Towards Optimal Capacity Segmentation with Hybrid Cloud Pricing

Transcription

1 Towards Opimal Capaciy Segmenaion wih Hybrid Cloud Pricing Wei Wang, Baochun Li, and Ben Liang Deparmen of Elecrical and Compuer Engineering Universiy of Torono Absrac Cloud resources are usually priced in muliple markes wih differen service guaranees. For example, Amazon EC2 prices virual insances under hree pricing schemes he subscripion opion (a.k.a., Reserved Insances), he pay-as-yougo offer (a.k.a., On-Demand Insances), and an aucion-like spo marke (a.k.a., Spo Insances) simulaneously. There arises a new problem of capaciy segmenaion: how can a provider allocae resources o differen caegories of pricing schemes, so ha he oal revenue is maximized? In his paper, we consider an EC2-like pricing scheme wih radiional pay-as-you-go pricing augmened by an aucion marke, where bidders periodically bid for resources and can use he insances for as long as hey wish, unil he clearing price exceeds heir bids. We show ha opimal periodic aucions mus follow he design of m+-price aucion wih seller s reservaion price. Theoreical analysis also suggess he connecions beween periodic aucions and EC2 spo marke. Furhermore, we formulae he opimal capaciy segmenaion sraegy as a Markov decision process over some demand predicion window. To miigae he high compuaional complexiy of he convenional dynamic programming soluion, we develop an approximae soluion ha has significanly lower complexiy and is shown o closely approach he opimal revenue. I. INTRODUCTION Cloud compuing ransforms a large par of he IT indusry by fulfilling he long-held ambiious vision of compuing as a uiliy. Users pay o access compuing resources delivered over he Inerne, jus as hey pay o use waer and elecriciy. Like oher uiliy providers, many cloud providers charge heir cusomers in a regular pay-as-you-go manner [], [2], [3], [4], [5], [6], [7]. A provider ses a saic or infrequenly updaed per-uni price, and users pay for only wha hey use. Along wih he pay-as-you-go offer, here are wo addiional pricing schemes widely adoped in cloud markes: he subscripion opion [2] and he spo marke []. In he former scheme, a user pays a one-ime subscripion fee o reserve one uni of resource for a cerain period of ime. A user can use he reserved resource whenever i wans during he subscripion period, under a significanly discouned usage fee. The spo marke, on he oher hand, is an aucion-like mechanism. Users periodically submi bids o he provider, who in urn poss a series of spo prices. Users gain resource access and can use he resources for as long as hey wish, unil he spo price rises above heir bids, a which ime hey are rejeced. Insead of exclusively selling compuing services via a single pricing channel, many providers use muliple pricing schemes simulaneously o charge for cloud services. For example, boh pay-as-you-go pricing and subscripion opion Price Pay-as-you-go demand Price Spo marke demand Cloud resources Subscripions Subscripion demand Fig.. The capaciy segmenaion problem: how do we allocae resources o each pricing model so ha he revenue is maximized? are offered in [2], [4], [7]. Amazon EC2, on he oher hand, leases virual insances hrough all hree pricing channels []. Compared wih leasing cloud resources via a single pricing channel, muliple caegories of pricing sraegies are more aracive o a provider for wo reasons. Firs, wih a combined pricing srucure, he deficiency of one business mode is compensaed by anoher. For example, he demand uncerainy of pay-as-you-go pricing is compensaed by risk-free income from subscripion users bearing long-erm usage commimen. Second, he use of muliple pricing caegories expands he marke demand by offering more flexible choices o accommodae differen ypes of users. For example, price-sensiive users who canno afford he pay-as-you-go price now have a chance o gain access o resources by bidding in he spo marke. The co-exisence of muliple pricing channels, as illusraed in Fig., raises a new, and challenging, quesion o a provider: wih limied resources available, how do we allocae hem o each pricing channel so ha he overall revenue is maximized? To answer his quesion, in his paper, we consider he problem of capaciy segmenaion wih wo pricing models applied in parallel, i.e., he regular pay-as-you-go offer and periodic aucions. The laer allows users o periodically bid for resources in a sequence of aucions. In each period, a uniform ake-i-or-leave-i price is posed o clear he marke. The winners can use he resources for as long as hey wish, unil he clearing price rises above heir bids. The provider s problem is o opimally allocae is resources o he wo pricing channels, based on supply and demand, o maximize he obained revenue. We choose he aforemenioned wo pricing models as building blocks of he capaciy segmenaion problem for wo reasons. Firs, due o he upfron usage commimen borne

2 by users, subscripion demand generaes long-erm risk-free income o he provider. In his sense, subscripion requess are more preferred o providers [7] and are always fulfilled a he highes prioriy. We herefore focus only on he remaining wo pricing models. Second, given ha Amazon reveals no deailed informaion on how he spo price is deermined [], i is unclear how he spo marke is operaed. We hence urn o periodic aucions as hey share similar pricing forms as he EC2 spo marke (i.e., boh are bid-based). To he bes of our knowledge, we are he firs o consider such a capaciy segmenaion problem in cloud markes wih hybrid pricing. We make wo original conribuions in his paper. Firs, we show ha an opimal design for he aucion channel mus follow he form of he m+-price aucion wih seller s reservaion price. Conrary o he well-known resul ha, in general, bidders end o underbid in a uniform-price aucion (including he m+-price aucion), we show ha, in cloud environmens, however, he m+-price aucion is essenially ruhful wih wo-dimensional bids. Ineresingly, such ruhfulness is also expeced in he EC2 spo marke. In his case, replacing Amazon s design wih periodic aucions does no change user behaviours, resuling in he same marke response. Second, we formulae he opimal capaciy segmenaion problem as a Markov decision process (MDP). However, opimally solving his MDP wih convenional dynamic programming is compuaionally prohibiive, especially for a large provider wih enormous capaciy. By uilizing some special bounding srucure of he opimizaion problem, we furher develop an approximae soluion ha significanly reduces he compuaional complexiy from O(C 3 ) o O(C 2 ), wih C being he capaciy of he provider. We condiionally bound he compeiive raio of he sub-opimal revenue o he opimal one. This analyical bound, ogeher wih exensive simulaions, suggess ha he approximaion closely approaches he opimal soluion. The remainder of his paper is organized as follows. In Sec. II, we briefly survey he relaed work. Our model and noaions are inroduced in Sec. III. In Sec. IV, we characerize he revenue obained in he aucion marke, and presen an opimal design wih maximum revenue. We also discuss is connecions o he EC2 spo marke. In Sec. V, we firs presen raionales for he use of muliple pricing models. We hen show ha achieving opimal segmenaion is compuaionally prohibiive. For his reason, we presen a near-opimal soluion where he compuaional complexiy is significanly reduced by idenifying some opimizaion srucures in he problem. Exensive simulaion sudies are presened in Sec. VI. Sec. VII concludes he paper. We emphasize ha periodic aucions are no equivalen o he EC2 spo marke. While spo users are price-akers unaware of how spo prices are produced, aucion bidders, on he oher hand, have full knowledge on pricing deails and can affec he clearing price via sraegic bidding. II. RELATED WORK Three pricing models are now widely adoped in cloud markes, i.e., he regular pay-as-you-go offer, he subscripion opion, and he newly invened spo marke. The providers adverise ha having muliple pricing schemes benefis cloud users by lowering heir coss [], [7]. Exising works also presen some user sraegies o swich beween differen pricing markes o cu he cos [8], [9]. However, lile has been addressed from he cloud provider s perspecive, on how heir resources should be allocaed o differen markes o maximize revenue. Relevan research works in he lieraure include [0], which invesigaes he resource allocaion problem in eiher saic pricing or variable pricing, by solving a saic opimizaion program, and [], which presens a dynamic aucion-based model for resource allocaions in he grid sysem. None of hese works considers he coexisence of muliple differen pricing markes. We believe he key o solve he capaciy segmenaion problem lies in undersanding he EC2 spo marke. However, since Amazon reveals no deailed pricing informaion, i remains unclear how he spo price is deermined. Despie Amazon s claim ha he price is calculaed based on marke demand and supply [], some recen works conjecure ha he price is in fac arificially se via some random process [9], [2]. In his paper, we consider periodic uniform-price aucions, as hey share similar pricing forms as he EC2 spo marke. Unlike general uniform-price aucions discussed in economics lieraure [3], [4], in he cloud environmen, parial fulfillmen is no acceped: a bidder is eiher rejeced or having all requesed insances [] being fulfilled. For his reason, our design avoids he well-known effec of demand reducion observed in general uniform-price aucions ha bidders have an incenive o bid lower han heir rue values [3], [4] and is proved o be ruhful wih wo-dimensional bids. We noe ha here exis some works in he lieraure of economics ha discuss a similar resource allocaion problem in he reail marke, where wo pricing channels are used o sell producs, he aucion marke and he regular pay-as-you-go pricing [5], [6]. However, neiher he model nor he analysis applies o he cloud environmen. Firs, heir models are based on sales markes, where resources are sold o cusomers and will never be reclaimed and made available o ohers. In conras, cloud insances are leased o users and can be reused by ohers once he resources are released by previous owners. Second, heir analysis relies on a srong assumpion ha each cusomer is resriced o ask for only one uni of produc, which is clearly no he case for cloud users. We noe ha [5] furher assumes ha he seller capaciy is infinie. I is worh menioning ha opimal periodic aucions have also been sudied in he reail marke in [7], [8], bu he same problems menioned above render hose works inapplicable in cloud markes.

3 III. SYSTEM MODEL Suppose a cloud provider has allocaed a fixed capaciy C for a cerain ype of virual insances, i.e., a any given ime, up o C insances of ha ype can be hosed. All hese insances are leased in wo pricing channels, an aucion marke and a pay-as-you-go marke, simulaneously. We ake discree ime horizons indexed by =, 2,... in he following analysis. A. User Model Pay-as-you-go users. The pay-as-you-go marke offers guaraneed services. Users can run heir insances for as long as hey wish, and are charged wha hey used based on a consan regular price p r. In paricular, denoe by i,j he running ime of insance j hosed for user i. User i is hen charged p r i,j for using ha insance. To make he analysis racable, we ake a echnical assumpion ha i,j s are i.i.d. exponenial. In discree seings, his implies ha i,j follows he geomeric disribuion wih p.m.f. P ( i,j = k) =q( q) k, where q is he probabiliy ha a currenly running insance will be erminaed by is user in he nex period. Therefore, he expeced overall paymen for using one insance is E[p r i,j ]=p r E[ i,j ]=p r /q. Alhough he i.i.d. exponenial insance running ime is a simple model and pracical user behaviours may no follow i, aking his echnical assumpion allows racable analysis and has been shown o give ineresing insighs ino pracical sysems. We also noe ha such exponenial resource usage ime is widely adoped in economics lieraures o analyze renal markes [9], [20]. Because p r is consan, pay-as-you-go users have no purchasing sraegy as he aucion bidders have. We assume here are Rr insance requess received a ime, and if he available capaciy allocaed o he pay-as-you-go marke is below Rr, some users do no receive heir requesed insances. The exac mechanism for user admission (e.g., firs-come, firs-served) is unimporan o he problem under consideraion since he same price p r is charged for each insance. Users in he aucion marke. Insances purchased in he aucion marke offer no service guaranees and will be erminaed by he provider whenever he bid has been exceeded by he clearing price. Suppose a ime, here are Na bidders joining he aucion. Each bidder i ( apple i apple Na) wishes o access n i insances and has a maximum affordable price v i, also known as he reservaion price, for using one insance a one period. User i hen submis a wo-dimensional bid (ri,b i ) requesing ri insances wih a bid price b i. Noe ha user i could sraegically misrepor is bid (i.e., b i 6= v i or ri 6= n i) as long as i believes ha his is more beneficial. Afer all bids are colleced, he cloud provider runs he aucion and charges a ake-i-or-leave-i clearing price p a o all winners: each user i wih b i >p a (resp. b i <p a) eiher has is new requess fulfilled (resp. rejeced) or has is running insances coninued (resp. erminaed). Those wih b i = p a may or may no be acceped depending on he specific aucion mechanism. The value of p a is calculaed based on some specified mechanism ha is publicly known o all bidders. We herefore define user i s uiliy a ime as follows: u i(ri,b ni v i ri i)= p a, if b i >p a and ri n i ; () 0, oherwise. Here, boh n i and v i are privae informaion known only o user i, and are disribued wih join p.d.f. f n,v and c.d.f. F n,v on he suppor [n, n] [v, v]. The user i s problem is o find he opimal bid such ha he uiliy is maximized, i.e., max r i,b u i i (r i,b i ). I is worh menioning ha he aucion described above is subsanially differen from he uniform-price aucion considered in he lieraure of economics [3], [4], as bidders in he laer mechanism accep parial fulfillmen and have differen uiliy funcions oher han (). B. The Problem of Opimal Capaciy Segmenaion The cloud provider aims o opimally allocae is available capaciy o boh he pay-as-you-go and aucion markes, o maximize is obained revenue. Le he available capaciy a ime be C. In addiion o knowing he exac number of requess in he curren ime slo, we assume ha he provider may predic he demand in he near fuure: i knows he disribuions of Na (he user number in he aucion marke) and Rr (he oal requess in he pay-as-you-go marke) for apple T = + w, wih w being some predicion window. Noe ha forecasing fuure demand has already been addressed in some lieraure [0], [2]. Given C a ime, denoe by (C ) he maximum expeced aggregae revenue obained from o T. Le a(c) and r(c) be he revenues of allocaing c insances o he aucion and he regular pay-as-you-go markes, respecively. The problem of opimal capaciy segmenaion is o find he opimal capaciy allocaions o he wo markes such ha he revenue colleced wihin he predicion window is maximized. This can be expressed in he following recursive form: apple (C )=E max a(ca)+ r (C Ca) 0appleCa applec + E C + + (C + ), (2) where Ca is he capaciy allocaed o he aucion marke, T and he boundary condiions are + (c) =0for all c = 0,,...,C. Since he pay-as-you-go price is infrequenly updaed [], in his work we consider only he shorer ime-scale problem of capaciy segmenaion given a fixed p r. Then we have pr c/q, if c apple R r(c) = r; p r Rr/q, oherwise. Noe ha a discussion on how o opimize p r can be conduced based on he proposed revenue maximizing mehod, bu i addiionally requires knowledge of he ye unknown supplydemand relaion and hence is lef open for fuure research. To deermine he value of C + in (2), we noe ha a ime, here are C a insances allocaed o he aucion marke and (3)

4 C Ca insances held for he pay-as-you-go users. Suppose ha righ before +, of hem are erminaed by payas-you-go users and are reurned o he sysem. As a resul, here are C + = Ca + insances being available for new requess a he beginning of +. From he assumpion of he exponenial running ime as explained in Sec. III-A, i is easy o see ha follows a binomial disribuion wih P ( = k) =B(C Ca,k,q), where B(n, k, q) = n k qk ( q) n k. We re-wrie (2) as apple (C )=E max a(ca)+ r (C Ca) 0appleCa applec + E + (C a + ). (4) Noe ha he capaciy segmenaion problem is essenially formulaed as a Markov decision process. The cloud provider s problem is o solve (4) o find he opimal capaciy segmenaion poin Ca. I is worh menioning ha he capaciy segmenaion problem saed in (4) is non-rivial: Neiher marke is always more profiable han he oher. In he aucion marke, here may be high-bid requess from users sarving for cloud resources, which can drive he aucion price above he regular payas-you-go price (i.e., p a >p r ), making he aucion marke more profiable for a provider. Such phenomenon has indeed been observed in he real world: In EC2 pricing, he spo price occasionally exceeds he regular price []. A cloud provider has o dynamically segmen is capaciy o maximize is revenue. For now, problem (4) is sill no well defined. One quesion remains: how o design he opimal aucion marke o maximize he revenue a(ca) given Ca insances are allocaed? We answer his quesion in he following secion. IV. OPTIMAL AUCTION DESIGN This secion addresses he quesion raised above. Given an allocaed capaciy C a, wha is he opimal design for he aucion marke described in Sec. III? We invesigae he srucure of he opimal aucion and characerize is revenue a(c a). We also discuss is connecions o Amazon EC2 Spo Insances. A. Preliminaries An aucion mechanism M is said o be ruhful if for every bidder, no maer how ohers behave, he opimal bidding sraegy is always o submi is rue bids. In our problem, his means ha for every i, u i (n i,v i ) u i (r i,b i ) for any (r i,b i ). By he Revelaion Principle [22], i suffices o focus only on ruhful aucion designs when revenue is of ineres. Lemma characerizes he revenue of any ruhful aucions by exending he Revenue Equivalence Theorem [23] o he wo-dimensional domain. The proof is similar o [23] and is given in Appendix A. Lemma : Le v =(v i ) and n =(n i ). Denoe by M he revenue of a mechanism M wih wo-dimensional bids. Then for any ruhful M, we have " N E n,v [ M ]=E n,v n i (v i )x i (n, v). (5) i= F Here, (v i )=v v(v i n i) i f v(v i n i), and x i (n, v) akes he value 0 or depending on wheher user i loses or wins, respecively. An imporan observaion from Lemma is ha he expeced revenue of a ruhful mechanism only depends on who is o win (i.e., x i s), no wha hey pay (i.e., p a). We now characerize he revenue of he aucion marke described in Sec. III. Wihou loss of generaliy, suppose bidders are sored in a decreasing order of heir bidding prices, i.e., v v 2 v N a. We have he following proposiion. Proposiion : Suppose M is a ruhful aucion offering a uniform ake-i-or-leave-i price. Le m be he number of winning bidders 2. Then he expeced revenue of M is characerized as follows: " m E n,v [ M ]=E n,v n i (v i ). (6) Proof: Since he aucion marke offers a uniform ake-i-orleave-i price p a, every winning bidder i mus have v i p a. In his case, he op m bidders win he aucion, i.e., x i = for i =, 2,...,m. Subsiuing his o (5) and applying Lemma, we see ha he saemen holds. Proposiion essenially indicaes ha maximizing he aucion revenue is equivalen o maximizing he RHS of (6), subjec o he capaciy consrain: max mapplen a s.. i= m n i (v i ) i= m n i apple Ca. i= For mahemaical convenience, we ake he sandard regulariy assumpion ha ( ) is increasing. This is no a resricive assumpion, as i generally holds for mos disribuions [23] and is widely adoped in he lieraure [3], [23], [24]. B. Opimal Aucion Marke Problem (7) can be opimally solved in a greedy fashion: sequenially accep bidders requess, from he op valued (i.e., he highes (v i )) o he boom, unil here is no longer capaciy for more. I suffices o assume ha all requess are posiively valued ( (v i ) > 0), as hose wih (v i ) apple 0 will never be fulfilled. The opimal aucion marke, described in Algorihm, is designed based on he above process. We noe ha Algorihm adops he similar design of he canonical m+-price aucion, wih a difference ha a seller now has a reservaion price (0). Though m+-price aucion is ruhful for he case where each bidder requess no more han one uni of he aucioned good [25], i is well known ha in general, he ruhfulness no longer holds when bidders 2 The value of m depends on n and v. (7)

5 Algorihm Opimal Aucion Marke wih Capaciy C a. if P N a i= r i apple C a hen 2. All bidders win (m = Na), wih p a = (0) 3. else 4. Top m bidders win, wih p a = b m+, where P m i= r i apple Ca < P m+ i= r i 5. end if have muli-uni demands [3], [4]. However, we show ha for he specific problem considered in his paper, m+-price aucion is wo-dimensionally ruhful in boh n i and v i. To see his, we require he following lemma. Lemma 2: For every bidder i, fix all ohers submissions. Denoe by p a(b i,r i ) he clearing price when i bids (b i,r i ). Then for all b i (resp. r i ), p a(b i,r i ) is increasing w.r.. r i (resp. b i ). Lemma 2 reflecs he basic principle of economics: Wih he same supply, he marke price rises as he bidders demand increases. The proof can be found in Appendix B. Lemma 2 immediaely suggess Lemma 3, whose proof is given in Appendix B. Lemma 3: For every bidder i, here is no advanage o overbook insances, i.e., given b i, u i (r i,b i ) apple u i (n i,b i ) for all ri >n i. Since no user has he incenive o reques fewer insances han needed (as u i (r i,b i )=0whenever r i <n i), Lemma 3 essenially indicaes ha he users always ruhfully repor heir n i value. This leads o he ruhfulness saemen as follows. Proposiion 2: Algorihm is wo-dimensionally ruhful, i.e., u i (n i,v i ) u i (r i,b i ) for all (r i,b i ), i =, 2,...,N a. The deailed proof is given in Appendix B. Inuiively, given ha all users repor n i ruhfully as dicaed by Lemma 3, he marke can be viewed as a second price aucion in erms of he bid price only, which is well-known o be ruhful. We poin ou ha he wo-dimensional ruhfulness of his special case of m+-price aucion in our problem is due o he specific characerisics of cloud markes ha parial fulfillmen is no allowed. The revenue opimaliy of Algorihm follows naurally from he proved ruhfulness (i.e., Proposiion 2): Proposiion 3: Among all mechanisms offering a uniform ake-i-or-leave-i price, Algorihm is opimal in erms of revenue maximizaion. Proof: Since Algorihm is ruhful, all bidders bid ri = n i and b i = v i. In his case, Algorihm opimally solves problem (7). By Proposiion, his implies ha Algorihm maximizes he revenue among all ruhful aucions offering uniform clearing prices. Due o he Revelaion Principle [22], imposing he ruhfulness o he aucion design does no hur he revenue. We herefore conclude ha he saemen generally holds. C. Opimal Revenue To derive he revenue obained from Algorihm, one has o deal wih wo cases, wih or wihou sufficien capaciy o accommodae all profiable requess. To combine boh cases in our subsequen discussion, we arificially inser a virual bidder o he marke, who requess an infinie amoun of insances a a price (0). Insering a virual bidder has no effec on he aucion resul, bu i significanly simplifies he revenue expression. Based on Algorihm, p a = v m+, and a(ca)=v P m m+ i= n i, where P m i= n i apple Ca < P m+ i= n i. By Proposiion, we have E[ a (Ca)] P m = E[v m+ i= n i]= E[ P m i= n i (v i )]. Therefore, in expecaion, i is equivalen o wrie m a(c) = i= n i (v i ), (8) where P m i= n i apple c< P m+ i= n i. In his sense, n i (v i ) can be viewed as he marginal revenue generaed by acceping he requess of bidder i. D. Connecions o EC2 Spo Marke I is ineresing o see some connecions beween he aucion marke discussed in his paper and he spo marke adoped by Amazon EC2 Spo Insances []. Similar o he aucion marke, spo users periodically submi bids (ri,b i ) o Amazon, requesing ri insances a a price b i. A uniform spo price p s is periodically posed by Amazon o charge he winners, i.e., hose who bid higher han he spo price (b i >p a). All winners can use he insances as long as he price does no rise above heir bids. Though similar in descripion, he pricing of Spo Insances is by no means an aucion marke. Since Amazon has revealed no deailed informaion regarding how he spo price p s is calculaed, here is no way for spo users o know wha p s is going o be, even wih he complee informaion of demand (i.e., users bids) and supply (i.e., he amoun of insances offered in he spo marke). This is no he case in a real aucion, where he mechanism deails are publicly known o every paricipan. We now invesigae he opimal bidding sraegy for Spo Insances. Wihou pricing deails, one valid approach for spo users is o view p s as a random variable, wih p.d.f. fs and c.d.f. Fs learned from he price hisory published by Amazon []. Suppose he uiliy defined for user i is similar o () wih he clearing price p a replaced by he spo price p s, i.e., u i(r i,b i)= ni v i r i p s, if b i >p s and r i n i ; 0, oherwise. The user s problem is o find he opimal bid so ha is expeced uiliy is maximized, i.e., max r i,b i E p s u i (r i,b i ). Proposiion 4: In he spo marke, he opimal bid for user i is o ruhfully submi (n i,v i ). Proof: Le A be he even ha i wins by bidding b i. Denoe by I he indicaor funcion of even. We have E[u i(ri,b i)] = P (A)(n i v i rie[p s A])I r i n " i = F s (b i) n i v i r i Z b i v xf s (x)dx I r i n i. (9)

6 I is easy o see ha bidding ri = n i dominaes all oher sraegies for every b i. Now subsiuing i back o (9) and applying he firs-order opimaliy condiions, we see ha he opimal bid price is v i. This concludes he proof. Inuiively, wihou knowing how he spo price reacs o differen submissions, no user has he incenive o sraegize over is bid. Therefore, by replacing a spo marke wih an aucion marke, he provider would expec he same user behaviour. In oher words, he wo markes are equivalen in erms of he marke reacion. Considering ha boh are of similar pricing srucures (i.e., boh are bid-based), we believe ha he aucion marke offers a good simulaion o he spo marke, and he analysis of he former sheds ligh on he laer. V. OPTIMAL CAPACITY SEGMENTATION Having characerized he revenue for he aucion marke, we are now ready o invesigae he marke segmenaion problem saed in (4). Before delving ino he deailed echnical discussions, we jusify he moivaion of having wo co-exising markes by aking a look a he simples scenario where no fuure informaion is available, i.e., he predicion window w is 0. A. Moivaions for Join Markes When w = 0, (4) is reduced o a one-sho opimizaion problem, i.e., apple (C )=E max a(ca)+ r (C Ca). (0) 0appleCa applec We are no ineresed in solving he problem above as an O(C) soluion rivially exiss (i.e., search all possible Ca s o find he opimal segmenaion). Insead, we show ha his simple scenario illusraes wo moivaing facors behind pricing via join markes. Firs, wih he aucion marke, low-valuaion users whose v i <p r are offered a chance for access o cloud insances. Therefore, having wo markes expands he poenial demand and increases he overall revenue. Second, users wih low olerance o inerrupions are offered an opion o increase heir reques prioriy, as saed below. Proposiion 5: To maximize revenue, he provider always ries o fulfill he requess of hose aucion bidders whose v i p r /q before i acceps any pay-as-you-go requess. Proof: Suppose he provider has sufficien capaciy o fulfill bidder i s requess n i. By Proposiion, he marginal revenue of accommodaing bidder i is n i (v i ). Now if he provider changes is mind and allocaes hese n i insances o pay-asyou-go users, hen he marginal revenue would be a mos n i p r /q. Noe ha his will no happen if (v i ) p r /q, as bidder i s requess bring more marginal revenue o he provider. We herefore conclude he proof by noicing ha v i (v i ) p r /q. In oher words, for high-valuaion users, he aucion marke is acually offering guaraneed services wih higher fulfillmen prioriy. Only low-valuaion users bear he risk of being inerruped. All above jusify he moivaion for using muliple markes: I benefis boh he provider and he users. However, hough Proposiion 5 reveals some basic crieria in allocaing resources, i alone is unable o guaranee he opimal revenue. In fac, opimal capaciy segmenaion is a complicaed problem. The following discussions are argeed for a general seing where shor predicion is available, i.e., w>0. B. Complexiy of Opimal Capaciy Segmenaion Since (4) describes an MDP problem, a sandard soluion is numerical dynamic programming via backward inducion. I proceeds by firs simulaing marke demand in he las sage T based on he prediced demand and calculaing he opimal segmenaion made in ha sage. Using his resul, i hen deermines how o segmen he capaciy in sage T, based on he prediced marke demand a ha ime. This process coninues backwards unil he opimal segmenaion Ca made in he curren sage is obained. In each sage, for each possible C and each demand realizaion (i.e., he aucion requess (n, v) and pay-as-you-go requess Rr ), compue (4), which akes O(C 2 ) operaions. Since he compuaion is aken over all C =0,,...,C, he complexiy of one-sage calculaion is O(C 3 ). By noing ha only shor predicion is possible and w is usually small, we see he overall compuaional complexiy of he above process is O(C 3 ). For large providers wih high capaciies, finding exac soluions o (4) is compuaionally inracable. As a ypical example, when C = 0 5, he compuaion above requires O(0 5 ) operaions, which is prohibiive when decisions need o be made in real ime. C. An Approximae Soluion The segmenaion decision needs o be made quickly afer he user demand has arrived. In pracice, his is ofen more imporan han pursuing exac opimaliy. Hence, we nex propose an approximae soluion o (4) ha significanly reduces he compuaional complexiy. In he aucion marke, he bidders requess do no always fi exacly wihin he allocaed capaciy. By (8), here are P m = c i= n i insances lefover as hese resources are no sufficien o accommodae bidder m + s requess, i.e., <n m+. However, if bidder m+ acceps parial fulfillmen, hen hose insances generae (v m+ ) addiional revenue o he provider. Le a( ) be he revenue obained as if parial fulfillmen were accepable, i.e., a(c) = a (c)+ (v m+ ) m = n i (v i )+ (v m+ ), () i= Clearly a is an upper bound of a. The following Proposiion bounds he gap beween a and a. Lemma 4: If c n for some, hen a (c) ( ) a(c). Proof: By (), we have a(c) a(c) = (v m+ ). I suffices o consider he following wo cases.

7 Case : n m+ =. In his case, m + is he virual bidder wih (v m+ )=0. We see he saemen holds wih a(c) = a (c). Case 2: n m+ <. In his case, m + is a regular bidder. We have < n m+ apple n apple c/. Hence a(c) a(c) = (v m+ ) apple c (v m+ )/ apple a(c)/, where he las inequaliy holds since (v ) (v m+ ). By Lemma 4, we see ha he upper bound a is a close approximaion o a in pracical seings, where he capaciy allocaed o he aucion marke is usually enormous compared wih a single bidder s requess (i.e., ). We herefore consider an approximae problem by replacing a wih a in (4), i.e., apple (C )=E max a(ca)+ r (C Ca) 0appleCa applec + E + (C a + ). (2) The boundary condiions are T + (c) = 0 for all c = 0,,...,C. Le C a be he opimal soluion o (2). The provider hen uses i as an approximae, sub-opimal soluion o (4), generaing revenue apple (C )=E a( C a)+ r (C C a ) + E + ( C a + ). (3) We jusify he inuiion of he approximaion above wih he following proposiion, which condiionally bounds he compeiive raio of o. Proposiion 6: In (2), if C a n for all =,..., T, hen ( ) (C ) apple (C ) apple (C ). Proof: I is rivial o show he second inequaliy as is he opimal soluion. To show he firs inequaliy, we have " T (C )=E a( C a )+ r (C C a ) ( = " T )E = a( C a )+ r (C C a ) =( ) (C ), (4) where he second inequaliy holds due o Lemma 4. The condiion of Proposiion 6 is frequenly saisfied in pracice. Due o he large number of bidders, he volume of each bidder s requess is much smaller han he oal capaciy. As a resul, he revenue obained from he approximaion does no deviae oo far away from he opimal one. We laer verify his poin via exensive simulaions in Sec. VI. We now show ha (2) has an imporan opimizaion srucure ha leads o an efficien soluion wihin O(C 2 ). Firs, we see ha a( ) is concave, as saed below. Lemma 5: Given n and v, a(c) defined in () is concave. Tha is, r a(c) = a(c) a(c ) is decreasing w.r.. c. Lemma 5 suggess he concaviy of ( ) as follows. Lemma 6: For every =,..., T, (C ) is increasing and concave for all C =0,,...,C. This concaviy finally leads o an ineresing srucure described in he following proposiion. Proposiion 7: For every realizaion n and v a ime =, +,...,T, le C a (C ) be he opimal soluion o (2). For all C =0,,...,C, we have C a (C + ) apple C a (C ) apple C a (C + ). (5) The deailed proofs of Lemmas 5 and 6, as well as Proposiion 7, are all given in Appendix C. Proposiion 7 indicaes ha previously calculaed resuls can be reused in subsequen compuaions. We herefore run dynamic programming from he las sage T and proceed backwards o. Wihin each sage, (C ) is sequenially compued as C = C, C,...,0. When compuing C a (C ), insead of exhausively searching he enire soluion space from 0 o C, one only needs o ry wo values, C a (C + ) and C a (C + ), and he one resuling in higher revenue is seleced as C a. The enire compuaion only akes O(C 2 ) operaions. In erms of compuaional efficiency, he approximae soluion significanly ouperforms he opimal one, as he oal capaciy of a provider is usually enormous in pracice. As an example, when C = 0 5, he approximaion is 0 5 imes faser han he exac soluion. VI. SIMULATION RESULTS We evaluae he revenue performance of he proposed approximae soluion via exensive simulaions. We adop a ypical scenario where C = 0 5. Tha is, he provider is able o hos up o 0 5 virual insances of a cerain ype simulaneously. We simulae he markes for 00 ime periods. In each period, cloud users arrive ino he sysem following a Poisson process wih inensiy, which are hen randomly spli ino he pay-as-you-go and aucion markes wih equal probabiliy. Our evaluaion adops hree demand paerns low, medium, and high, wih being 00, 200, and 500, respecively. For he pay-as-you-go marke, each user s demand is modeled by a random variable uniformly disribued in [, 000], he price p r is normalized o, and he insance reurn probabiliy q is aken as 0.5. For he aucion marke, each bidder i s reques n i is modeled by an i.i.d. random variable uniformly disribued in [, 000], and is affordable price v i is i.i.d. exponenial wih mean E[v i ]=0.5p r =0.5. We enable shor predicions and se he predicion window w =5. Each resul below has been averaged over 000 runs. A. Revenue Performance We firs evaluae he proposed near-opimal segmenaion scheme by comparing is revenue (i.e., defined in (3)) agains he heoreical upper bound (i.e., defined in (2)). The resuls are illusraed in Fig. 2a, where all daa is normalized by he maximum upper-bound revenue. Fig. 2a shows ha our approximaion design closely approaches he opimal soluion. Even compared wih he heoreical revenue upper

8 Normalized Revenue Approxn., λ = 00 UB, λ = 00 Approxn., λ = 200 UB, λ = 200 Approxn., λ = 500 UB, λ = Time (a) Normalized revenue vs. ime. CDF λ = 00 λ = 200 λ = Marke share of periodic aucions (%) (b) CDF of he marke share of periodic aucions. CDF λ = 00 λ = 200 λ = Aucion price (c) CDF of he aucion price. Revenue conribuion (%) Aucion (approxn.) Aucion (UB) λ (d) Revenue conribuion of periodic aucions. Fig. 2. Performance evaluaion of he approximae capaciy segmenaion algorihm, where UB sands for upper bound while approxn. is shor for approximaion. TABLE I REVENUE GAP BETWEEN THE APPROIMATION AND THE UPPER BOUND. = 00 = 200 = 500 Revenue gap 0.8% 0.52% 0.38% bound, he gap is almos negligible, less han % in all cases, as summarized in Table I. This confirms our conclusion from Proposiion 6. B. Capaciy Segmenaion and Aucion Prices We now analyze how he capaciy is allocaed o he wo markes under he near-opimal segmenaion sraegy. Fig. 2b illusraes he CDF of he marke share of periodic aucions in all hree demand paerns. Here, he marke share is defined as he raio, beween he capaciy allocaed o he aucion marke and he enire capaciy ha he provider has. I is worh menioning ha he allocaed capaciy migh no be fully used o accommodae aucion bidders, even for he case where he aucion demand exceeds he supply. The provider would sraegically reserve some insances by rejecing lowbid requess, since acceping hem lowers he clearing price, which may decrease he revenue. As illusraed in Fig. 2b, when demand is low (i.e., = 00), abou half of he capaciy is allocaed o he aucion marke, leading o a 50% marke share. Fig. 2c shows he corresponding clearing price ha is around he mean bid E[v i ] of aucion bidders. In his case, since cloud insances are overprovisioned, some of hem are aucioned a a discouned price o increase he revenue. I is worh menioning ha hough aucion bidders enjoy using he resources a a lower price, hey bear he risks ha he services migh be inerruped. As demand increases, he marke share drops, while he aucion price rises. For he case where = 200, Fig. 2b shows ha almos all insances are hosed o accommodae pay-as-you-go requess, wih less han 0% capaciy allocaed o aucion markes. This is essenially due o he simulaion seings ha insances are less valued in periodic aucions han hey are in pay-as-you-go marke, as he mean bid is only half of he pay-as-you-go price (i.e., E[v i ]=0.5p r ). In his case, pay-as-you-go requess are considered more profiable han aucion bids. Only a few high-value bids are acceped by he provider, resuling in a higher clearing price in he aucion channel as illusraed in Fig. 2c. I is ineresing o observe ha, when demand keeps increasing, he marke share of periodic aucions rebounds, which is shown in Fig. 2b wih = 500. In his case, he enire marke demand significanly exceeds he provider s capaciy. As a resul, more high-bid requess are received from he aucion marke. Since hese requess are more profiable han hose in he pay-as-you-go marke, he provider fulfills hem by allocaing more resources o he aucion marke. The clearing price is also observed o rise in Fig. 2c. All discussions above show ha augmening pay-as-yougo pricing wih periodic aucions essenially increases he provider s abiliy o respond o demand uncerainies. Periodic aucions help o fulfill some lefover revenue when resources are over-provisioned in he pay-as-you-go marke. On he oher hand, i exracs more revenue by charging high prices o highbid requess when demand exceeds supply. C. Comparisons Beween Pay-as-You-Go and Aucions The wo markes do no make equal revenue conribuions. As presened in Fig. 2d, he pay-as-you-go marke conribues more han 85% revenue o he provider in all hree demand paerns. Noe ha he pay-as-you-go marke akes up only 66% of he overall demand 3. Therefore, i provides a disproporionaely large share of revenue. Similar observaions are made when differen demand raios beween he wo markes are considered. By offering guaraneed services wih a saic price, insances in he pay-as-you-go marke ofen demand a higher premium han hose in he aucion marke. For his reason, pay-asyou-go requess are usually more profiable han mos aucion bids, and are acceped a a higher prioriy for revenue maximizaion. Table II furher validaes his poin, where he reques accepance raes are lised for all hree demand paerns. We see ha pay-as-you-go requess are generally acceped wih a considerably higher probabiliy han aucion bids. However, his does no mean ha aucion bidders are always secondary cusomers. As saed in Proposiion 5, hose 3 New demand arrivals are equal for boh markes, bu each new pay-asyou-go insance requires wice he capaciy of each new aucion insance since q =0.5.

9 TABLE II AVERAGE REQUEST ACCEPTANCE RATES Bid price Bid price Pay-as-you-go users Aucion users = 00 00% 36.9% = % 5.7% = % 5.2% p ri a ri + 0 C Capaciy 0 a Ca (a) Increasing ri raises he clearing price. p a Capaciy who bid sufficienly high will always be accommodaed firs. In our simulaion, hese are he op 5% bidders. As illusraed in Table II, heir requess are leas affeced by he specific demand paern. Therefore, he aucion marke offers an opion o he users o increase he prioriy of heir requess. VII. CONCLUSIONS In his paper, we invesigae he problem of opimal capaciy segmenaion in an EC2-like cloud marke wih he regular pay-as-you-go pricing augmened by periodic aucions. To his end, we analyically characerize he revenue of uniformprice aucions, and presen an opimal design wih maximum revenue. Conrary o he well-known resul ha uniform-price aucions have suffered from he demand reducion in general, our design achieves ruhfulness in cloud environmens where parial fulfillmen is unaccepable o users. We furher connec our design o he EC2 spo marke, showing ha he wo are equivalen in erms of heir marke response. Based on he esablished analysis for he aucion channel, we formulae he capaciy segmenaion problem as a Markov decision process. Realizing ha he exac soluion is compuaionally prohibiive in pracical seings, we presen a near-opimal approximaion ha reduces he compuaional complexiy from O(C 3 ) o O(C 2 ), which is significan for cloud providers wih large capaciies. All our heoreical resuls are furher validaed by exensive simulaion sudies. APPENDI A ETENSION OF THE REVENUE EQUIVALENCE THEOREM Proof of Lemma : " N E n,v [ M ]=E n E v p i n i i= " N = E n E v i E vi v i [p i n i ], (6) i= where v i is obained by removing he ih componen v i from v. Now consider E vi v i [p i n i ]. I has been shown in [22] ha, when n and v i are given, a ruhful mechanism M always offers a ake-i-or-leave-i paymen 4, say i, for bidder i, who wins when bidding higher han i. Therefore, E vi v i [p i n i ]=n i i ( F v ( i n i )) Z v = n i z i F v (z n i ) f v (z n i ) f v (z n i )dz = E vi v i [n i (v i )x i (n, v)], (7) 4 Such paymen relies on M, n and v i, bu is independen of v i. Bid price 0 C a r i b i + b i = p a Capaciy Bid price r i b i + 0 C a (b) Increasing b i raises he clearing price. p a >b i Capaciy Fig. 3. Effec of demand r i and bid price b i on he clearing price p a. where he second equaliy can be verified by performing inegraion by pars on he righ-hand side (RHS). We conclude he proof by subsiuing (7) back ino (6). APPENDI B TRUTHFULNESS ANALYSIS OF ALGORITHM In his secion, we show ha he proposed aucion mechanism (Algorihm ) is ruhful. We sar o prove Lemma 2. Proof of Lemma 2: We explain he picorial proof hrough Figs. 3a and 3b. Firs we show ha p a is increasing w.r.. ri, i.e., p a(b i,r i ) apple p a(b i,r i + ) for all > 0. I suffices o consider he following wo cases. Case : Bidder i loses by requesing ri insances. Noe ha increasing a bidder s reques does no change is ranking (as hey are sored based on heir bid prices). I is easy o verify ha having his bidder requesing more insances, say, ri + insances, resuls in he same clearing price, i.e., p a(b i,r i )= p a(b i,r i + ). Case 2: Bidder i wins by requesing ri insances. Suppose i now increases is reques by insances. Fig. 3a illusraes he changes of clearing price p a, from which we see ha having a winning bidder requesing more insances essenially raises he clearing price, i.e., p a(b i,r i ) apple p a(b i,r i + ). We nex prove ha p a is also increases w.r.. b i, i.e., p a(b i,r i ) apple p a(b i +,r i ) for all > 0. Sill, i suffices o consider wo cases below. Case : Bidder i loses by bidding b i. Suppose i now raises is bid by. We consider wo cases. () Bidder i wins by bidding b i +. This is shown in Fig. 3b, from which we see ha he clearing price is raised. (2) Bidder i loses by bidding b i +. In his case, if bidder i s new bid is used as he new clearing price (i.e., p a = b i + ), hen his new price mus be higher han he original one (because is value is updaed). Oherwise, he clearing price p a remains unchanged. In summary, p a(b i,r i ) apple p a(b i +,r i ) for all > 0. Case 2: Bidder i wins by bidding b i. In his case, raising he bid price has no effec on he clearing price he laer remains unchanged wih he value equal o he m+-h highes bid, i.e., p a(b i,r i )=p a(b i +,r i ).

10 Wih Lemma 2, we are now ready o prove ha no bidder has he incenive o overbook insances. Proof of Lemma 3: I suffices o consider he case where bidder i wins by submiing (r i,b i ), wih r i >n i. In his case, b i p a(r i,b i) p a(n i,b i), (8) where he firs inequaliy holds since i wins by bidding (ri,b i ), while he second inequaliy is derived from Lemma 2. This implies ha i also wins by bidding (n i,b i ). As a resul, u i(r i,b i)=n i v i r ip i(r i,b i) apple n i v i n i p i(n i,b i) = u i(n i,b i). Lemma 2 and 3 leads o he ruhfulness of Algorihm, as proved below. Proof of Proposiion 2: We consider all possible oucomes of bidding ruhfully or unruhfully. Since every bidder i chooses o ruhfully repor n i, any unruhful submission is of he form (n i,b i ) where b i 6= v i. Suppose he unruhful submission leads o he bidder losing, hen he proposiional saemen rivially holds for b i. Therefore, in he following we only need o consider he case where he unruhful submission leads o he bidder winning. Suppose bidding ruhfully leads o he bidder winning, hen i is easy o verify ha p a(n i,b i )=p a(n i,v i )=p, where p is eiher (0) or b m+, depending on wheher here is sufficien capaciy o accommodae all requess. As a resul, we see u i (n i,v i )=u i (n i,b i ). On he oher hand, if bidding ruhfully leads o he bidder losing, hen p a(n i,v i ) v i. Since by changing he submission o (n i,b i ) user i wins, we mus have b i >v i. By Lemma 2, p a(n i,b i ) p a(n i,v i ) v i. Hence, u i (n i,b i ) = n i(v i p a(n i,b i )) apple 0=u i (n i,v i ). APPENDI C ANALYSIS OF THE APPROIMATE SOLUTION This secion invesigaes he opimizaion srucure of (3). Firs we show ha a is concave. Proof of Lemma 5: By (), we have (vm ), if =0; r a(c) = (9) (v m+ ), oherwise. In he firs case ( =0), r a(c + ) = (v m+ ) apple (v m )= r a (c). In he second case ( >0), r a(c+) = (v m+ )= r a (c). To show is increasing and concave, we need he following wo echnical lemmas. Lemma 7: Le f be a decreasing and concave funcion. Define g by g(n) = P n B(n, k, q)f(k). Then g is also decreasing and concave. Proof: For noaional simpliciy, we wrie f(n) by f n and g(n) by g n. Wihou loss of generaliy, we assume f 0 =0. Le rg n = g n g n and r 2 g n = rg n rg n be he firsand second-order forward differences of g( ), respecively. Similar definiion also applies o rf n and r 2 f n. When here is no confusion, we simply wrie B(n, k, q) as B n,k. For convenience, we define B n,k =0for all k<0. We firs show ha g is decreasing by proving ha rg n apple 0. By definiion, we have rg n = g n g n n n = B n,k f k n B n,k f k = q [B n,k B n,k ] f k + q n f n. k= n = q B n,k rf k+ (20) Here he hird equaliy holds because f 0 =0and B n,k = qb n,k +( q)b n,k. (2) Since f is decreasing, rf k+ apple 0 for all k, which implies rg n apple 0. We nex show he concaviy of g by proving n r 2 g n+ = q 2 B n,k r 2 f k+2. (22) The following equaion is obained by plugging (20) ino (22) and applying (2): q 2 r2 g n+ n = [B n,k 2 + B n,k 2B n,k ]f k k= +[B n,n 2 2B n,n ]f n + B n,n f n+. (23) Fixing n, we define m by m m = [B n,k 2 + B n,k 2B n,k ]f k, (24) and define k= m by m =[B n,m 2 2B n,m ]f m + B n,m f m+. (25) Noicing ha m =0for all m<2, we can wrie (23) by q 2 r2 g n+ = n + n. (26) We now invesigae he srucure of m + m for all m. Firs, we have m =[B n,m 2 B n,m ]f m + B n,m rf m+ = B n,m f m + B n,m 2 f m + B n,m r 2 f m+. (27) Subsiuing (27) back o m + m leads o he following

11 equaliies: m + m = m + m +[B n,m 3 + B n,m 2B n,m 2 ]f m = m + B n,m r 2 f m+ +[B n,m 3 2B n,m 2 ]f m + B n,m 2 f m = m + m + B n,m r 2 f m+ (28) = m B n,k r 2 f k+2, (29) where (29) is obained by recursively applying (28). Wih (29) and (26), we see (22) holds, which concludes he proof. Lemma 8: Le f and f 2 be wo funcions ha are increasing and concave. Define f as follows. f(n) = max {f (k)+f 2 (n k)}. (30),,...,n Le k n be he opimal soluion o (30). We have he following hree saemens. () f is increasing and concave. (2) k n apple k if and only if f (k) apple f 2 (n k ), where f(n) =f(n + ) f(n) for all discree f. (3) k n+ apple k n apple k n+. Proof of (): For any discree funcion h, le h be is linear inerpolaion. Tha is, for all x, le n = bxc. We have h(x) =h(n)+ h(n) (x n). (3) Given n, we define g k = f (k)+f 2 (n k). I is easy o see ha ḡ(x) = f (x)+ f 2 (n x). Since ḡ is a linear inerpolaion of a discree funcion g, we know is maximum value is achieved a he inegeer poin, i.e., Le f 0 (y) be defined by f(n) = max k,,...,n = max 0applexapplen = max (x)+ f 2 (n 0applexapplen x)}. (32) f 0 (y) = max 0applexappley { f (x)+ f 2 (y x)}. (33) Since boh f and f 2 are increasing and concave, heir linear inerpolaions, f and f 2, are also increasing and concave. By [26] (Rule 9 of Theorem 3..5), we know f 0 is increasing, concave, and piecewise linear. Also noe ha f 0 (n) =f(n) for all ineger n. We conclude ha f 0 is a linear inerpolaion of f, i.e., f 0 = f. In his case, f is increasing and concave. Proof of (2): Le f(n) = max 0applekapplen g(k) where g(k) = f (k)+f 2 (n k). We know g is concave since boh f and f 2 are concave. In his case, kn apple k if and only if g(k) apple 0. We hence conclude he proof by noicing ha g(k) = f (k) f 2 (n k ). Proof of (3): Firs we show kn apple kn+. By saemen (2), his is equivalen o show f (kn+) apple f 2 (n kn+ ). This is indeed he case as f (k n+) apple f 2 (n k n+) apple f 2 (n k n+ ). Here he firs inequaliy is derived by applying saemen (2) o he case of n +, while he second inequaliy is due o he fac ha f 2 is concave. We nex show ha k n+ apple k n +. By saemen (2), his is equivalen o prove f (k n + ) apple f 2 (n k n ), which is rue because f (k n + ) apple f (k n) apple f 2 (n k n ), where he firs inequaliy is due o he concaviy of f and he second inequaliy is derived from saemen (2). We are now ready o prove Lemma 6. Proof of Lemma 6: We prove by inducion. Basis: Show ha he saemen holds for = T. In his case, (2) becomes T (C T ) = max a(c 0appleCa T a T )+ r (C T Ca T ). applect Since boh a( ) and r( ) are increasing and concave, by Lemma 8, T ( ) is also increasing and concave. Inducive sep: Suppose he saemen holds for. We now show i also holds for. Le g (c) =E [ (c + )]. Given n and v, (2) can be rewrien as (C ) = max 0appleC a applec a(c a )+ r (C C a )+g (C a ). We firs show ha g ( ) is increasing and concave. Le n = C c. We derive as follows. C c g (C n) = B(C c, k, q) (c + k) = = n B(n, k, q) (C n + k) n B(n, k, q) (C k). Since ( ) is increasing and concave (due o he inducion assumpions), we see g ( ) is also increasing and concave by applying Lemma 7. Now ha a( ), r( ), and g ( ) are all increasing and concave, by saemen () of Lemma 8, is also increasing and concave. This concludes he proof. The lemmas above immediaely sugges he proof of Proposiion 7. Proof of Proposiion 7: Le f(ca ) = a(ca ) + E [ + (Ca + )]. When n and v are given, (2) can be rewrien as (C ) = max 0appleCa applec {f(c a )+ r (C Ca )}. Noing ha boh f and r are concave, and applying saemen (3) of Lemma 8, we see he saemen holds. REFERENCES [] Amazon EC2 Pricing, hp://aws.amazon.com/ec2/pricing/.

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