CLoud computing technologies have enabled rapid

Transcription

1 1 Cost-Mnmzng Dynamc Mgraton of Content Dstrbuton Servces nto Hybrd Clouds Xuana Qu, Hongxng L, Chuan Wu, Zongpeng L and Francs C.M. Lau Department of Computer Scence, The Unversty of Hong Kong, Hong Kong, {xqu,hxl,cwu,fcmlau}@cs.hku.hk Department of Computer Scence, Unversty of Calgary, Canada, zongpeng@ucalgary.ca Abstract Wth the recent advent of cloud computng technologes, a growng number of content dstrbuton applcatons are contemplatng a swtch to cloud-based servces, for better scalablty and lower cost. Two key tasks are nvolved for such a move: to mgrate the contents to cloud storage, and to dstrbute the web servce load to cloud-based web servces. The man ssue s to best utlze the cloud as well as the applcaton provder s exstng prvate cloud, to serve volatle requests wth servce response tme guarantee at all tmes, whle ncurrng the mnmum operatonal cost. Whle t may not be too dffcult to desgn a smple heurstc, proposng one wth guaranteed cost optmalty over a long run of the system consttutes an ntmdatng challenge. Employng Lyapunov optmzaton technques, we desgn a dynamc control algorthm to optmally place contents and dspatch requests n a hybrd cloud nfrastructure spannng geo-dstrbuted data centers, whch mnmzes overall operatonal cost over tme, subect to servce response tme constrants. Rgorous analyss shows that the algorthm ncely bounds the response tmes wthn the preset QoS target, and guarantees that the overall cost s wthn a small constant gap from the optmum acheved by a T-slot lookahead mechansm wth known future nformaton. We verfy the performance of our dynamc algorthm wth prototype-based evaluaton. Index Terms Hybrd Cloud, Content Dstrbuton, Dynamc Mgraton, Lyapunov Optmzaton 1 INTRODUCTION CLoud computng technologes have enabled rapd provsonng and release of server utltes (CPU, storage, bandwdth) to users anywhere, anytme. To explot the dversty of electrcty costs and to provde servce proxmty to users n dfferent geographc regons, a cloud servce often spans multple data centers over the globe, e.g., Amazon CloudFront [1], Mcrosoft Azure [], Google App Engne [3]. The elastc and on-demand nature of resource provsonng has made cloud computng attractve to provders of varous applcatons. More and more new applcatons are beng created on the cloud platform [4][5][6], whle many exstng applcatons are also consderng the cloud-ward move [7][8], ncludng content dstrbuton applcatons [9][1]. As an mportant category of popular Internet servces, content dstrbuton applcatons, e.g., vdeo streamng, web hostng and fle sharng, feature large volumes of contents and demands that are hghly dynamc n the temporal doman. A cloud platform wth multple, dstrbuted data centers s deal to host such a servce, wth substantal advantages over a tradtonal prvate or publc content dstrbuton network (CDN) based soluton, n terms of more aglty and sgnfcant cost reducton wth respect to machnes, bandwdth, and management. In ths way, the applcaton provders can focus ther busness more on content provsonng, rather than IT nfrastructure mantenance. Two maor components exst n a typcal content dstrbuton applcaton, namely back-end storage for keepng the contents, and front-end web servces to serve the requests. Both can be mgrated to the cloud: contents can be stored n storage servers n the cloud, and requests can be dstrbuted to cloud-based web servces. Therefore, the key challenge for cloud-ward move of a content dstrbuton applcaton s how to effcently replcate contents and dspatch requests across multple cloud data centers, as well as the provder s exstng prvate cloud, such that good servce response tme s guaranteed and only modest operatonal expendture s ncurred. It may not be too hard to desgn a smple heurstc for dynamc content placement and load dstrbuton n the hybrd cloud; however, proposng one wth guarantee of cost optmalty over a long run of the system, s an ntrgung yet ntmdatng challenge, especally when arbtrary arrval rates of requests are consdered. Some exstng work [7][8][9][1] have advocated optmal applcaton mgraton nto clouds, but none focus on guaranteeng over-tme cost mnmzaton wth a dynamc algorthm. In ths paper, we present a generc optmzaton framework for dynamc, cost-mnmzng mgraton of content dstrbuton servces nto a hybrd cloud (.e., prvate and publc clouds combned), and desgn a ont content placement and load dstrbuton algorthm that mnmzes overall operatonal cost over tme, subect to servce response tme constrants. Our desgn s rooted n Lyapunov optmzaton theory [11][1], where cost mnmzaton and response tme guarantee are acheved smultaneously by effcent schedulng of content m-

2 graton and request dspatchng among data centers. Lyapunov optmzaton provdes a framework for desgnng algorthms wth performance arbtrarly close to the optmal performance over a long run of the system, wthout the need for any future nformaton. It has been extensvely used n routng and channel allocaton n wreless networks [11][13], and has only recently been ntroduced to address resource allocaton problems n a few other types of networks [14][15]. We talor Lyapunov optmzaton technques n the settng of a hybrd cloud, to dynamcally and ontly resolve the optmal content replcaton and load dstrbuton problems. The contrbuton of ths work can be summarzed as follows: We propose a generc optmzaton framework for dynamc, optmal mgraton of a content dstrbuton servce to a hybrd cloud consstng of a prvate cloud and publc geo-dstrbuted cloud servces. We desgn a ont content placement and load dstrbuton algorthm for dynamc content dstrbuton servce deployment n the hybrd cloud. Provders of content dstrbuton servces can practcally apply t to gude ther servce mgraton, wth confdence n cost mnmzaton and performance guarantee, regardless of the request arrval pattern. We demonstrate optmalty of our algorthm wth rgorous theoretcal analyss and prototype-based evaluaton. The algorthm ncely bounds the response tmes (ncludng queueng and round-trp delays) wthn the preset QoS target n cases of arbtrary request arrvals, and guarantees that the overall cost s wthn a small constant gap from the optmum acheved by a T-slot lookahead mechansm wth nformaton nto the future. In the rest of the paper, we dscuss related work n Sec., present the optmzaton framework n Sec. 3, desgn a ont content placement and load dstrbuton algorthm n Sec. 4, and rgorously analyze ts performance n Sec. 5. We evaluate the algorthm wth prototypebased evaluaton n Sec. 6, and conclude the paper n Sec. 7. RELATED WORK Mgraton of applcatons nto clouds: A number of research proects have emerged n recent years that explore the mgraton of servces nto a cloud platform. Haat et al. [7] develop an optmzaton model for mgratng enterprse IT applcatons onto a hybrd cloud. Ther model takes nto account enterprse-specfc constrants, such as transacton delays and securty polces. Onetme optmal servce deployment s consdered, whle our work nvestgates optmal dynamc mgraton over tme, to acheve the long-term optmalty. Zhang et al. [8] propose an ntellgent algorthm to factor workload and dynamcally determne the servce placement across the publc cloud and the prvate cloud. Ther focus s on desgnng an algorthm for dstngushng base workload and trespassng workload. Mgraton of content delvery servces nto clouds: Some research efforts have been put nto mgratng generc content delvery servces onto clouds. MetaCDN by Pathan et al. [16] s a proof-of-concept testbed, experments on whch show that deployng content delvery based on storage clouds can mprove utlty, based on prmtve content placement and request routng mechansms. Chen et al. [17] propose to buld CDNs n the cloud n order to mnmze cost under the constrants of QoS requrement, but they only propose greedy-strategy based heurstcs wthout provable propertes. In contrast, we target an optmzaton framework whch renders optmal mgraton solutons for long run of the system. Some work focuses on mgratng specfc types of content delvery servces onto clouds, e.g., socal networkng servce, or vdeo streamng servce. Cheng et al. [1] study the partton of socal data and ther storage onto a number of cloud servers, to mgrate a socal networkng applcaton nto the cloud. It focuses on balancng the data access load, by consderng socal relatonshps and user access patterns n the data storage. L et al. [9] advocate cost savng by partal mgraton of a VoD servce to a content cloud. Heurstc strateges are proposed to decde the update of cloud contents, whch are verfed by trace-drven evaluatons. Our work focuses on cost mnmzaton n mgraton of a generc content dstrbuton applcaton, based on dfferentated chargng models of dfferent data centers. Applcaton of Lyapunov optmzaton theory: Lyapunov optmzaton was developed from the stochastc network optmzaton theory [11][1], and has been appled n routng and channel allocaton n wreless networks [18][11][13], as well as n a few other types of networks ncludng peer-to-peer networks [15]. Magulur et al. [19] propose varous VM confguraton schedulng algorthms for cloud computng platforms, that acheve arbtrary fracton of the capacty regon of the cloud. But ther model does not take nto consderaton delay guarantee, whch s an mportant component n our optmzaton framework. The work of Ren et al. [] also consders an onlne scheduler that dspatches workloads across multple geographcally dstrbuted data centers subect to delay requrements. It assumes where each ob s data s stored s fxed and known. However n our work we further ncorporate the decson on data mgraton nto the schedulng. The work of Amble et al. [14] s close to ours n that t also utlzes Lyapunov functon to study request routng and content cachng, but n the settng of CDNs wth capactated caches and lnks. They nvestgate the optmalty of dfferent cachng polces. Gven a workload wthn the capacty regon, they prove that several types of cachng and content evcton methods can each provde a throughput equal to the workload. Instead, our study focuses on optmal mgraton of content dstrbuton servces onto a hybrd cloud, such that the operatonal cost s mnmzed whle

3 3 servce delay bound s guaranteed. Related servce placement problems: Placement of servces onto dfferent stes has been nvestgated [1][] based on the theores of Faclty Locaton Problems (FLP) [3], ncludng the k-medan Problem (kmp) [4] and k-component Mult-Ste Placement Problem (k-cmsp) []. Such a problem typcally nvolves an NP-hard nteger program, and can only be solved by approxmaton algorthms; t focuses on one-tme optmzaton wth fxed servce demands, rather than onlne optmzaton over a long run of the system. Our work s sgnfcantly dfferent, whch apples Lyanopuv optmzaton theory to pursue the global optmalty wth dynamc request arrvals over tme. 3 THE SERVICE MIGRATION PROBLEM 3.1 System Model We consder a typcal content dstrbuton applcaton, whch provdes a collecton of contents (fles), denoted as set M, to users spreadng over multple geographcal regons. There s a prvate cloud owned by the provder of the content dstrbuton applcaton, whch stores the orgnal copes of all the contents. The prvate cloud has an overall upload bandwdth of b unts for servng contents to users. There s a publc cloud consstng of data centers located n multple geographcal regons, denoted as set N. One data center resdes n each regon. There are two types of nter-connected servers n each data center: storage servers for data storage, and computng servers that support the runnng and provsonng of vrtual machnes (VMs). Servers nsde the same data center can access each other va a certan DCN (Data Center Network). The provder of the content dstrbuton applcaton (applcaton provder) wshes to provson ts servce by explotng a hybrd cloud archtecture, whch ncludes the geo-dstrbuted publc cloud and ts prvate cloud. The maor components of the content dstrbuton applcaton nclude: () back-end storage of the contents and () front-end web servce that serves users requests for contents. The applcaton provder may mgrate both servce components nto the publc cloud: contents can be replcated n storage servers n the cloud, whle requests can be dspatched to web servces nstalled on VMs on the computng servers. An llustraton of the system archtecture s gven n Fg. 1. Our obectve n ths paper s to desgn a dynamc, optmal algorthm for the applcaton provder to strategcally make the followng decsons for servce mgraton nto the hybrd cloud archtecture: () content replcaton: whch content should be replcated n whch data center at each tme? () request dstrbuton: How many requests for a content should be drected to the prvate cloud and to each of the data centers that store ths content at the tme? The goal s to pursue the mnmum operatonal Fg. 1. The system archtecture. TABLE 1 Notatons M Fle set N Regon set v (m) Sze of fle m, n bytes a (m) (t) No. of requests for fle m from regon at tme slot t (t) Sze of request queue for fle m n regon at t A Max. no. of requests for fle m from regon n a tme slot (t) No. of requests dspatched from to prvate cloud at t (t) No. of requests dspatched from to data center at t y (m) (t) Bnary var: store fle m on data center or not at t. b Max. no. of requests the prvate cloud can serve n a tme slot µ Max. no. of requests dspatched from each request queue to a data center n a tme slot,.e., (t) µ g Charge for uploadng a byte from data center o Charge for downloadng a byte nto data center f Charge for rentng one VM nstance n data center r No. of requests a VM n data center can serve n a tme slot p Charge for storng a byte on data center q (m) Charge for uploadng fle m from data center h Tme-averaged charge for uploadng a byte from the prvate cloud w (m) Charge for mgratng fle m to data center Bound of queueng delay of requests n queue ǫ (m) Pre-set constant for controllng queueng delay n d Round-trp delay between regon and the prvate cloud e Round-trp delay between regon and data center α Bound of tme-averaged round-trp delay G(t) Vrtual queue for boundng tme-averaged round-trp delay (t) Vrtual queue for boundng queueng delay n V A non-negatve parameter to control the tradeoff between the operatonal cost and request response delays cost for the applcaton provder over tme, whle ensurng the servce qualty of content dstrbuton. We next develop an optmzaton framework to characterze the optmal content dstrbuton servce mgraton problem. Important notatons are summarzed n Table 1 for ease of reference.

4 4 3. Cost-Mnmzng Servce Mgraton Problem We suppose that the system runs n a tme-slotted fashon. Each tme slot s a unt tme whch s enough for uploadng any fle m M wth sze v (m) (bytes) at the unt bandwdth. In tme slot t, a (m) (t) requests are generated for downloadng fle m M, from users n regon. We assume that the request arrval s an arbtrary process over tme, and the number of requests arsng from one regon for a fle n each tme slot s upper-bounded by A. The cost of uploadng a byte from the prvate cloud s h. The charge for storage at data centersp per byte per unt tme. g and o per byte are charged for uploadng from and downloadng nto data center, respectvely. The cost for rentng a VM nstance n data center s f per unt tme. These charges follow the chargng model of leadng commercal cloud provders, such as Amazon EC [5] and S3[6]. We assume that the storage capacty n each data center s suffcent for storng contents from ths content dstrbuton applcaton. We also assume that each request s served at one unt bandwdth, and the number of requests that a VM n data center can serve per unt tme s r. Decson varables. The decson varables n our optmzaton framework are formulated as follows: (1) For content replcaton, bnary varable y (m) (t) ndcates whether fle m s stored n data center n tme slot t or not. If y (m) (t 1) = and y (m) (t) = 1, fle m s coped from the prvate cloud to the data center at t; f y (m) (t 1) = 1 and y (m) (t) =, fle m s removed from data center. In other cases, the storage status reman the same. In case of mgraton, we assume that the vdeo s always coped from the prvate cloud to the destnaton data center. () For dspatchng requests from regon for fle m, (t) be the number of requests to be served by the prvate cloud n tme slot t, and (t) denote the number of requests dspatched to data center n tme slot t, wth an upper bound of µ. Based on the elastcty of clouds, we reasonably assume that A < µ. Requests for fle m can only be dspatched to data center when t stores the fle,.e., (t) > let (t) = 1. We assume that a data center can serve a fle to users n the tme slot when the fle s beng coped to the data center, snce replcatng the fle from the prvate cloud and servng chunks of the fle can be carred out n parallel: after recevng a small porton of the fle, a data center can already start to serve the receved chunks of the fle to users. We assume that upload bandwdth s reserved for replcatng fles to data centers from the prvate cloud, and ths bandwdth s not counted n b, the mum unts of bandwdth that the prvate cloud can use to upload contents to users. Not all requests arsng n one tme slot are dspatched n the same tme slot, subect to capacty constrants. only f y (m) A queue s mantaned to buffer requests for fle Fg.. The queueng model. m generated from users n regon over tme, N,m M. The backlog sze of queue at tme t,.e., the number of requests generated n regon for fle m but not dspatched yet by t, s denoted by (t). The update of the request queue sze s gven as the followng queueng law [1]: (t+1) = [ (t) (t) (t),]+a (m) (t). (1) Fg. llustrates the queueng model n our system. Servce qualty. The servce qualty experenced by users s evaluated by request response delay, consstng of two maor components: queueng delay n the request queue, and round-trp delay from when the request s dspatched from the queue to the tme the frst byte of the requested fle s receved. We gnore the processng delay nsde a data center, due to the hgh nter-connecton bandwdth and CPU capactes nsde a data center. Let d and e denote the round-trp delay between regon and the prvate cloud, and between regon and data center, respectvely. Letαbe the upper-bound of the average round-trp delay per request, whch the applcaton provder wshes to enforce n ths content dstrbuton applcaton. We reasonably assume α > e, N,.e., ths bound s larger than the round-trp delay between a user and the data center n the same regon. We wll show that our dynamc optmal servce mgraton algorthm can bound both the average round-trp delay and queueng delay experenced by users. Operatonal cost. Our algorthm focuses on mnmzng recurrng operatonal cost of the content dstrbuton system, not one-tme costs such as the purchase of machnes n the prvate cloud and contents. The recurrng costs n each tme slot t nclude the followng categores: ) Bandwdth charge at the prvate cloud for uploadng contents to users, at the total amount of

5 5 m M N v(m) (t)h. N c(m) (t) ) Storage cost at data center, N, for cachng replcated contents, at the total amount of m M v(m) y (m) (t)p. ) Request servce cost at data center for uploadng replcated fles to users. The cost for servng fle m ncludes VM rental cost r f and upload bandwdth cost N v(m) (t)g. Let q (m) = f r + v (m) g denote the unt cost to serve each request for fle m on data center. The total cost of servng requests at data center s m M N c(m) (t)q (m). v) Mgraton cost for copyng fles from the prvate cloud to data center. Let w (m) denote the mgraton cost to copy fle m nto data center, whch ncludes costs of upload and download bandwdths from the prvate cloud to data center,.e., w (m) = v (m) (h + o ). The total mgraton cost ncurred at data center s m M [y(m) (t) y (m) (t 1)] + w (m) x f x and [x] + = f x <., where notaton[x] + = We wll not consder any recurrng storage cost on the prvate cloud (the purchase of storage dsks by the applcaton provder s a one-tme nvestment). In addton, the removal of contents from a data center s cost-free. Therefore, the overall operatonal cost to the applcaton provder n tme slot t s M(t) = v (m) (t)h+ v (m) y (m) (t)p m M N m M + (t)q (m) m M N + [y (m) (t) y (m) (t 1)] + w (m). () m M Optmzaton formulaton. The optmzaton pursued by our dynamc algorthm s formulated as follows, whch mnmzes the tme-averaged operatonal cost whle guaranteeng servce qualty. We use x(t) = 1 T lm T T t=1x(t) to represent the tme-averaged value of x(t). subect to: m M N mnm(t) (3) (t) b, t, (4) (t) µ y (m) (t), N, N,m M, t, (5) a (m) (t) (t)+ (t), N, m M, (6) ( N m M (t)d + α N m M (t)e ) ( (t)+ (t)), (7) (t) Z + {}, N,m M, t, (8) (t) Z + {}, N, N,m M, t, (9) y (m) (t) {,1}, N,m M, t. (1) Recall that each request s served by a unt bandwdth. (4) corresponds to the upload bandwdth lmt at the prvate cloud. (5) states that requests for a fle are only dspatched to data centers storng ths fle, and the mum number of requests dspatched from each request queue to a data center n each tme slot s no larger than µ. (6) represents each request queue s rate stable. In m M (s(m) (7), Γ 1 = N (t)+ c(m) (t)) s the average number of overall requests n the system per unt tme, and Γ = N m M (s(m) (t)d + c(m) (t)e ) s the total round-trp delay experenced by requests n the system per unt tme. Therefore, ths constrant specfes that the average round-trp delay per request, Γ Γ 1, should be bounded by α. Though queueng delay s not explctly modeled n the constrants, we wll show n Sec. 5 that our algorthm can smultaneously solve ths optmzaton and bound the queueng delay of each request wthn a pre-set threshold as well. 4 DYNAMIC MIGRATION ALGORITHM In ths secton, we desgn a dynamc control algorthm usng Lyapunov optmzaton technques, whch solves the optmal mgraton problem n (3) and bounds the tme-averaged round-trp delays and queueng delays for each request. We also dscuss ts practcal mplementaton. 4.1 Boundng Delays The optmzaton problem n (3) ncludes a constrant on tme-averaged varable values,.e., nequalty (7). Our dynamc algorthm wll only be able to adust varables n each tme slot. How can we guarantee ths nequalty by controllng the varable values over tme? To satsfy constrant (7), we resort to the vrtual queue technques n Lyapunov optmzaton [1]. We ntroduce a vrtual queue G, wth arrval rate m M (s(m) of N (t)d + c(m) (t)e ),.e., the overall round-trp delay experenced by all requests n t, and departure rate of α N m M (s(m) (t) + c(m) (t)),.e., the total number of requests n t multpled by the round-trp delay bound α. G s updated as follows: ( m M (t)e ) (t)),]. (11) G(t+1) = [G(t)+ (t)d + N α ( (t)+ N m M If queue G s stable, ts tme-averaged arrval rate should not exceed ts tme-averaged departure rate (accordng to Theorem.5 n [1]),.e., constrant (7) s satsfed. Therefore, we can adust the request dstrbuton strateges (t) s and (t) s n each tme slot to guarantee that ths vrtual queue s always stable, n order to satsfy constrant (7). Intutvely, when the sze of G s large,.e., when a rsk arses for constrant (7) to be volated,

6 6 requests should be dspatched more to the prvate cloud or data centers wth small round-trp delay from users; when the queue sze s small, requests can be dstrbuted based more on cost consderatons. Recall that our dynamc mgraton algorthm also seeks to bound queueng delays n the request queues Q m, N,m M. To bound the worst-case queueng delay of each request n all queues, N,m M, the ǫ-persstent servce queue technque [7] can be appled. Ths technque features carefully desgnng a set of vrtual queues. Any schedulng algorthm keepng that set of vrtual queues bounded ensures worst-case queueng delay of each request bounded. In partcular, we desgn a set of vrtual queues by assocatng queue, updated by: (t+1) = [ (t)+1 (m) {Q (t)>} (ǫ(m) wth a vrtual (t) (t)) 1 (m) {Q µ,], (1) (t)=} where ǫ (m) ( < ǫ (m) < µ ) s a constant that can be gauged to control the queueng delay bound, whch further renders a tradeoff between the queueng delay bound and the cost optmalty acheved by our algorthm (to be dscussed n Sec. 5). The ratonale behnd (1) can be explaned ntutvely: If request queue empty n tme slot t (.e., s not (t) > ), then a constant number of arrvals ǫ (m) are added nto vrtual queue, whle the departure rate from the vrtual queue, (t)+ c(m) (t), s the same as the departure rate from request queue. If the request queue s empty n t (.e., queue (t) = ), the length of the vrtual decreases by µ. In the algorthm desgn gven n Sec. 4., we wll strategcally decde (t) s and (t) s to keep the vrtual queue bounded. In ths way, requests are expedently dspatched from the request queue, resultng n lmted queueng delay per request. Detaled analyss s gven n Theorem n Sec Dynamc Algorthm Desgn va Drft-Plus- Penalty Mnmzaton Method Next we desgn a dynamc algorthm whch stablzes all queues and solves optmzaton (3). In our dynamc algorthm for the cost mnmzng problem, three types of queues are needed,.e., request queues ( N, m M), vrtual queue G, and vrtual queues ( N, m M). Let Θ(t) = [Q(t),G(t),Z(t)] be the vector of all queues n the system. Defne our Lyapunov functon as L(Θ(t)) = 1 [ m M N ( (t) + (t)) +G(t) ]. (13) The one-slot condtonal Lyapunov drft s (Θ(t)) = E{L(Θ(t+1)) L(Θ(t)) Θ(t)}. Followng the drft-plus-penalty framework n Lyapunov optmzaton (Chapter 5 n [1]), we can make the tmeaveraged operatonal cost M(t) wthn an upper bound of optmalty and stablze all queues, by mnmzng an upper bound of the followng tem n each tme slot: (Θ(t))+VM(t), where V s a non-negatve parameter set by the applcaton provder to control the tradeoff between the operatonal cost and request response delays. The ratonale s as follows: f n every tme slot greedly mnmzng Lyapunov drft (Θ(t)), backlogs are consstently pushed towards smaller, whch ntutvely mantans queues stable. Addng VM(t) (a weghted term of ncurred operatonal cost at tme slot t) onto (Θ(t)) allows a tradeoff between backlog reducton and operatonal cost mnmzaton at tme slot t. Although we do not mnmze (Θ(t)) + VM(t) drectly, mnmzng one of ts upper bound can have smlar effects. In the long term wth ths carefully desgned local optmzng obectve we stablze all queues and make the tme-averaged operatonal cost M(t) wthn an upper bound of optmalty. Now we derve an upper bound of (Θ(t))+VM(t) by squarng the queueng laws (1), (11) and (1) as follows (the detals are n Appendx D): (Θ(t))+VM(t) = B (t)[ (t)+(α d )G(t) +1 {Q (m) [ m M N (t)>} Z(m) m M (t) v (m) Vh] (t)+(α e )G(t)+1 (m) {Q +V [v (m) y (m) (t)p +[y (m) + m M N + m M N (t)[1 {Q (m) (t)a (m) (t), (t)>} ǫ(m) m M N (t) (t)>} Z(m) (t) Vq (m) ] (t) y (m) (t 1)] + w (m) ] 1 (m) µ] {Q =} (14) where B = 1 M N [A + ǫ + (b + Nµ ) ] ( M N µ e +bd ) + 1 α ( M N µ +b) s a constant, wth d = {d N}, e = {e N, N}, and ǫ = {ǫ (m) N,m M}. The mpact of constant B wll be shown n Theorem 3. In the followng we desgn an algorthm that mnmzes the the rght-hand-sde of nequalty (14), and wll dscuss queue stablty and cost optmalty n Sec. 5. To mnmze the rght-hand-sde of nequalty (14), the algorthm observes the queues (t), G(t) and (t) n each tme slot t, and decdes optmal values of (t) and (t), N, N,m M. We smplfy the notaton by defnng γ (m) (t) = (t)+1 (m) {Q (t)>} Z(m) whch s a constant n tme slot t, and η (m) (t) = (t)+1 (m) {Q (t)>} Z(m) (t) Vv (m) h+(α d )G(t), (t) Vq (m) +(α e )G(t),

7 7 whch s also a constant n t, and φ (m) (t) = V(v (m) p +1 (m) {y (t 1)=} w(m) ), whch s a constant n t as well, when y (m) (t 1) s gven. Therefore, mnmzng the rght-hand-sde of (14) s equvalent to: F(t) = (t)γ (m) (t)+ m M N m M N (t)η (m) (t) m M subect to: constrants (4) (5) (8) (9) (1). φ (m) (t)y (m) (t) (15) It s equvalent to solve the followng problems separately: (t)γ (m) (t) (16) m M N s.t. : constrant (8) and, for each m M and each N, (t)η (m) N (t) φ (m) s.t. : (t) µ y (m) (t), N constrants (9)(1) The soluton of (16) s: (t)y (m) (17) b f (m,) = arg (m M, N){φ (m ) (t)} (t) = and γ (m) (t) otherwse (17) can be solved by choosng the larger soluton value from the followng two cases: Case 1: y (m) (t) =. =, N. The optmal value of the obectve functon (17) s. Case : y (m) (t) = 1. (17) s equvalent to: (t)η (m) (t) φ (m) (18) N s.t. : (t) µ, N c Z + {}. The optmal soluton { of (18) s 1 (t) = µ f η (m) (t) otherwse Now we compare the optmal values of the obectve functon (17) n Case 1 and Case : If 1 N c(m) (m) (t)η (t) φ (m) >, the optmal soluton s y (m) = and (m) = c, N. Otherwse, the optmal soluton s y (m) = 1 and (m) 1 = c, N. The soluton means that for each data center, t wll choose mgratng the fle and uploadng the fle to users f the correspondng queues s long enough (η (m) (t) mples that the request queue and assocated vrtual queue s long) and the related weght surpasses the cost of mgratng the fle and/or keepng the fle n storage ( N c(m) (t)η (m) (t) φ (m) > ). 4.3 Dscussons on Practcal Implementaton Our dynamc algorthm s to be deployed by the applcaton provder to optmally dstrbute ts content dstrbuton servce onto the hybrd cloud. The applcaton provder deploys one or multple web servers provdng portal servce of the content dstrbuton applcaton, n a centralzed or dstrbuted fashon. The portal aggregates user requests and sends the collected request nformaton to a control center, whch executes our algorthm perodcally. The control center mantans a content placement table wth entres y (m), N,m M, ndcatng whether fle m s currently replcated on data center. The entres are ntalzed to be at the system ntalzaton stage. In each tme slot, receved requests for fle m orgnated from regon are placed n request queue. Vrtual queues and G are mantaned smply as counters. The control center observes lengths of the queues and request arrval rates, and calculates the optmal content placement and load dstrbuton strateges by solvng (15). Based on the derved content placement strateges, t updates the placement table, and compares the optmal soluton y (m) (t) aganst the current value of y (m) (t 1) n the table, N,m M: If y (m) (t 1) = and y (m) (t) = 1, the control center nstructs data center to request a copy of fle m from the prvate cloud; f y (m) (t 1) = 1 and y (m) (t) =, t sgnals data center to remove fle m from ts storage. Based on the request dstrbuton decsons, the control center dspatches (t) requests from queue to the prvate cloud, and (t) requests from the queue to data center,, N,m M. Vrtual queue and G are updated accordngly. The sketch of our complete dynamc, ont content placement and load dstrbuton algorthm s presented n Algorthm 1. As an engneerng parameter, the length of ntervals between two executons of the algorthm can be set by the applcaton provder, based on update frequences of contents, szes of the fles, as well as ts targeted performance optmalty. 5 PERFORMANCE ANALYSIS We next analyze the performance guarantee provded by our dynamc algorthm, wth respect to bounded queueng delay and optmalty n cost mnmzaton. 5.1 Bound of Queueng Delay Theorem 1: (Bound of Queue Length) Defne where = V(v (m) p +w (m) = V(v (m) p +w (m) +q (m) )+A, (19) +q (m) )+ǫ m, ()

8 8 Algorthm 1: Control Algorthm on the Control Center Intalzaton: Set up request queue, vrtual queues G and, N,m M, and ntalze ther backlogs to ; In every tme slot t: 1. Enqueue receved requests to request queues s);. Solve optmzaton (15) to obtan optmal content placement and load dstrbuton strateges (t), ( (t), y (m) (t),, N,m M; 3. Update content placement table wth y (m) and mgrate fles as follows: for N,m M do f y (m) (t 1) = and y (m) (t) s, (t) = 1 then nstruct data center to request fle m from prvate cloud; f y (m) (t 1) = 1 and y (m) (t) = then sgnal data center to remove fle m; 4. Dspatch (t) requests from queue to prvate cloud, (t) requests to data center,, N,m M; 5. Update vrtual queue and G accordng to Eqn. (1) and (11); = argmn {v (m) p +w (m) +q (m) α e >, N}. (1) Then we have (t) (m) and Z (t) (m) (m), N, m M,.e., Q and Z s the mum sze of queue t respectvely. and at any tme The proof s gven n Appendx A. Based on Theorem 1, we have Theorem : (Bounded Queueng Delay): For each request queue, N, m M, defne = Q(m) (m) +Z ǫ (m) where s defned as n (1). The queueng delay of each request n s bounded by. The proof s gven n Appendx B. 5. Optmalty aganst the T-Slot Lookahead Mechansm Snce request arrval rates are arbtrary n our system, t s dffcult to fnd the global cost optmum, wth whch to compare the tme-averaged cost M(t) acheved by our algorthm. Therefore we utlze a local optmum target, whch s the optmal (obectve functon) value of a smlar cost mnmzaton problem wthn known nformaton (e.g., request arrvals) for T tme slots nto the future,.e., a T-slot lookahead mechansm [1]. We wll show that the optmal value obtaned by our algorthm s close to that of the T-slot lookahead mechansm, even f our algorthm does not assume any future nformaton. In the T-slot lookahead mechansm, tme s dvded nto successve frames, each consstng of T tme slots. Denote each frame as F k = {kt + 1,kT +,...,kt + T}, where k =,1,... In each tme frame, consder the followng optmzaton problem on varables (t), (t),y (m) (t), N, N,m M, t F k : mn 1 kt+t M(t) () T subect to: m M N t=kt+1 (t) b, t F k, (3) (t) µ y (m) (t), N, N,m M,t F k, (4) kt+t [a (m) t=kt+1 1 T kt+t [ t=kt+1 kt+t m M( t=kt+1 N α (t) (t) (t)], N,m M, (5) (t)+ (t)] ǫ (m), N,m M (6) kt+t m M( t=kt+1 N (t)e + (t)d ) (t)+ (t)), (7) (t) Z + {}, N,m M,t F k, (8) (t) Z + {}, N, N,m M,t F k (9) y (m) (t) {,1}, N,m M,t F k. (3) Assumng full knowledge of request arrvals n the T tme slots n F k, ths optmzaton derves the optmal content placement and load dstrbuton decsons n each of the T tme slots, whch mnmze the average cost per tme slot n the obectve functon. We show the tme-averaged cost M(t) acheved by our algorthm s wthn a constant gap BT V from that acheved by solvng the above optmzaton: Theorem 3: (Optmalty of Cost) Let M k denote the optmal obectve functon value n the T-slot Lookahead problem () n tme frame F k. The mnmum operatonal cost derved wth our algorthm s M(t) n tme slot t. Suppose the tme lasts for KT tme slots, where K s a constant. We have KT 1 1 M(t) 1 KT K t= K 1 k= M k + BT V, (31).e., our algorthm acheves a tme-averaged cost wthn constant gap BT V from that by assumng full knowledge n T tme slots n the future.

9 Workload Fg. 3. The key modules of our prototype. Theorem 3 s proved n detals n Appendx C. Theorems and 3 show that when V ncreases, worstcase queueng delay ncreases, whle the gap between the operatonal cost of our algorthm and that of the T-Slot lookahead mechansm s reduced. ǫ (m) has a smlar effect: when ǫ (m) ncreases, the worst-case queueng delay decreases, and B ncreases such that the gap to optmalty ncreases. We wll nvestgate proper values to assgn to these tradeoff control parameters n our evaluaton n Sec PERFORMANCE EVALUATION 6.1 Experment Setup We evaluate the performance of the dynamc algorthm wth a prototype deployed on sx VMs that are located n sx data centers (n the ctes of Dallas, Fremont, Atlanta, Newark, London, Tokyo) of Lnode Cloud [8] and a cluster resdng n our lab n the Unversty of Hong Kong. We deploy the web portal and the control center on one Lnode VM and use t to emulate the prvate cloud at the same tme, whle we use the remanng fve Lnode VMs to emulate fve data centers of the publc cloud. When nstructed by the control center, the Requester module at the publc cloud s responsble for requestng a copy of a fle from the Uploader module at the prvate cloud. The communcaton among the control center, the prvate cloud and the publc cloud s va our customzed protocol encapsulated n TCP. We emulate the users of the content dstrbuton servce usng our lab cluster. Each user s emulated by an ndependent thread, whch communcates wth the portal, the publc cloud and the prvate cloud va HTTP protocol. We mplement a web-based fle download applcaton. Users ssue requests n the form of HTTP requests for the fles. A user wll be responded wth a URL redrecton command pontng to the data center from whch the user can download a copy of the requested fle. The key modules of the prototype are llustrated n Fg. 3. The dstrbuton of the fle szes follows the measurement result of YouTube vdeos n [9], wth a mean sze of 7.6 MB and an upper bound of 5 MB (because 99.1% of YouTube vdeos are less than 5 MB n sze). The total number of fles s 1. The total number of requests for all fles follows a Posson dstrbuton over tme, wth a mean of.15 request per fle per tme slot. All the concurrent fle uploadngs share the dynamcal overall bandwdth of a VM, whch s generally larger than 15M bps. The number of fles and the mean number of requests for a fle are so set due to the bandwdth capacty of Lnode VMs avalable to us. Especally, the scale of experment s approxmately restrcted by the followng: Mean fle sze Mean # of requests per fle per tme slot # of fles # of seconds per tme slot Bandwdth per VM # of VMs Nevertheless, we beleve that the system can scale up smoothly wthout degraded performance when the system s deployed on real elastc clouds. We brefly dscuss how the system wll behave when the system s scaled up: When the average fle sze becomes larger, the applcaton provder adusts the system parameters b (mum number of requests that the prvate cloud can serve n a tme slot) and r (number of requests a VM n data center can serve n a tme slot), or use VMs that have hgher bandwdth capacty, to make sure each fle can be uploaded wth a tme slot. When the workload s more ntensve, more VMs wll be rented from the publc cloud to serve more requests. Snce our control center s mplemented n a dstrbuted fashon, t would not become the bottleneck of the system. We wll empercally study the case when the number of fles ncreases n Sec We splt the total number of requests onto dfferent fles n dfferent regons, n each tme slot. The relatve access frequency of the fles follows a Zpf-lke dstrbuton [3] wth parameter.8. We emulate the geographcal dversty of requests by splttng the requests among regons followng a bnomal dstrbuton. We assgn the probablty that a request s dspatched to regon ( =,1,,3,4,5) to be the probablty of obtanng exactly successes out of 6 Bernoull trals. For dstrbutng the requests for each fle, we assgn each of the 6 regons an ndex by arrangng them n a random order. The mum number of requests arsng from each regon for each fle n one tme slot, A, s, and the mum number of requests dspatched from a queue to a data center per tme slot, µ, s 4. The mpact of the settng of A and µ s neglgble as long as they are sgnfcantly larger than the mean number of requests for a fle n each tme slot (.15 as set above).

10 Cost We emulate a chargng mechansm n the prototype as follows, nstead of relyng on natve chargng method of Lnode Cloud (Lnode allows us to transfer 4TB of data per month for free based on our two-year rental contract). All charges below are n the unts of US dollars. The cost of uploadng data from the prvate cloud s $1 1 1 per byte, whch s the average prce for Internet access provded by typcal hostng servce provders [31][3]. The charges by the publc cloud are extracted from real settngs of Amazon EC and S3 [5][6]. We estmate VM rental cost per fle accordng to the followng: VM rental cost per fle = Rental cost of a VM nstance Upload bandwdth per VM/Mean sze of a fle. Snce the rental cost of ndvdual VM nstances s not avalable n the Lnode chargng model, we set t accordng to comparable charges n Amazon EC, as the rental cost of a typcal Amazon EC VM wth upload bandwdth of 5 Mbps [33], whch s $.7 per hour. The charge of storng a byte on a data center s n the range [$ ,$ ]. The cost of uploadng from a data center n the publc cloud s randomly selected from the set {$.5 1 1,$.7 1 1,$.9 1 1,$ } (per byte), whch are prces of the data upload servce offered by Amazon EC for dfferent scales of purchases [5]. Accordng to the current cloud busness model [6], there s no charge for downloadng data nto the cloud,.e., o =, N Delays In the control center we set the round-trp delay between users n a regon and a data center n another regon as the real latency we obtaned by pngng the respectve Lnode VMs. We set the round-trp delay between a user and the data center n the same regon e to be 5 ms for all regons. The average round-trp delay bound set by the applcaton provder,.e., α, s ms, snce a RTT more than ms wll brng users poor experence [34]. ǫ (m) s are set proportonal to (m) + Z make the queueng delay bounds to s the same for each request. By default the target s are and the mpact of ts other values wll be evaluated n Sec Other Parameters The duraton of a tme slot s 1 seconds. The duraton of a tme slot s set based on the followng practcal consderatons: On one hand, runnng the optmzaton solver too frequently s too costly, and snce fle mgraton s nvolved, t s unlkely to be done n a tme scale smaller than a few seconds; on the other hand, the duraton of a tme slot should not be too long, as otherwse queueng delays experenced by requests tend to be too long. After some trals, we fnd 1-seconds s an approprate value. The default value of V s 1, and ts mpact on the system performance wll be evaluated n Sec Cost per mnute ($) Our dynamc algorthm IPMW wth Mn-Weght Evctons Myopc schedulng algorthm 4 6 Tme(mnutes) Fg. 4. Cost comparson among our dynamc algorthm, IPMW wth Mn-Weght Evctons, and the Myopc Schedulng Algorthm (average costs are.5, 6.8, 35.6 dollars respectvely). Intally, the fles are not deployed n the data centers,.e., y (m) () =, N, m M. Under each confguraton, we run the prototype system for 6 mnutes, for multple tmes. The data presented n Sec. 6. and Sec. 6.3, whch show temporal dynamcs of the system, are collected n a representatve run of the experment, because we observe that the results of multple runs reveal the same pattern, whle the data presented n Sec. 6.4 are the average of the results n 1 runs of the experments. 6. Cost Optmalty 6..1 Comparson wth Exstng Algorthms We frst compare our dynamc algorthms aganst a smple Myopc Schedulng Algorthm and another exstng algorthm for content placement and request routng for tradtonal CDN [14], named Iteratve Perodc Max- Weght Schedulng wth Mn-Weght Evctons (abbrevated as IPMW wth MWE). The Myopc Schedulng Algorthm processes all requests n the tme slot when they arrve wthout bufferng them n queues, and decdes content replcaton and request dstrbuton by mnmzng overall operatonal cost (3) under constrants (4)(5)(7)(8)(9)(1) by changng all tme-average expressons to that of the current sngle tme slot. Smlar to our work, IPMW wth MWE Algorthm models the system of frontend source nodes and backend cache (backend cache n [14] s a synonym for space of the storage server n ths paper) of the CDN as the nput and output nodes of a swtch. It bulds queues for requests for dfferent fles at the source nodes, makes decsons on request routng n every tme slot, and refreshes contents of backend cache perodcally. But dfferent from our algorthm, the sze of the cache s statc, whch renders a trade-off between the storage cost and queueng delays. To make the comparson far, we do a bnary search for the optmal sze of ts backend cache whch leads to the smallest cost, under the constrant that the queueng delays of more than 9% of requests are wthn the

11 11 Cost per mnutes ($) T-slot look ahead mechansm Our dynamc algorthm 4 6 Tme(mnutes) Fg. 5. Cost comparson among our dynamc algorthm and the T-slot lookahead mechansm ( M = 1) (average costs are.6 and 3. dollars respectvely). specfc target. Another parameter that needs to be set for IPMW wth MWE s the perodcty when the content of the backend cache s allowed to be refreshed. To compare farly, we set the perodcty to be 1, the same as what s allowed n runnng our dynamc algorthm. Fg. 4 shows the overall cost ncurred at each tme slot when each method s appled. We observe that the cost ncurred by our dynamc algorthms s lower than that by the Myopc Schedulng Algorthm and IPMW wth MWE at all tmes. Our dynamc algorthm outperforms IPMW wth MWE not only n terms of cost reducton, but also n that our dynamc algorthm can guarantee the queueng delays of 1% of requests are wthn the specfc QoS whle under the same setup IPMW wth MWE can only guarantee 9%. Ths s because (1) our algorthm s aware of worst-case queueng deadlnes whle IPMW wth MWE s not; () our dynamc algorthm flexbly occupes caches of sutable szes on the fly due to ts deployment n an elastc cloud, whle IPMW wth MWE occupes caches of fxed szes, whch tends to ncur more cost; (3) our dynamc algorthm ams to strke a good trade-off between delays and cost, whle IPMW wth MWE only shortens the queue lengths n the best effort fashon. Cost per mnute ($) V=1 V=5 V=1 V=3 4 6 Tme(mnutes) Fg. 6. Cost wth dfferent V (from V = 1 to 3, average costs are 31.6, 7.4, 5.3, 3. dollars respectvely). Avg. servce response delay (seconds) V=1 V=5 V=1 V=3 4 6 Tme (mnutes) Fg. 7. Average servce response delay wth dfferent V (from V = 1 to 3, average delays are 91.5, 116.6, 137.9, seconds respectvely). To derve the optmal soluton, we use an open source tool MOSEK [35]. We fnd that t s very tme consumng to solve the T-slot lookahead problem n (3) usng the MOSEK solver n our default system scale, even when we are merely optmzng the decsons n T = 6 tme slots. Therefore, n ths set of experments, we reduce the total number of fles to 1. Fg. 5 shows that the gap between the costs acheved by our algorthm and those by the T-slot lookahead mechansm wth known future nformaton, s close. 6.. Comparson wth T-slot Lookahead Mechansm We next compare our algorthm aganst the T-slot lookahead mechansm. In order to solve () whch contans non-lnear tems [y (m) (t) y (m) (t 1)] +, we convert the problem () to an equvalent problem as follows: kt+t mn 1 [ T + m M N t=kt+1 m M N subect to: (t)q (m) v (m) + (t)h+ m M m M v (m) y (m) (t)p y (m) (t)w (m) ] (3) (3)(4)(5)(6)(7)(8)(9)(3) and, for N, m M, t, y (m) (t) y (m) y (m) (t) (t) y (m) (t 1) 6.3 Impact of Algorthm Parameters We next study the mpact of mportant parameters n our algorthm, on the tradeoff between cost optmalty and servce response delays Impact of V Fg. 6 shows that when V ncreases, the overall operatonal cost becomes smaller. Fg. 7 reveals that the average servce response delay per request (queueng delay+round-trp delay) ncreases wth the ncrease of V, whle Fg. 8 shows the ncrease of the average of mum request queue lengths (.e., the average of the mum lengths that the request queues have ever reached untl each tme slot) wth the ncrease of V as well. These fgures clearly show a tradeoff n V s settng. Selectng V = [5, 1] can acheve a good tradeoff between cost optmalty and servce qualty.

12 1 Avg. of. queue backlogs 1 5 V=1 V=5 V=1 V=3 4 6 Tme (mnutes) Fg. 8. Average of mum queue lengths wth dfferent V (from V = 1 to 3, average lengths are 9., 13.9, 4.6, 34.1 respectvely). Cost per mnute ($) = 3 = 5 = = 15 = Tme (mnutes) Fg. 9. Cost wth dfferent s (from = 3 to 1, average costs are 1.3, 3.7, 6.1, 8.6, 33.5 dollars respectvely). Avg. servce response delay(seconds) = 3 = 5 = = 15 = Tme (mnutes) Fg. 1. Average servce response delay wth dfferent s (from = 3 to 1, average delays are 31.4, 1.5, 143.9, 7,9, 39.6 seconds respectvely) Impact of ǫ (m) The parameters ǫ (m) queueng delay bound are controlled by preset target s. In Fg. 9 and Fg. 1, we observe that when decreases, the overall operatonal cost ncreases whle the average servce response delay per request decreases. Ths shows that the value of also renders a tradeoff between cost optmalty and servce qualty. When s smaller than, the cost ncreases sgnfcantly. Therefore selectng ǫ around would be a good choce. Bandwdth (Kbps) 6 4 Our dynamc algorthm IPMW wth MWE No. of fles Fg. 11. Tme-averaged control messagng bandwdth. CPU tme (ms) 3 1 Our dynamc algorthm IPMW wth MWE No. of data centers n the publc cloud Fg. 1. Average computaton tme consumed by the control center. Memory consumpton (MB) Fg. 13. center Our dynamc algorthm IPMW wth MWE No. of fles Average memory consumpton by the control 6.4 Overhead Comparson Now we compare the overhead between our dynamc algorthm and IPMW wth MWE, n terms of control messagng bandwdth, computaton tme, and memory consumpton respectvely. In ths secton, n order to see how the overhead grows when the number of fles grows, we proportonally shrnk down the sze of each fle, n order to accommodate more fles wth our bandwdth-lmted Lnode VMs Control Messagng Bandwdth Fg. 11 shows that IPMW wth MWE consumes less bandwdth for control messagng than our dynamc algorthm, and the overhead of both grows lnearly wth the number of fles. We analyze the reason by decomposng the control messages nto the followng

13 13 categores: (1) HTTP requests from users to the portal, () URL redrecton, (3) commands for fle replcaton, and (4) requests for fles from the publc cloud to the prvate cloud. In terms of (1) and (), both algorthms generate smlar numbers of messages. In terms of (3) and (4), our algorthm generates more messages, because our algorthm makes use of the elastcty of clouds to flexbly mgrate fles so as to mnmze the operatonal cost. However, we notce that ncreased control messagng overhead only consttutes less than.5% of bandwdth consumpton of the whole system, whch s neglgble Computaton Tme Fg. 1 plots the CPU tme consumed when the control center runs the respectve schedulng algorthm once n each tme slot. It shows that when the number of data centers ncreases, the CPU tme by both algorthms grows, but our algorthm consumes less CPU tme than IPMW wth MWE. Ths s because the computaton tme of our dynamc algorthm s O( N M ) whle the computaton tme of IPMW wth MWE Algorthm s O( N! N M ) Memory Consumpton Fg. 13 llustrates the memory consumpton of the control center. It shows that the memory consumed by both algorthms grows lnearly wth the number of fles, and our algorthm consumes slghtly less memory than IPMW wth MWE. Memory consumpton manly conssts of space for storng temporary results for the schedulng decsons, and space for the queues. Both algorthms have queues to buffer the requests. Although our dynamc algorthm has more queues (except request queues, t has vrtual queues), each vrtual queue s only represented by a sngle real number, wth neglgble memory consumpton. Our algorthm ncurs smaller average queue backlogs, and hence smaller overall memory consumpton. 7 CONCLUSION Ths paper nvestgates optmal mgraton of a content dstrbuton servce to a hybrd cloud consstng of a prvate cloud and publc geo-dstrbuted cloud servces. We propose a generc optmzaton framework based on Lyapunov optmzaton theory, and desgn a dynamc, ont content placement and request dstrbuton algorthm, whch mnmzes the operatonal cost of the applcaton wth QoS guarantees. We theoretcally show that our algorthm approaches the optmalty acheved by a mechansm wth known nformaton n the future T tme slots by a small gap, no matter what the request arrval pattern s. Our prototype-based evaluaton verfes our theoretcal fndngs. We ntend to extend the framework to specfc content dstrbuton servces wth detaled requrements, such as vdeo-on-demand servces or socal meda applcatons, n our ongong work. ACKNOWLEDGMENT The research was supported n part by a grant from Hong Kong RGC under the contract HKU 71781E. APPENDIX A PROOF OF THEOREM 1 A.1 Provng (t), N, m M, t Proof: We prove t by nducton. Inducton Bass: Accordng to the assumpton of our model, we have () =, N, m M. Inducton steps: We assume that (t) and then we show If V(v (m) p +w (m) If. (t+1) (t) V(v (m) p+w (m) +q (m) ), then (t+1) )+A = +q (m) (t) > V(v (m) p+w (m) +q (m). ), then accordng the defnton of, we have α e >. Therefore η(m) (t) > φ (m) (t). Accordng the soluton to 17), we have y (m) (t) = 1 and (t) = µ. (t+1) = [ (t) (t) (t) (t) (t) (t)+a (m) = (t) µ +a (m) (t) µ +A < (t) (t)+a (m) (t),]+a (m) Ths means that the length of can not ncrease n tme slot t+1. Therefore, (t+1) <. A. Provng (t), N, m M, t Proof: We prove t by nducton. Inducton Bass: Accordng to the assumpton, we have () =. Inducton Steps: We assume that (t) (t+1) + q (m) (t + 1) V(v (m) p + w (m) and then we show If (t) V(v (m) p + w (m) to (1),.., ), then accordng + q (m) ) + ǫ (m) = If (t) > V(v (m) p + w (m) + q (m) cases, whch we are gong to dscuss respectvely: ), there are two Case (1). When (t) > : We have η (m) (t) > φ (m) (t). Accordng the soluton to (17), we have (t) = µ. Then (t+1) = [ (t)+ǫ (m) (t)+ǫ (m) (t)+ǫ (m) < (t) (t) (t),] (t) (t) µ

14 14 Case (). When (t) = : Accordng (1), (t+ 1) = [ (t) µ,] (t). APPENDIX B PROOF OF THEOREM Proof: Suppose a (m) (t ) requests arrve at queue at tme slot t. We prove these requests can depart the queue by tme t + by contradcton. If these requests haven t left the queue by t + must be that (t) > for all t {t +1,...,t + Then for all t {t +1,...,t + (t+1) = [ (t)+ǫ (m) Therefore we have (t+1) (t)+ǫ (m) } we have:, t }. (t) (t),]. (t) (t). Summng up the above over t {t +1,...,t + } yelds: (t + +1) (t +1) Because, we have: t + t=t +1 ǫ (m) (t + t +W (m) t=t +1 +1) [ (t)+ (t)] ǫ (m) [ (t)+ (t)]. and (m) Z (t +1) On the other sde, because we assume that not all (t ) requests depart by the tme t +, we have: a (m) t + t=t +1 [ (t)+ (t)] < (t +1) Therefore we have (m) > ǫ < Q(m) (m) +Z ǫ (m).,.e.,. Ths contradcts the defnton of. Therefore, any request that arrved at tme t wll be dspatched by tme slot t +. APPENDIX C PROOF OF THEOREM 3 T(Θ(t))+V BT + m M N +G(t)( N + m M N t+t 1 τ=t M(τ) t+t 1 (t) m M α N τ=t t+t 1 τ=t (a (m) ( m M( t+t 1 (t) τ=t (τ) (τ) (τ)) (τ)e + (τ)d) (m) (τ)+s (τ))) (1 (m) {Q (τ)>} (ǫ(m) τ=t (τ)) 1{ µ) (τ)=} t+t 1 +V ( + + m M N m M N + m M N [y (m) m M N (m) (τ)q v (m) v (m) y (m) (τ)h (τ)p (τ) (m) (τ) y (τ 1)] + w (m) ), where (m) (m) (τ), c (τ), and y (τ),, N,m M, are any alternatve decsons that can be made n tme slot τ wthn the feasble set. Proof of Theorem 3: Because (m) (τ), c (τ), and y (m) (τ),, N,m M, are any alternatve decsons that can be made n tme slot τ wthn the feasble set, apparently, (m) (m) (τ), c (τ), and y (τ),, N,m M, can be the optmal soluton of the problem wth nformaton of T tme slots nto future that mnmzes Eqn. (). Then, combnng wth Lemma 1, we derve T(Θ(t))+V t+t 1 τ=t M(τ) BT +V t+t 1 τ=t M (τ). Consderng the total K frames and summng the above over k {,...,K 1} and then dvdng the sum by VKT, we get L(Θ(KT)) L(Θ()) VKT + 1 KT 1 M(t) BT KT V + 1 K 1 M k. K t= k= Rearrangng the terms n the above nequalty, and notng that L(Θ(KT)) and L(Θ()) =, we derve Defne T-slot Drft as T (Θ(t)) = L(Θ(t+T)) L(Θ(t)). Based on Lemma 4.11 n [1], we have Lemma 1: (T-slot Drft) Wth our dynamc algorthm, for all t, all Θ(t), and for any nteger T > we have: KT 1 1 M(t) 1 KT K τ= K 1 k= M k + BT V.

15 15 APPENDIX D DERIVATION OF (14) (Θ(t)) = E{ 1 m M N + 1 m M N = 1 m M N + 1 m M N We have 1 m M N = 1 m M N (t) ] 1 m M N + m M N [ (t+1) (t) ]+ 1 [G(t+1) G(t) ] [ (t+1) (t) ] [Q(t),G(t),Z(t)]} [ (t+1) (t) ]+ 1 [G(t+1) G(t) ] [ (t+1) (t) ] [ (t+1) (t) ] [([ (33) (t) (t) (t),]+a (m) (t)) [( (t)+ (t)) +a (m) (t) ] (t)[a (m) (t) (t) 1 [ M N (b+nµ) +A ] + (t)[a (m) (t) (t) m M N and 1 [G(t+1) G(t) ] = 1 [([G(t)+ ( (t)d + N m M α ( (t)+ N m M 1 [ ( (t)d + (t)e )] N m M + 1 α [ ( (t)+ (t))] N m M +G(t)[ m M( N (t)] (t)], (t)e ) (t)),]) G(t) ] (t)e + (t)d ) (34) 1 ( M N µ e +bd ) + 1 α ( M N µ +b) +G(t)[ m M( (t)e + (t)d ), N (35) and 1 m M N = 1 m M N [ (t+1) (t) ] [([ (t)+1 {Q (m) (t)>} (ǫ(m) (t) (t)) 1 (m) {Q (t)=} µ,]) (t) ] 1 [1 { + m M N (t)>} (ǫ(m) 1 (m) µ] {Q =} (t) (t)[1 {Q (m) 1 M N [ǫ +(b+nµ ) ] + (t)[1 (m) {Q m M N (t)>} (ǫ(m) (t)>} (ǫ(m) (t)) 1 (m) {Q (t)=} µ] (t) (t)) (t) (t)) 1 (m) µ]. {Q =} (36) We defne a constant B = 1 M N [A + ǫ + (b + Nµ ) ] ( M N µ e + bd ) + 1 α ( M N µ +b), where d = {d N}, e = {e N, N}, and ǫ = {ǫ (m) N,m M}. Combnng (34)(35)(36) together, we have (Θ(t)) B + (t)[a (m) (t) (t) m M N +G(t)[ m M( (t)e + (t)d ) N α m M( (t)+ (t))]+ N m M N [1 {Q (m) (t)>} (ǫ(m) Therefore, we get (t) (t)] (t) (t)) 1 (m) µ] {Q =} (Θ(t))+VM(t) B + (t)[a (m) (t) (t) m M N +G(t)[ m M( (t)e + (t)d ) N α m M( (t)+ (t))]+ N m M N [1 (m) {Q +V[ m M (t)>} (ǫ(m) (t) + (t)q (m) m M N + v (m) y (m) (t)p m M N + [y (m) = B m M N +1 {Q (m) (t)] (t) (t)) 1 (m) µ] {Q =} m M N (t) y (m) (t 1)] + w (m) ] (t)[ (t)+(α d )G(t) (t)>} Z(m) (t) v (m) Vh] m M v (m) (t)h N (t) [ (t)+(α e )G(t)+1 (m) {Q (t)>} Z(m) (t) Vq (m) ] +V [v (m) y (m) (t)p +[y (m) (t) y (m) (t 1)] + w (m) ] m M + (t)[1 (m) {Q (t)>} ǫ(m) 1 (m) µ] {Q =} m M N + (t)a (m) (t). m M N

16 16 REFERENCES [1] Amazon CloudFront, [] Mcrosoft Azure, [3] Google App Engne, [4] Dropbox, [5] Mcrosoft Offce Web Apps, [6] Google docs, [7] M. Haat, X. Sun, Y. E. Sung, D. Maltz, and S. Rao, Cloudward Bound: Plannng for Benefcal Mgraton of Enterprse Applcatons to the Cloud, n Proc. of IEEE SIGCOMM, August 1. [8] H. Zhang, G. Jang, K. Yoshhra, H. Chen, and A. Saxena, Intellgent Workload Factorng for a Hybrd Cloud Computng Model, n Proc. of the Internatonal Workshop on Cloud Servces (IWCS 9), June 9. [9] H. L, L. Zhong, J. Lu, B. L, and K. Xu, Cost-effectve Partal Mgraton of VoD Servces to Content Clouds, n Proc. of IEEE CLOUD, July 11. [1] X. Cheng and J. Lu, Load-Balanced Mgraton of Socal Meda to Content Clouds, n Proc. of NOSSDAV, June 11. [11] L. Georgads, M. J. Neely, and L. Tassulas, Resource allocaton and cross-layer control n wreless networks, Foundatons and Trends n Networkng, vol. 1, no. 1, pp , 6. [1] M. J. Neely, Stochastc Network Optmzaton wth Applcaton to Communcaton and Queueng Systems. Morgan & Claypool, 1. [13], Energy optmal control for tme varyng wreless networks, IEEE Tran. on Informaton Theory, no. 7, pp , July 6. [14] M. M. Amble, P. Parag, S. Shakkotta, and L. Yng, Content- Aware Cachng and Traffc Management n Content Dstrbuton Networks, n Proc. of IEEE INFOCOM, Aprl 11. [15] M. J. Neely and L. Golubchk, Utlty Optmzaton for Dynamc Peer-to-Peer Networks wth Tt-For-Tat Constrants, n Proc. of IEEE INFOCOM, Aprl 11. [16] M. Pathan, J. Broberg, and R. Buyya, Maxmzng Utlty for Content Delvery Clouds, n Proc. of the 1th Internatonal Conference on Web Informaton Systems Engneerng, 9. [17] F. Chen, K. Guo, J. Ln, and T. L. Porta, Intra-cloud Lghtnng: Buldng CDNs n the Cloud, n Proc. of IEEE INFOCOM, 1. [18] H. L, W. Huang, C. W. abd Z. L, and F. C. Lau, Utlty- Maxmzng Data Dssemnaton n Socally Selfsh Cogntve Rado Networks, n Proc. of IEEE Internatonal Conference on Moble Ad-hoc and Sensor Systems (IEEE MASS 11), Oct 11. [19] S. T. Magulur, R. Srkant, and L. Yng, Stochastc Models of Load Balancng and Schedulng n Cloud Computng Clusters, n Proc. of IEEE INFOCOM, 1. [] S. Ren, Y. He, and F. Xu, Provably-Effcent Job Schedulng for Energy and Farness n Geographcally Dstrbuted Data Centers, n Proc. of IEEE ICDCS, 1. [1] N. Laoutars, G. Smaragdaks, K. Okonomou, I. Stavrakaks, and A. Bestavros, Dstrbuted Placement of Servce Facltes n Large-Scale Networks, n Proc. of IEEE INFOCOM, 7. [] J. Leblet, Z. L, G. Smon, and D. Yuan, Optmal Network Locaton n Dstrbuted Vrtualzed Data-Centers, Computer Communcatons, no. 16, pp , 11. [3] S. H. Owen and M. S. Daskn, Strategc Faclty Locaton: A Revew, pp , [4] M. Charkar, S. Guha, E. Tardos, and D. B. Shmoys, A constantfactor approxmaton algorthm for the k-medan problem, n Proc. of the 31st Annual ACM Symposum on Theory of Computng (STOC 99), [5] Amazon Elastc Compute Cloud, [6] Amazon Smple Storage Servce, [7] M. J. Neely, Opportunstc Schedulng wth Worst Case Delay Guarantees n Sngle and Mult-Hop Networks, n Proc. of IEEE INFOCOM, 11. [8] Lnode,, [9] X. Cheng, J. Lu, and C. Dale, Understandng the Characterstcs of Internet Short Vdeo Sharng: A YouTube-Based Measurement Study, IEEE Transactons on Multmeda, vol. 15, no. 5, pp , 13. [3] L. Breslau, P. Cao, L. Fan, G. Phllps, and S. Shenker, Web Cachng and Zpf-lke Dstrbutons: Evdence and Implcatons, n Prof. of IEEE INFOCOM, [31] SoftLayer, [3] Layered Tech, [33] X. Xng, J. Dang, S. Mshra, and X. Lu, A Hghly Scalable Bandwdth Estmaton of Commercal Hotspot Access Ponts, n Proc. of IEEE INFOCOM, 11. [34] R. Kuschng, I. Koer, and H. Hellwagner, Improvng Internet Vdeo Streamng Performance by Parallel TCP-based Request- Response Streams, n Proc. of CCNC, Jan. 1. [35] MOSEK,