (2 w (4 w ( (a Convergence to Global Optmalty w w (1 w w ( (b Dvergence w w (c 1: Perodc; 2: Convergng. w Fgure 2: Behavor of prce update based on chaotc equatons n a one dmensonal case. o represents the startng pont(w (,k (. that locates {k } untl t s confned n a small regon. We gve a suffcent condton under whch chaotc prce update can acheve rapd convergence. The cuttng-plane method, whch s guaranteed to converge, s used to compensate chaotc prce update when the latter s not convergng. Note that our algorthms apply to a general convex cost functon c(k(w (n terms of w under any servce technology (e.g., multplexng tenant demands n groups. 5.1 Chaotc Prce Update Chaotc prce update s based on alternated phases of prce updates va the cost-prce relatonshp (2 and the tenant surplus maxmzaton equaton (16. Chaotc Prce Update. Denote w (t := [w (t 1,...,w(t N ]T. Set p b. Set k ( := β(µ + θ ( ǫσ for all and w ( = 1. For t =,1,..., repeat (1 Dstrbuted Surplus Maxmzaton. Passpandk (t to each tenant, whch returns: w (t+1 (2 Prce Update. Set k (t+1 := arg max w 1 E[ S (w,p,k (t ],. (26 := c ( K(w (t+1 K(w w. (27 w=w (t+1, (3 If w (t+1 w (t ξ, returnw = w (t+1,k = k (t+1. The above algorthm starts by settngk ( to be thek n the cost functon wthout multplexng. It then updatesk andw alternately by settng the current prcesk (t to be the margnal reservaton cost wth the current w (t, and by collectng the next w (t+1 from tenants who maxmze ther surpluses gven the current prces. Applyng the KKT condtons to (16, step (1 can also be vewed as solvng an equaton (1 ǫu (w (t+1 = p(1 ǫµ +k (t, (28 for w (t+1. Snce each tenant maxmzes ts expected surplus locally, the cloud provder does not have to know the utlty functon of each tenant, leadng to an teratve dstrbuted soluton. Compared wth Lagrangan dual decomposton based on consstency prcng [1,21], chaotc prce update represents a new way of handlng coupled objectves. Snce prce updates are based on equaton (27 rather than on updatng Lagrangan multplers, the algorthm s not concerned wth the choce of step szes that are requred by subgradent methods. We observe that the sequence{k (t } produced by equatons (27 and (28 could demonstrate sgnfcantly dfferent behavor under dfferent ntal values w ( and dfferent forms of utlty and cost functons. In other words, our algorthm demonstrates chaotc behavor, whose eventual outcome s senstve to ntal condtons and the structure of updatng equatons. Our objectve gong forward s to analyze the behavor of{k (t } and {w (t } and fnd out the condtons under whch the algorthm can acheve fast convergence. To smplfy notatons, we defne h ( (w (1 ǫ U (w bµ, (29 ν c( K(w (w. w (3 Recall that w := [w 1,...,w N] T, where w = w (p,k s the guaranteed porton chosen by tenant under optmal prcng. By the defnton of w, the optmalty condton s h (w = k = ν (w,. (31 Snce we have setp b, (28 can be wrtten as h (w (t+1 := (1 ǫ ( U (w (t+1 (t bµ = k. (32 Thus, the updatng rules (27 and (28 n chaotc prce update can be rewrtten as h (w (t+1 = k (t = ν (w (t,. (33 Let us llustrate the algorthm behavor usng the specal case of a sngle tenant. There are three scenaros where the algorthm can produce dramatcally dfferent results, as shown n Fg. 2. Snce utlty u (w s strctly concave and monotoncally ncreasng n [,1], and cost c ( K(w s strctly convex and monotoncally ncreasng n[,1], we have for all : { h(w > bµ, h (w <, w [,1] ν (w >, ν (w (34 w >, w : w 1. All three cases n Fg. 2 satsfy (34. In Fg. 2(a, {w (t 1 } always converges to w1 for any ntal value w ( 1 [,1], whereas n

7 Fg. 2(b, {w (t 1 } always dverges regardless of ts ntal startng pont. In Fg. 2(c, however, the behavor of {w (t 1 } crtcally depends on ts ntal value w ( 1. If w( 1 takes ntal value 1, w (t 1 wll eventually hop between two values alternately, wthout beng able to approachw1. On the other hand, fw ( 1 takes ntal value 2,w (t 1 wll converge to w1. We now gve a suffcent condton for the convergence of chaotc prce update n Theorem 1. Theorem 1. If for each = 1,...,N, we have mn(1 ǫ U (x x > N 2 c ( K(w j=1 w w j, (35 for all w between w ( = 1 and w (1, then usng chaotc prce update, k (t converges tok andw (t converges tow. Proof: Please refer to our techncal report [15] for the proof. The economc mplcaton behnd Theorem 1 s that the algorthm wll converge when the margnal utlty gan decreases faster than the margnal cost ncreases, asw ncreases. Ths techncal assumpton can be easly justfed, snce the margnal costc (K = β for addng network capacty (routers and swtches s decreasng at a fast pace n our economy. From the Proof of Theorem 1 n our techncal report [15], the convergence speed of {k (t } n chaotc prce update s dctated by P (w and Q (w, whch depend on h (w and ν (w, the margnal utlty gan and margnal cost n terms of w. Intutvely speakng, the larger the gap between the rates at whch the margnal utlty gan decreases and the margnal cost ncreases, the faster the convergence speed. As a result, n systems where U (w exceeds 2 c ( K(w / w 2 by a substantal margn, the step-szeoblvous chaotc prce update can acheve extremely fast convergence. 5.2 The Cuttng-Plane Method Chaotc prce update acheves fast convergence when condton (35 s met. A natural queston arses: Can we desgn an algorthm that s step-sze-free whle convergng under a wder range of condtons? Now we present such an algorthm that converges for arbtrary concave utlty functon U (w and arbtrary convex cost functonc ( K(w, wth a guaranteed convergence speed. Our basc dea s to apply the cuttng-plane method [8] to the dual problem of socal welfare maxmzaton, leadng to an alternatve formulaton of the optmal prcng problem Dual Problem of Socal Welfare Maxmzaton We ntroduce the dual problem of socal welfare maxmzaton, followng the framework of consstency prcng [1, 21]. Note that the socal welfare maxmzaton problem (8 can be rewrtten as max w,v (1 ǫ ( ( U(w bw µ c K(v (36 s.t. w = v, where the auxlary vectorvs ntroduced to facltate dual decomposton. To derve the dual problem of (36, we defne the Lagrangan L(w,v,k=(1 ǫ ( ( U(w bw µ c K(v +k T (v w =(1 ǫ ( U(w bw µ k w + k v c ( K(v. (37 Herek s the Lagrange multpler assocated wth theth equalty constrant w = v ; k can be nterpreted as a consstency prce, as t wll eventually steer v towards w, as explaned n [1, 21]. The Lagrange dual functon s q(k = supl(w,v,k, (38 w,v and the dual problem of socal welfare maxmzaton (36 s mn q(k. (39 k Note that there s no dualty gap between the dual problem (39 and prmal problem (36 by the strong dualty theorem [8], snce the prmal problem s convex optmzaton for any concaveu (w and convex c ( K(v. As a result, t suffces to solve the dual problem nstead of the prmal problem. In fact, the dual problem (39 s preferable because t enables dstrbuted algorthms due to a natural decomposton of q(k: q(k= sup ((1 ǫ ( U (w bw µ kw w ( +sup k v c ( K(v. v Ths dual decomposton decouples the objectve functon q(k so that the value of q(k can be found by solvng a surplus maxmzaton problem at each tenant : max w (1 ǫ ( U (w bw µ kw, for all, (4 and a proft maxmzaton problem at the cloud provder: max k v v c(k(v. (41 As these subproblems are ndependent of each other, the dual problem enables dstrbuted solutons by chargng each tenant a reservaton prce k and usage prce p = b (the revenue and cost related to usage cancel each other n (41. In contrast, the prmal problem (36 s not decomposable because of the coupled term c(k(v Dstrbuted Solutons va Cuttng Planes A tradtonal subgradent method wll fnd a subgradent g k of q(k at pontk, and update the prceskusng ths subgradent tmes a small step sze. For example, one of such subgradents s gven as below: Lemma 1. For any pont k R N, let the vector w = [ w ] be the optmal solutons to problem (4 and ṽ be the optmal soluton to problem (41. That s, w =argmax (1 ǫ ( U w (w bw µ kw,, (42 ṽ=argmax k v v c(k(v. (43 Then a subgradent of q(k at pont k s gven byg k = ṽ w. Proof: Please refer to our techncal report [15] for the proof. However, to ensure the convergence speed, the subgradent method requres tunng the step szes strategcally, whch s dffcult to mplement n realty. In contrast, the cuttng-plane method s stepsze-free: t s essentally a search algorthm based on the followng fact (see [8] for a proof: Lemma 2. Let vector g k R N be a subgradent of the objectve functon q(k at pont k R N,.e.,g k must satsfy q(x q(k+g T k(x k, x R N. (44

8 Bandwdth (Mbps Channel 295 Channel 241 Channel Tme (unt: 1 mnutes (a Three large channels Bandwdth (Mbps Channel 22 Channel Tme (unt: 1 mnutes (b Two small channels Fgure 3: Bandwdth consumpton tme seres of 5 representatve channels over a 2.5-day perod. Let k be the optmal soluton of the dual problem (39. Then k satsfesg T k(k k. Lemma 2 mples that f we know a pont k wth ts subgradent g k, we can confne our search for k wthn the half-space {x : g T k(x k }, snce the other half-space does not contan k. Hence, we can locate k up to a certan accuracy by teratvely rulng out a suffcent number of half-spaces, as descrbed below. The Cuttng-Plane Method. Set k ( = k, where k s an optmal soluton for the non-multplexng case. Set the ntal polyhedron to be P = {k k k ( }. It s clear that P contans an optmal soluton k. For t = 1,2,..., repeat: (1 Choose a pont k (t,whch s the center of gravty of P t 1, denoted k (t = CG(P t 1; (2 Fndng a subgradent g k (t of q(k atk (t ; (3 If g k (t ξ 1, returnk (t ; else, contnue; (4 Add a new cuttng plane g T k (t (k k (t to form the new polyhedron P t := P t 1 {k g T k (t(k k(t }. (45 Intutvely speakng, the above algorthm attempts to shrnk the volume of polyhedronp t that contans the optmal solutonk one teraton after another, untlk s contaned n a trvally small ball. In Step (2, the subgradent can be found usng Lemma 2. Now we can quantfy the communcaton cost of the above algorthm a crucal factor n a cloud envronment. Snce k (t s the center of gravty of P t 1, about half of the uncertanty s ruled out n each teraton. It can be proved that ( vol(p t 1 1 vol(p t 1.63 vol(p t 1. (46 e Therefore, the above algorthm converges exponentally fast. Furthermore, Lemma 2 shows that, n order to obtan a subgradent g k (t at k (t, the cloud provder can smply charge each tenant a usage fee p = b and a reservaton fee k = k (t, and expect a return w from each tenant; t obtansṽlocally. Such prce notfcaton and response are performed n each teraton for all the tenants n parallel. In other words, each executon of Step (2 ntroduces only one round of message passng between the cloud and tenants. Snce cloud-tenant communcaton happens only n Step (2, the total rounds of message-passng can be bounded as follows: Proposton 2. If the cuttng-plane method termnates when the dameter of the smallest Eucldean ball that contansp t s no greater than d, then n the worst case, the cuttng-plane method requres R = 1.51N log 2 ( max k ( d rounds of message passng between the cloud and tenants. (47 The above proposton s a well-known property of the cuttngplane method [8]. In contrast, the worst-case communcaton cost of subgradent methods are of the formo(n 1/α, whereαs the step sze. Clearly, f d s comparable to α, the cuttng-plane method may lead to much faster convergence due to thelog 2 ( operaton. It s worth notng that fndng the center of gravty CG(P t 1 requres heavy computaton. However, computaton cost does not pose a challenge for data centers that have superor computng power, whereas communcaton cost (convergence speed s the major bottleneck. The cuttng-plane method converges faster, reducng the rounds of message passng between tenants and the cloud, yet at the expense of computatonal cost. Such a property s n fact desrable n the cloud. To summarze, the cuttng-plane method converges for arbtrary concave utltyu (w and convex costc ( K(w, and can be used to compensate chaotc prce update when the latter s not convergng. 6. DEMAND STATISTICS ESTIMATION WITH A FACTOR MODEL Recall that both algorthms for fndng the optmal prcng polcy need µ, σ and Σ as nputs: µ and σ are used for the surplus maxmzaton at each tenant n (26 and (42, whle the demand covarance matrx Σ s used to calculate the servce cost c(k(w n the presence of multplexng n (27 and (43. In ths secton, we address the practcal ssue of predctng demand statstcs based on demand hstory, whch can be obtaned from cloud montorng servces such as Amazon CloudWatch [2] at a fne granularty (e.g., at a frequency of 5 mnutes n CloudWatch. As has been mentoned, n ths paper, we target applcatons whose bandwdth demand patterns are tractable and predctable to some extent. Vdeo access s one example of such applcatons, wth clear durnal patterns and the tme-of-the-day effect [17], n the sense that a popular vdeo almost always sees ts peak (or trough demand around the same tme of day. Our study s based on a large dataset collected from thousands of on-demand vdeo channels n a commercal VoD system [3] durng the 28 Summer Olympcs. The vdeo genres are not lmted to Olympcs, but range from TV epsodes to sports and from moves

9 Mbps C3 C2 C Tme (unt: 1 mnutes Fgure 4: The frst 3 prncpal components C 1 C 3 n bandwdth consumpton seres of 468 channels durng 2 days va prncpal component analyss. Varance Explaned (% Prncpal Component Fgure 5: Varance explaned n bandwdth consumpton seres of 468 channels n 2 days. 2nd Prncpal Component Channel Channel 298 Channel 317 Channel 241 Channel st Prncpal Component Fgure 6: Data projected onto the frst 2 prncpal components. Each pont s one of the 468 channels. to news. Fg. 3 shows the aggregate bandwdth demand n each channel for 5 representatve channels over a 2.5-day perod. We have four observatons about the dataset. Frst, the workload dataset conssts of a large number of small unpopular channels, such as those n Fg. 3(b, domnated by a small number of large popular channels, such as those n Fg. 3(a. Second, there s a durnal perodcty n the access pattern of each vdeo channel. Thrd, a channel s popularty evoluton may follow some trends over days. For example, bandwdth consumpton n channels 241 and 317 exhbts a downward trend over the 2.5 days n Fg. 3(a, wth 144 tme perods representng one day. Fnally, both the durnal perodcty and daly trends become vague n small channels, such as n channel 22 n Fg. 3(b. Our pror work has proposed to use seasonal ARIMA processes to predct bandwdth seres n each ndvdual channel [17] at a fne granularty of 1 mnutes. The predcton s based on a regresson of hstorcal demand n the most recent tme perods, as well as demand around the same tme n prevous days. However, ths method has a shortcomng that a separate statstcal model needs to be traned for every channel and thus does not scale to a large number of channels. Also, ths approach performs poorly for small channels, e.g., channel 22, or ll-behaved channels, e.g., channel 317. In both types of channels, the daly repetton pattern s obscured by varous random factors. To tackle these problems, we use a factor model to account for demand evolutons,.e., the demand seres{d (t} of each channel can be vewed as drven by M uncorrelated underlyng factors C 1(t,...,C M(t wth a zero-mean random shock e(t: D (t = α 1C 1(t+...+α MC M(t+e(t,. (48 Mbps (Log Scale Ch Ch Ch Mean Bandwdth Consumpton 1 RMSE of PCA based Predcton RMSE of Indvdual Channel Predcton Channel Index n Log Scale (Sorted by Channel Sze Fgure 7: Root mean squared errors (RMSEs of the two predcton methods over test perod (1.25 days, compared to mean bandwdth consumpton n each channel. The three largest channels are channels 295, 241 and 317. If the coeffcentsα 1,...,α M can be learned statstcally, we wll be able to forecast {D (t} by predctng factor movements frst. As noted from Fg. 3, channel demands exhbt co-movements. Ths nspres us to mne the factors whle learnng ther coeffcents from the collectve demand hstory of all the channels. We use prncpal component analyss (PCA to fnd such underlyng factors. Gven N demand seres {D 1(t},...,{D N(t}, PCA apples an orthogonal transformaton to these N demand seres to obtan a small number of uncorrelated tme seres {C 1(t},..., {C M(t} called the prncpal components. Ths transformaton s defned n such a way that data projected onto the frst prncpal component has as hgh a varance as possble (that s, accounts for as much of the varablty n data as possble, and each succeedng component n turn has the hghest varance possble. We perform PCA for all the 468 channels that are onlne n a 2-day perod (tme Fg. 4 plots the frst 3 prncpal component seres n the data. We can see the frst component C 1 explans the durnal perodcty shared by all the channels. The second component C 2 accounts for the downward daly trend, whch s salent n channel 317 as popularty dmnshes and less salent n channel 295. A further check of Fg. 5 reveals that the frst 1 prncpal component seres explan 99% of the data varablty, whch are suffcent to model the factors underlyng all the demand evoluton. However, dfferent channels have dfferent dependences on each factor. Fg. 6 shows the 468 demand seres projected onto the frst 2 components,.e., the pont (α 1,α 2 for all. Wthout surprse, the dependence on the frst component,α 1, ndcates how large the channel s. In contrast, the dependence on the second component, α 2, accounts for how fast end-users may lose nterest n channel. Channel 295 has a low α 2, ndcatng almost no decrease n popularty over days. Channel 241 has a moderate α 2, showng a slghtly downward trend. Channel 317 has a largeα 2, exhbtng a dramatc decrease of popularty just on the second day. To predct the demand means µ(t and covarances Σ(t for {D (t : = 1,...,N} at tme t, we frst predct the prncpal components C 1(t,...,C M(t for M = 1 based on the hstory and obtan forecasts about ther meansĉ(t = [Ĉ1(t,...,ĈM(t]T and ther covarances ˆΣ C(t. We further predct the error seres e(t to obtan ts forecast ê(t and error varance ˆσ e(t. 2 Note that ˆΣ C(t s a dagonal matrx because the prncpal components are uncorrelated. Denote the coeffcent matrx as A N M = [α m], 468

10 Bandwdth (Mbps 1 5 Orgnal data PCA based Predcton Tme (unt: 1 mnutes Bandwdth (Mbps Orgnal data Indvdual Predcton Tme (unt: 1 mnutes Fgure 8: 1-mnute-ahead condtonal mean predcton n channel 172 over a test perod of 1.25 days. Mbps Departure from Mean Forecast Predcted Standard Devaton Tme (unt: 1 mnutes (a PCA-based Predcton Mbps Departure from Mean Forecast Predcted Standard Devaton Tme (unt: 1 mnutes (b Indvdual Predcton Forecast Error Quantles Forecast Error Quantles Standard Normal Quantles Standard Normal Quantles (a PCA-based Predcton (b Indvdual Predcton Fgure 9: The departure of actual bandwdth consumpton from ts condtonal mean forecast and the predcted standard devaton of bandwdth consumpton n channel 172. Fgure 1: Q-Q plot of condtonal mean forecast errors n channel 172 over the test perod (1.25 days, n reference to Gaussan quantles. = 1,...,N, m = 1,...,M. We can therefore forecast µ(t and Σ(t as ˆµ(t=AĈ(t+ê(t 1, (49 ˆΣ(t=AˆΣ C(tA T + ˆσ e(t [1,...,1], 2 (5 where 1 s an all-one column vector of length N and [1,...,1] s an all-one matrx of szen N. We model each prncpal component seres usng a low-order seasonal AIRMA model [9]. Snce {C 1(t} clearly shows daly perodcty, we model {C 1(t C 1(t 144} as an ARMA(1,1 process, so that the forecastĉ1(t s regressed from both the prevous valuec 1(t 1, the values one day beforec 1(t 144,C 1(t 145, and random nose terms. All other components {C (t} for 2 do not exhbt perodcty. We thus use ARMA(1,1 processes to model these prncpal components. The condtonal varances of all the component seres are forecasted usng GARCH(1, 1 models [9, 16]. Snce the components are orthogonal, we do not need to forecast ther covarances. For detals of usng seasonal AIRMA models and GARCH models for vdeo traffc forecast, please refer to [16, 17]. We compare PCA-based predcton wth ndvdual channel predcton over a test perod of 1.25 days. Each predcton s made based on the tranng data of only the prevous 1.25 days, whch are a lttle more than one day to ncorporate perodcty. The root mean squared errors (RMSEs of both approaches for all the 468 channels are summarzed n Fg. 7. The channel ndces are sorted n descendng order of the channel sze. We can see that the PCA-based approach outperforms ndvdual predctons regardless of channel szes. For large channels, the rato of RMSE over mean bandwdth consumpton s less than 15% n most cases usng the PCA-based approach. To zoom n, we take channel 172 as an example. Fg. 8 compares the condtonal mean predctons produced by the PCA-based approach wth those produced by ndvdual predcton. We ob- serve that ndvdual predcton tends to oscllate drastcally, whle the PCA-based approach can better dentfy both perodcty and downward trends. One reason s that the drvng factors found by PCA are weghted averages over all the channels, wth channelspecfc erratc noses smoothed out, exhbtng co-movements of all the channels. The standard devaton forecast of channel 172 s plotted n Fg. 9. Even though there s a bg gap between real demand and ts condtonal mean forecast around tme 17, the GARCH(1, 1 model s able to forecast a larger demand varance at ths tme, whch wll be leveraged by the cloud to allocate more capacty to guard aganst performance rsks, usng the technologes n Sec It s worth notng that we do not assume that demand can always be perfectly forecasted: the entre pont of varance or volatlty forecast va GARCH s to estmate the devaton of actual demand from the condtonal mean predcton and enable rsk management n a probablstc sense. Fg. 1 shows the Q-Q plot of forecast errors. We observe that wth PCA, the actual demand wll oscllate around ts condtonal mean forecast more lke a Gaussan process. Ths also substantates the belef that each D behaves lke a Gaussan random varable. Last but not least, the PCA-based approach has a lower complexty: t nvolves tranng a seasonal ARIMA model for each of the 1 prncpal components, together wth fndng these components from the 468 channels usng PCA. Once the models are traned, forecastng s smply a lnear regresson wth neglgble runnng tme. In contrast, ndvdual predcton has to tran a seasonal ARIMA model for each of the 468 channels separately, leadng to a much hgher complexty. 7. TRADING SIMULATIONS In ths secton, we smulate a computerzed bandwdth reservaton and tradng envronment based on our proposed algorthms. The smulaton operates n rounds of 1 mnutes. Before the start

11 Mean Prce Dscount of Tenants (% Chaotc Prce Update Cuttng Plane Method Subgradent Method Tme (unt: 1 mnutes Tme Averaged Guaranteed Porton Chaotc Prce Update Cuttng Plane Method Subgradent Method Tenants Fgure 11: The mean prce dscount of all 1 tenants vs. tme. of each 1-mnute perod, the cloud provder has predcted the demand mean and covarances n ths perod and nformed each tenant about ts specfc µ and σ. When the perod starts, the dstrbuted prce negotaton process mmedately starts untl convergence. Snce n our partcular problem, the cloud provder has superor computaton power (even for fndng polyhedra centrod, the delay s manly due to the teratve message passng of prces and guaranteed portons between tenants and the cloud. We compare three algorthms: chaotc prce update, the cuttng-plane method and subgradent method, n terms of the convergence speed and optmzaton accuracy. We consder 1 vdeo channels of dfferent szes and statstcs n the UUSee demand traces over a test perod of 81 mnutes. We assume each channel s a tenant that reles on the cloud for servcng the vdeo requests from ts end-users. We nput such demand traces to our prcng framework and check the algorthm effcency n the challengng case that predcton and optmzaton are to be carred out every 1 mnutes. If the algorthms work for a 1-mnute frequency, they wll be competent for lower operatng frequences, such as on an hourly bass. We consder utlty functons of the form (2. Under a Gaussan approxmaton ofd, each tenant wll have an expected utlty E[U (w ] = α w µ e A (1 w µ A2 (1 w 2 σ 2. (51 k t 1 The frst term on the rghthand sde corresponds to the expected revenue of each tenant made from servng the demandw D, whle the second term models a reputaton loss whch s convex and ncreasng n terms of the unfulflled demand. In our smulaton, we set α = 1 and A =.5. Snce dfferent tenants have dfferent µ and σ, ther utltes are heterogeneous. We set the margnal cost of allocatng bandwdth capacty to be β := c (K =.5, and assume that the cloud provder has an outage probablty ofǫ =.1. We set the algorthm termnaton condtons as follows. In each teraton, f the change n etherw,k org (g k (t = [g (t 1,...,g(t N ] s below some threshold, ts w s not updated (usng messagepassng. Chaotc prce update wll stop f w (t w t 1 <.1 or t has run for 1 teratons. The cuttng-plane method wll stop f k (t <.5 or t has run for 1 teratons. The subgradent method, as the benchmark, wll stop f g (t g (t 1 <.5. In order to be generous to the benchmark algorthm, we set the maxmum number of teratons for the subgradent method to 2. Note that n the subgradent method, the step sze of prce updates cannot be too small, whch ncurs slow convergence; t cannot be too bg ether, n whch case the fnal output wll be far away from Fgure 12: The guaranteed porton of each tenant averaged over all test perods Emprcal CDF Chaotc Prce Update Subgradent Method Convergence Iteraton of the Last Tenant Fgure 13: The CDF of the maxmum convergence teraton of all tenants n each test perod. the real optmal value. We optmze such a step sze and set t to.1 for prce updates. The other two algorthms are step-sze-free. We frst compare the algorthm outputs upon termnaton. Note that wth multplexng, the fnal optmal prce k for each tenant should be lower than ts ntal value k ( = β ( µ + θ(ǫσ, whch s also the optmal prce wthout multplexng. We defne 1 k/k ( as the prce dscount that tenant enjoys from multplexng. Fg. 11 plots the mean prce dscount averaged over all the tenants n each test perod. We observe that both chaotc prce update and the cuttng-plane method brng more dscounts to tenants than the subgradent method. We further check the mean guaranteed porton chosen by each tenant averaged over all test perods n Fg. 12, whch shows that most tenants choose a guaranteed porton close to 1 and the three algorthms are close to each other. Ths means although the three algorthms may reach a smlar level of socal welfare, the subgradent algorthm s not so good at fne-tunng the optmal prces for tenants wth a guaranteed porton close to 1. Fnally, we check the communcaton overhead of all three algorthms. We defne an teraton of message passng as a round-trp communcaton n whch the cloud provder passes the prces to a tenant, whch returns a chosen guaranteed porton. We observe that for chaotc prce update, the convergence teraton of the last tenant (worst-case convergence teraton n each test perod s almost always less than 1. The CDF of the worst-case convergence teraton of chaotc prce update s plotted n Fg. 13. In the same fgure, we can observe that 4% of the tme, the subgradent method needs 2 rounds to converge, whle 6% of the tme, t converges between 25 and 2 rounds. The cuttng plane method always takes 1 rounds to converge, whch are half of the maxmum rounds needed by the subgradent method.

12 Takng both performance and speed nto consderaton, chaotc prce update largely outperforms the other two. The cuttng-plane method can also acheve better performance than the subgradent method wth better prce dscounts reached wthn 1 rounds. Therefore, a practcal strategy s to use chaotc prce update frst, and f t does not converge n 2 rounds, swtch to the cuttng-plane or subgradent method. 8. CONCLUDING REMARKS Current-generaton cloud computng platforms do not provde bandwdth guarantees, mpedng the cloud adopton by tenants runnng QoS senstve applcatons. Recent advancements n datacenter engneerng augment the cloud-tenant nterface wth bandwdth reservaton enabled. As bandwdth reservaton becomes techncally feasble, new models are needed to prce the bandwdth guarantees to compensate the pay-as-you-go model whch only prces the usage. In ths paper, we propose a guaranteed cloud servce model, where each tenant does not have to estmate the absolute amount of bandwdth t needs to reserve t smply specfes a percentage of ts demand from end-users that t wshes to serve wth guaranteed performance, whch we call the guaranteed porton, whle the rest of the demand wll be served wth best efforts as the current cloud provders do. The cloud provder wll estmate tenant demands through workload analyss and guarantee the performance n a probablstc sense. The above process s repeated n small perods such as hours or tens of mnutes. Our man contrbuton s to farly prce such guaranteed servces n each perod. In contrast to the unform usage prcng model, we prce bandwdth reservatons heterogeneously for tenants based on ther workload statstcs such as burstness and correlaton. It turns out to be computatonal challenge to fnd the optmal prces that maxmze the expected socal welfare under demand uncertanty. To address ths challenge, we propose two novel dstrbuted algorthms based on teratve equaton updates and cuttng-plane methods, whch are oblvous to the choce of step szes. We also propose practcal algorthms to predct demand statstcs based on a factor model. Trace-drven smulatons show that both algorthms acheve faster convergence and better performance than subgradent methods. Gven the abundant computng power and workload data n the cloud, our bandwdth reservaton and algorthmc prcng system operates effectvely at a fne granularty of as small as 1 mnutes. 9. REFERENCES [1] Amazon Cluster Compute, applcatons/. [2] Amazon Web Servces. [3] UUSee Inc. [Onlne]. Avalable: [4] Four Reasons We Choose Amazon s Cloud as Our Computng Platform. The Netflx Tech Blog, Dec [5] D. Applegate, A. Archer, V. G. S. Lee, and K. Ramakrshnan. Optmal Content Placement for a Large-Scale VoD System. In Proc. ACM Internatonal Conference on Emergng Networkng Experments and Technologes (CoNEXT, 21. [6] M. Armbrust, A. Fox, R. Grffth, A. D. Joseph, R. Katz, A. Konwnsk, G. Lee, D. Patterson, A. Rabkn, I. Stoca, and M. Zahara. A Vew of Cloud Computng. Communcatons of the ACM, 53(4:5 58, 21. [7] H. Ballan, P. Costa, T. Karaganns, and A. Rowstron. Towards Predctable Datacenter Networks. In Proc. of SIGCOMM 11, Toronto, ON, Canada, 211. [8] D. P. Bertsekas, A. Nedc, and A. E. Ozdaglar. Convex Analyss and Optmzaton. Athena Scentfc, 23. [9] G. E. P. Box, G. M. Jenkns, and G. C. Rensel. Tme Seres Analyss: Forecastng and Control. Wley, 28. [1] M. Chang, S. H. Low, A. R. Calderbank, and J. C. Doyle. Layerng as optmzaton decomposton: A mathematcal theory of network archtectures. Proceedngs of the IEEE, 95(1: , Jan. 27. [11] R. L. Grossman. The Case for Cloud Computng. IT Professonal, 11(2:23 27, March-Aprl 29. [12] C. Guo, G. Lu, H. J. Wang, S. Yang, C. Kong, P. Sun, W. Wu, and Y. Zhang. SecondNet: a Data Center Network Vrtualzaton Archtecture wth Bandwdth Guarantees. In Proc. of ACM Internatonal Conference on Emergng Networkng Experments and Technologes (CoNEXT, 21. [13] G. Gürsun, M. Crovella, and I. Matta. Descrbng and Forecastng Vdeo Access Patterns. In Proc. of IEEE INFOCOM Mn-Conference, 211. [14] D. Kossmann, T. Kraska, and S. Loesng. An Evaluaton of Alternatve Archtectures for Transacton Processng n the Cloud. In Proc. of Internatonal Conference on Management of Data (SIGMOD, 21. [15] D. Nu, C. Feng, and B. L. Prcng Cloud Bandwdth Reservatons under Demand Uncertanty. Techncal report, Unversty of Toronto, [16] D. Nu, B. L, and S. Zhao. Understandng Demand Volatlty n Large VoD Systems. In Proc. of the 21st Internatonal workshop on Network and Operatng Systems Support for Dgtal Audo and Vdeo (NOSSDAV, 211. [17] D. Nu, Z. Lu, B. L, and S. Zhao. Demand Forecast and Performance Predcton n Peer-Asssted On-Demand Streamng Systems. In Proc. of IEEE INFOCOM Mn-Conference, 211. [18] D. Palomar and M. Chang. A tutoral on decomposton methods for network utlty maxmzaton. IEEE J. on Sel. Areas n Communcatons, 24(8: , Aug. 26. [19] J. C. Panzar and D. S. Sbley. Publc Utlty Prcng under Rsk: The Case of Self-Ratonng. The Amercan Economc Revew, 68(5: , Dec [2] J. Schad, J. Dttrch, and J.-A. Quane-Ruz. Runtme Measurements n the Cloud: Observng, Analyzng, and Reducng Varance. In Proc. of VLDB, 21. [21] C. W. Tan, D. Palomar, and M. Chang. Dstrbuted optmzaton of coupled systems wth applcatons to network utlty maxmzaton. In Proc. of IEEE Internatonal Conference on Acoustcs, Speech and Sgnal Processng, Toulouse, France, May 26. [22] C. Wlson, H. Ballan, T. Karaganns, and A. Rowstron. Better Never Than Late: Meetng Deadlnes n Datacenter Networks. In Proc. of SIGCOMM, 211. [23] C. Wu, B. L, and S. Zhao. Mult-Channel Lve P2P Streamng: Refocusng on Servers. In Proc. of IEEE INFOCOM, 28. [24] H. Yn, X. Lu, F. Qu, N. Xa, C. Ln, H. Zhang, V. Sekar, and G. Mn. Insde the Brd s Nest: Measurements of Large-Scale Lve VoD from the 28 Olympcs. In Proc. ACM Internet Measurement Conference (IMC, 29. [25] M. Zahara, A. Konwnsk, A. D. Joseph, Y. Katz, and I. Stoca. Improvng MapReduce Performance n Heterogeneous Envronments. In Proc. of OSDI, 28.