Resource Auto-Scaling and Sparse Content Replication for Video Storage Systems

Transcription

1 1 Reource Auto-Scaling and Spare Content Replication for Video Storage Sytem DI NIU 1, HONG XU 2 and BAOCHUN LI 3, Univerity of Alberta 1, City Univerity of Hong Kong 2, Univerity of Toronto 3 Many video-on-demand (VoD) provider are relying on public cloud provider for video torage, acce and treaming ervice. In thi paper, we invetigate how a VoD provider may make optimal bandwidth reervation from a cloud ervice provider to guarantee the treaming performance while paying for the bandwidth, torage and tranfer cot. We propoe a predictive reource auto-caling ytem that dynamically book the minimum amount of bandwidth reource from multiple erver in a cloud torage ytem, in order to allow the VoD provider to match it hort-term demand projection. We exploit the anticorrelation between the demand of different video for tatitical multiplexing to hedge the rik of under-proviioning. The optimal load direction from video channel to cloud erver without replication contraint i derived with provable performance. We further tudy the joint load direction and pare content placement problem that aim to reduce bandwidth reervation cot under pare content replication requirement. We propoe everal algorithm, and epecially an iterative L 1 -norm penalized optimization procedure to efficiently olve the problem while effectively limiting the video migration overhead. The propoed ytem i backed up by a demand predictor that forecat the expectation, volatility and correlation of the treaming traffic aociated with different video baed on tatitical learning. Extenive imulation are conducted to evaluate our propoed algorithm, driven by the real-world workload trace collected from a commercial VoD ytem. Categorie and Subject Decriptor: Information ytem [Information torage ytem]: Storage architecture Cloud baed torage; Network [Network performance evaluation]: Network performance modeling General Term: Performance Evaluation; Algorithm; Optimization. Additional Key Word and Phrae: Video-on-Demand; Cloud computing; Auto-caling; Content placement; Load direction; Optimization; Spare deign; Prediction. ACM Reference Format: ACM Tran. Appl. Percept. 1, 1, Article 1 (January 1111), 3 page. DOI = / INTRODUCTION Cloud computing i redefining the way many Internet ervice operate, including Video-on-Demand (VoD). Intead of buying rack of erver and building private datacenter, it i now common for VoD ervice provider to leverage the computing, network and torage reource of cloud ervice provider for video torage and treaming. A an example, Netflix place it video data tore, treaming erver, Author contact information: D. Niu, dniu@ualberta.ca; H. Xu, henry.xu@cityu.edu.hk; B. Li, bli@ece.toronto.edu. Part of thi work ha been preented at IEEE INFOCOM 212. Permiion to make digital or hard copie of part or all of thi work for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie how thi notice on the firt page or initial creen of a diplay along with the full citation. Copyright for component of thi work owned by other than ACM mut be honored. Abtracting with credit i permitted. To copy otherwie, to republih, to pot on erver, to reditribute to lit, or to ue any component of thi work in other work require prior pecific permiion and/or a fee. Permiion may be requeted from Publication Dept., ACM, Inc., 2 Penn Plaza, Suite 71, New York, NY USA, fax +1 (212) , or permiion@acm.org. c 1111 ACM /1111/1-ART1 $15. DOI /.

2 1:2 D. Niu, H. Xu and B. Li encoding oftware, and other cutomer-oriented API all in Amazon Web Service (AWS) [Netflix 21]. One of the mot important economic appeal of cloud computing i it elaticity and auto-caling in reource proviioning. Traditionally, after careful capacity planning, an enterprie make long-term invetment on it infratructure to accommodate it peak workload. Over-proviioning i inevitable while utilization remain low during mot non-peak time. In contrat, in the cloud, the number of computing intance launched can be changed adaptively at a fine granularity with a lead time of minute. Thi convert the up-front infratructure invetment to operating expene charged by cloud ervice provider. A the cloud auto-caling ability enhance reource utilization by cloely matching upply with demand, the overall expene of the enterprie may be reduced. Unlike web erver or cientific computing, VoD i a network-bound ervice with tringent bandwidth requirement. A uer mut download at a rate no maller than the video playback rate to moothly watch video tream online, bandwidth contitute the performance bottleneck. Thank to the recent advance in datacenter network virtualization [Bari et al. 213], bandwidth reervation i likely to become a near-term value-added feature offered by cloud ervice to appeal to cutomer with bandwidth-intenive application like VoD. In fact, there have already been propoal from the perpective of datacenter engineering to offer bandwidth guarantee for egre traffic from a virtual machine (VM), a well a among VM themelve [Guo et al. 21], [Ballani et al. 211], [Xie et al. 212]. In thi paper, we analyze the benefit and addre open challenge of cloud reource auto-caling for VoD application. The benefit of auto-caling for a video torage and treaming ervice i intuitive and natural. A hown in Fig. 1(a), traditionally, a VoD provider acquire a monthly plan from ISP, in which a fixed bandwidth capacity, e.g., 1 Gbp, i guaranteed to accommodate the anticipated peak demand. A a reult, reource utilization i low during non-peak time of demand trough. Alternatively, a pay-a-you-go charge model may be adopted by a cloud provider a hown in Fig. 1(b), where a VoD provider pay for the total amount of byte tranferred. However, the bandwidth capacity available to the VoD provider i ubject to variation due to contention from other application, incurring unpredictable quality-of-ervice (QoS) iue. Fig. 1(c) illutrate bandwidth auto-caling and reervation to match demand with appropriate reource, leading to both high reource utilization and QoS guarantee. Apparently, the more frequently the recaling happen, the more cloely reource upply will match the demand. However, a number of important challenge need to be addreed to achieve auto-caling in a video torage and treaming ervice. Firt, ince reource recaling require a delay of at leat a couple of minute to update configuration and move object if neceary, it i bet to predict the demand with a lead time greater than the update interval, and cale the capacity to meet anticipated demand. Such a proactive, rather than reactive, trategy for reource proviioning need to conider not only conditional mean demand but alo demand fluctuation in order to prevent under-proviioning rik. Second, a tatitical multiplexing can mooth traffic, a VoD provider may reerve le bandwidth to guard againt fluctuation if it jointly reerve bandwidth for all it video accee. However, in a cloud torage ytem, the content i uually replicated on multiple erver to introduce reliability in the preence of failure and to enable load balancing. The key quetion i how hould a VoD provider optimally plit and direct it workload acro the cluter of erver (whether virtual or phyical) provided by the cloud ervice, in order to ave the overall bandwidth reervation cot? Furthermore, video content mut be replicated acro different erver in a pare way to avoid a high torage cot. In thi paper, we propoe a bandwidth auto-caling facility that dynamically reerve reource from a tightly connected erver cluter for VoD provider, with everal ditinct characteritic. Firt, it i predictive. The facility track the hitory of bandwidth demand for each video uing cloud monitor-

3 Bandwidth Auto-Scaling and Content Placement for Video-on-Demand in the Cloud 1:3 Bandwidth Capacity Demand 1 2 (a) Over-proviioning Day Bandwidth Capacity Demand 1 2 (b) Pay a you go Day Bandwidth 1 2 (c) Auto Scaling Day Capacity Demand Fig. 1: Bandwidth auto-caling with quality aurance, a compared to proviioning for the peak demand and pay-a-you-go. ing ervice, and periodically etimate the expectation, volatility, and correlation of demand for all video for the near future uing tatitical analyi. We propoe a novel video channel interleaving cheme that can even predict demand for new video that lack hitorical demand data. Second, it provide QoS aurance by judiciouly deciding the minimum bandwidth reervation required to atify the demand with high probability. Third, it optimally mixe demand baed on tatitical anticorrelation to ave the aggregate bandwidth capacity reerved from all the erver, under the condition that the content mut be parely replicated with limited content migration. Given the predicted demand tatitic a input, we formulate the bandwidth minimization problem to jointly decide load direction and pare content placement a a combinatorial problem involving L norm that model content placement parity. We derive the theoretically optimal load direction acro erver when full replication i permitted, and propoe everal approximate olution to the joint load direction and pare content placement problem, triking a balance between bandwidth and torage cot. In particular, a a highlight, we novelly apply an iteratively reweighted L 1 -norm relaxation technique to approximately olve the L -norm penalized optimization problem. Our technique not only yield pare content placement deciion but alo effectively reduce the content migration overhead. We have performed extenive evaluation of the propoed autocaling trategie for video torage ytem, through trace-driven imulation baed on the video treaming trace of 1693 video channel collected from UUSee [Liu et al. 21], a production VoD ytem, over a 21-day period. 2. SYSTEM ARCHITECTURE Conider a VoD ervice provider hoting N video and relying on S (collocated) erver in a cloud torage ytem for ervice. We propoe an unobtruive auto-caling ytem that make prediction about

4 1:4 D. Niu, H. Xu and B. Li Bandwidth Reervation Load DirectionW Bandwidth Uage Monitor Server 1 Server 2... w 11 w 21 w 1 Demand Hitory Channel 1 w 1i... Load Optimizer Demand Projection Server w i w 2i Demand Predictor Demand Hitory Channel i Fig. 2: The ytem decide the bandwidth reervation from each erver and a matrix W = [w i ] every t minute, where w i i the proportion of video channel i requet directed to erver. future demand of all video and reerve minimal neceary reource from the erver cluter to atify the demand. Our ytem architecture i hown in Fig. 2, which conit of three key component: bandwidth uage monitor, demand predictor, and load optimizer. Bandwidth recaling i performed proactively every t minute, with the following three tep: Firt, before time t, the ytem collect bandwidth demand hitory of all video up to time t, which can eaily be obtained from cloud monitoring ervice. A an example, Amazon CloudWatch provide a free reource monitoring ervice to AWS cutomer at a 5-minute frequency [AWS ]. Second, the bandwidth demand hitory of all video i fed into the demand predictor to predict the bandwidth requirement of each video for the next t minute, i.e., for the period [t, t + t). Our predictor not only forecat the expected demand, but alo output a volatility etimate, which repreent the degree that demand will be fluctuating around it expectation, a well a the demand correlation between different video in thi period. Our volatility and correlation etimation i baed on multivariate GARCH model [Bollerlev 1986], which ha gained ucce in tock analyi and forecat in the pat decade. Finally, the load optimizer take predicted tatitic a the input, calculate the bandwidth capacity to be reerved from each erver in the available erver pool and determine how many erver hould be ued. It alo output a load direction matrix W = [w i ], where w i repreent the portion of video i requet directed to erver. Apparently, we hould have w i = 1 if the aggregate erver capacity i ufficient. It i worth noting that the matrix W alo indicate the content placement deciion: a copy of video i i placed on erver only if w i >. In practice, the load direction W can be readily implemented by routing the requet for video i to erver with probability w i. The ytem finihe the above three tep before time t, o that a new bandwidth reervation can be made at time t for the period [t, t + t). The above proce i then repeated for the next period [t + t, t + 2 t). Apparently, the key to uch a reource autocaling framework for video torage i the load optimizer, which need to jointly determine a load direction matrix a well a a pare content placement trategy to limit both torage and content tranfer overhead in each t-minute time period. The optimizer

5 Bandwidth Auto-Scaling and Content Placement for Video-on-Demand in the Cloud 1:5 Bandwidth Reerved Capacity A 1 Reerved Capacity A 2 Channel 1 Channel 2 Aum Aum/2 Aum/2 Server 1 Server 2 Time (a) 1 Minute Time (b) 1 Minute Time (c) 1 Minute Time (d) 1 Minute Time (e) 1 Minute Fig. 3: By exploiting demand correlation between different video channel, we can ave the total bandwidth reervation, even within each 1-minute period, while till providing quality aurance to each video channel. hould alo determine load direction W in a way o a to puh workload onto a few erver a poible, which will autocale the number of erver ued. Bandwidth Reervation v. Load Balancing. One may be tempted to think that periodic bandwidth reervation i unneceary, ince requet can be flexibly directed to whichever erver that ha available capacity by a load balancer. However, the latter will exactly fall in the range of pay-a-yougo model with no quality guarantee to VoD uer, wherea bandwidth reervation enure that the proviioned reource can atify the projected demand with high probability. Furthermore, ince the content placement i pre-determined in a traditional load balancing ytem, it i hard to achieve reource autocaling it i impoible to puh all demand onto a few erver a poible when the total demand hrink, i.e., the number of erver ued i alway fixed. Neither can a traditional load balancer adjut content placement dynamically to maximize the multiplexing gain baed on demand tatitic, a will be dicued ubequently. Quality-Aured Bandwidth Multiplexing. The bandwidth demand of each video channel can fluctuate dratically even at mall time cale. To avoid performance rik, the bandwidth reervation made for each channel in each period hould accommodate uch fluctuation, inevitably leading to low utilization at trough, a illutrated in Fig. 3(a) and (b). Trough filling within a hort period uch a 1 minute i hard with too many random hock in demand. However, our load optimizer trive to enhance utilization even when t i a mall a 1 minute by multiplexing demand baed on their correlation. The uefulne of anti-correlation i illutrated in Fig. 3(c): if we jointly book capacity for two negatively correlated channel, the total reerved capacity i A um < A 1 +A 2. Beide aggregation, we can alo take a part of demand from each channel, mix them and reerve bandwidth for the mixed demand from multiple erver. A an example, in Fig. 3(d) and (e), the aggregate demand of two channel i plit onto two erver, each erving a mixture of demand, which till lead to a total bandwidth reervation of A um. In each t period, we leverage the etimated demand correlation to optimally direct workload acro different erver o that the total bandwidth reervation neceary to guarantee quality i minimized. Finally, in the cae that the actual demand exceed the reerved bandwidth capacity, the additional requet can till be erved in the traditional bet-effort fahion. 3. OPTIMAL LOAD DIRECTION AND BANDWIDTH RESERVATION In thi ection, we focu on the load optimizer. Suppoe before time t, we have obtained the etimate about demand in the upcoming period [t, t + t). Our objective i to decide load direction W o a to minimize the total bandwidth reervation while controlling the under-proviion rik in each erver. The quetion of how to make demand prediction will be the ubject of Sec. 5.

6 1:6 D. Niu, H. Xu and B. Li We firt introduce a few ueful notation. Since we are conidering each individual time period, without lo of generality, we drop ubcript t in our notation. Recall that the VoD provider run N video channel. The bandwidth demand of channel i i a random variable D i with mean µ i and variance σ 2 i. For convenience, let D = [D 1,..., D N ] T, µ = [µ 1,..., µ N ] T and σ = [σ 1,..., σ N ] T. Note that the random demand D 1,..., D N may be highly correlated due to the correlation between video genre, viewer preference and video releae time. Denote ρ ij the correlation coefficient of D i and D j, with ρ ii 1. Let Σ = [σ ij ] be the N N ymmetric demand covariance matrix, with σ ii = σ 2 i and σ ij = ρ ij σ i σ j for i j. The VoD provider will book reource from S erver. Denote C the upper bound on the bandwidth capacity that can be reerved from erver, for = 1,..., S. C may be limited by the available intantaneou outgoing bandwidth at erver, or may be intentionally et by the VoD provider to pread it workload acro different erver and avoid booking reource from a ingle erver. Let C um = C be the aggregate utilizable bandwidth capacity of all S erver. Throughout the paper, we aume that C um i ufficiently large to atify all the demand in the ytem. 1 Let Φ = [φ i ] be the content placement matrix, where φ i = 1 if video i i replicated on erver, and φ i = otherwie. We define a load direction deciion a a weight matrix W = [w i ], = 1,..., S, i = 1,..., N, where w i repreent the portion of video i demand directed to and erved by erver, with w i 1 and w i = 1. Apparently, if φ i =, we mut have w i =. We oberve that w = [w 1,..., w N ] T repreent the workload portfolio of erver. Given w, the aggregate bandwidth load impoed on erver i a random variable L = i w i D i = w T D. (1) We ue A to denote the amount of bandwidth reerved from erver for thi period. Clearly, we mut have A C. Let A =: [A 1,..., A S ] T. To control the under-proviion rik, we require the load impoed on erver to be no more than the reerved bandwidth A with high probability, i.e., Pr(L > A ) ɛ,, (2) where ɛ > i a mall contant, referred to a the under-proviion probability. 3.1 Load Direction under Full Replication Suppoe φ i = 1 for all, i, i.e., each video i replicated on every erver. Then, every w i may take nonzero value. Specifically, given demand expectation µ and covariance Σ, and the available capacitie C 1,..., C S, the load optimizer can decide the optimal bandwidth reervation A and load direction W by olving the following optimization problem: minimize W,A A (3) ubject to A C,, (4) Pr(L > A ) ɛ,, (5) w i = 1, i. (6) Through reaonable aggregation, we believe that L follow a Gauian ditribution. We will empirically jutify thi aumption in Sec. 6 uing real-world trace. When L i Gauian, contraint (2) i 1 A rigorou condition for upply exceeding demand i given in Theorem 3.1.

7 Bandwidth Auto-Scaling and Content Placement for Video-on-Demand in the Cloud 1:7 equivalent to A E[L ] + θ Var[L ], (7) with θ := F 1 (1 ɛ), where F ( ) i the CDF of normal ditribution N (, 1). For example, when ɛ = 2%, we have θ = 2.5. Since { E[L ] = µ 1 w µ N w N = µ T w, Var[L ] = i,j ρ ijσ i σ j w i w j = w T Σw, it follow that (2) i equivalent to A µ T w + θ w T Σw. (8) Therefore, the bandwidth minimization problem (3) under full replication i equivalent to minimize A (9) W ubject to A = µ T w + θ w T Σw, (1) µ T w + θ w T Σw C,, (11) w = 1, (12) w 1,, (13) where 1 = [1,..., 1] T and = [,..., ] T are N-dimenional column vector. Under full replication, we can derive nearly cloed-form olution to problem (9) in the following theorem: THEOREM 3.1. If C um µ T 1 + θ 1 T Σ1, an optimal load direction matrix [wi ] i given by where α 1,..., α S can be any olution to α = 1, w i = α, i, = 1,..., S, (14) α min { 1, C µ T 1+θ 1 T Σ1 },. If C um < µ T 1 + θ 1 T Σ1, there i no feaible olution that atifie contraint (11) to (13). Proof Sketch: Firt, f(w ) = w T Σw i a cone and thu a convex function. Hence, we have ( ) w1 + w 2 f f(w 1) + f(w 2 ), 2 2 or equivalently, (w 1 + w 2 ) T Σ(w 1 + w 2 ) w1 TΣw 1 + w2 TΣw 2. (15) By induction, we can prove ( w T Σw w T ) ( Σ w ). (16)

8 1:8 D. Niu, H. Xu and B. Li If w = 1 i feaible, by (11) and (16) we have ( C µ T 1 + θ = µ T 1 + θ 1 T Σ1. w T ) ( ) Σ w If C µ T 1 + θ 1 T Σ1, it i eay to verify (15) i feaible. When w i = α given by (14), we find (11), (12) and (13) are all atified. Hence, (14) i a feaible olution and w = 1 i feaible. By (16), the objective (9) atifie µ T ( ) µ T w + θ w T Σw w + θ =µ T 1 + θ 1 T Σ1. ( w T ) ( ) Σ w We find that [wi ] given by (14) achieve the above inequality with equality, and thu i alo an optimal olution to (9). Theorem 3.1 implie that in the optimal olution, each video channel hould plit and direct it workload to all S erver following the ame weight α 1,..., α S, which can be found by olving the linear contraint (15). Moreover, the optimal workload portfolio of each erver ha a imilar tructure of w = α 1, where α depend on it available capacity C through the contraint (15). Under the optimal load direction, the aggregate bandwidth reervation reache it minimum value: A = ( µ T w + θ = µ T 1 + θ 1 T Σ1, which doe not depend on S, the number of erver. Thi mean that having demand erved by multiple erver intead of one big erver doe not increae bandwidth reervation cot a long a w i = α, i given by (14). Therefore, the load optimizer can firt aggregate all the demand and then plit the aggregated demand into different erver ubject to their capacitie. w T Σw ) (17) 3.2 Load Direction under Limited Replication Although olution (14) i optimal in term of bandwidth reervation, it encounter two major obtacle in practice. Firt, a long a α >, w i = α > for all i, which mean that erver ha to tore all N video. In other word, a video ha to be replicated on to every erver that ha α >. Thi incur ignificant additional torage cot. Second, each video channel i plit it workload onto all S erver according to the weight α 1,..., α S. When S i large and D i i mall, uch fine-grained plitting will not be feaible from an engineering perpective. Therefore, in practice, each video hould only be replicated on a few erver to maintain a reaonable torage overhead. Thu, for each video i, we hould have φ i = 1 only for a ubet of all. If the content

9 Bandwidth Auto-Scaling and Content Placement for Video-on-Demand in the Cloud 1:9 placement matrix Φ i given, the optimal load direction problem become minimize A (18) W ubject to A = µ T w + θ w T Σw, (19) µ T w + θ w T Σw C,, (2) φ i w i = 1, i, (21) (1 φ i )w i =, i, (22) w 1,, (23) A compared to the load direction optimization problem (9) under full replication, problem (18) ha the new contraint (21) and (22) due to limited replication, which are clearly equivalent to { w i = 1, i, w i =,, if φ i = Problem (18) i a convex problem, and in particular, a econd-order cone program (SOCP) that can be olved efficiently uing tandard convex optimization olver uch a interior-point algorithm or active-et method. Apparently, the minimum achievable total bandwidth reervation A i lowerbounded by the A under full replication, which i µ T 1 + θ 1 T Σ1. 4. SPARSE CONTENT PLACEMENT DESIGN In thi ection, we conider the joint deign of content placement matrix Φ and load direction matrix W, given the workload tatitic µ and Σ. A ha been mentioned in Sec. 2, given a W, Φ can be determined in the following way: { 1, if wi >, φ i (w i ) = (24), if w i =, which mean that video i need to be replicated on to erver only if w i >. Therefore, to determine load direction with a limited replication overhead, we can ue W a a ingle deciion variable and contrain the number of non-zero entrie in it when minimizing bandwidth reervation, leading to the following optimization problem: minimize W A (25) ubject to A = µ T w + θ w T Σw, (26) µ T w + θ w T Σw C,, (27) w = 1, (28) w 1,, (29) w k,, (3)

10 1:1 D. Niu, H. Xu and B. Li where w i the l norm of w, which repreent the number of non-zero component in w. Contraint (3) eentially ay that each erver hould only tore up to k video. Apparently, problem 25 involve L norm and i non-convex. 4.1 Per-Server Heuritic We propoe a uboptimal olution to problem (25) that can handle the torage overhead contraint. Firt, we preent a heuritic outlined in Algorithm 1, which output w1,..., ws for each erver one after another. ALGORITHM 1: Per-Server Optimal. b 1 for = 1,..., S do Solve the following problem to obtain w : b b w. Exit if b. end maximize w µ T w (31) ubject to µ T w + θ w T Σw A C, (32) w b (33) Algorithm 1 pack the random demand into each erver, one after another, by maximizing the expected demand µ T w each erver can accommodate ubject to the probabilitic performance guarantee in (32). A a reult, the total amount of reource needed to guard againt demand variability i reduced. Clearly, with Algorithm 1, the aggregate bandwidth reervation from all erver i A = S (µ T w =1 + θ w T Σw ). (34) Note that Algorithm 1 i alo computationally efficient ince (31) i a tandard econd-order cone program. Now we handle the contraint w k, which require each erver to tore at mot k video. We modify Algorithm 1 to cope with thi contraint, leading to Algorithm 2, which output w 1,..., w S for each erver one after another. ALGORITHM 2: Per-Server Limited Channel. b 1 for = 1,..., S do Solve problem (31) to obtain w Chooe the top k channel with the larget weight and olve problem (31) again only for thee k channel to obtain w b b w. Exit if b. end

11 Bandwidth Auto-Scaling and Content Placement for Video-on-Demand in the Cloud 1:11 With Algorithm 2, the aggregate bandwidth reerved i A = S (µ T w + θ w T Σw ). (35) =1 In Sec. 6, we will how through trace-driven imulation that Algorithm 2, though uboptimal, effectively limit the content replication degree, thu balancing the aving on torage cot and bandwidth reervation cot for VoD provider. 4.2 Relaxation through Iteratively Weighted L 1 Norm We now propoe to ue another algorithm baed on L 1 -norm relaxation to olve (25) iteratively. Our idea i to adapt a o-called Log-det heuritic in pare recovery to our pare deign problem. The Logdet heuritic ha previouly been applied to cardinality minimization [Fazel et al. 23], i.e., finding a vector x = (x 1,..., x n ) with the minimum cardinality in a convex et C, or equivalently, minimizing x = i φ(x i) ubject to x C, where φ(x i ) = if x i = and φ(x i ) = 1 if x i >. The baic idea i to replace the -1 valued objective φ(x i ) for each x i by a mooth log function log( x i +δ) and to minimize a linearization of log( x i + δ) iteratively, which lead to iteratively reweighted L 1 -norm approximation to L norm. In pare recovery, it i hown that uch iteratively reweighted L 1 norm can yield more accurate recovery reult than one-time L 1 norm approximation [Cande et al. 28]. To adapt the iteratively reweighted L 1 -norm approximation to our pare deign problem, in each iteration, we ue a carefully deigned convex contraint to replace the L -norm contraint (3) and olve the modified problem (25). A iteration proceed, the deigned convex contraint i expected to approach the L -norm contraint (3) eventually. The algorithm i decribed in Algorithm 3. ALGORITHM 3: Iterative L 1-Contrained. Initially, replace (3) by i wi k, for all, and olve the modified problem (25) under the new contraint to obtain W. for t = 1,..., maximum iteration do Given the olution W t 1 in the previou iteration, define ˆφ t i a ˆφ t w i i(w i) = w t 1,, i (36) i + δ Solve the following modified problem to obtain W t : minimize W Break if W t and W t 1 are approximately equal. end Return W W t. A (37) ubject to A = µ T w + θ w T Σw, (38) µ T w + θ w T Σw C,, (39) w = 1, (4) w 1,, (41) ˆφ t i(w i) k,. (42) i

12 1:12 D. Niu, H. Xu and B. Li Let u explain the rationale behind Algorithm 3. Initially, we replace the contraint w k with i w i k, which i a tandard L 1 -norm relaxation, ince w 1 = i w i = i w i. It i not hard to ee that the bandwidth reervation achieved by W i a lower bound on the optimal value of the original problem (25). The reaon i that we have and thu φ i (w i ) w i, if w i 1, w = φ i (w i ) w i. i i Therefore, i w i k form a larger region than w k, and the optimal value achieved by W in the relaxed (convex) problem i a lower-bound of the original optimal value. Subequently, in each iteration, the contraint w k i replaced by an inequality involving a weighted um, i.e., w i w t 1 i i + δ k, which i a generalized verion of L 1 -norm relaxation with a different weight for each variable. Note that for a ufficiently mall δ, upon convergence, i.e., when w t 1 i = wi t = w i, we have { ˆφ i (wi) = w i if w wi + δ i =, 1 if wi >, which i approximately φ i (wi ). Thu, the modified contraint ˆφ i t i (w i) k eventually approache the L -norm contraint w k in the original problem, and the generated W t will almot be feaible for the original problem (25). 4.3 Reducing Migration Overhead and Iterative L 1 -Penalized Optimization A common iue faced by the above cheme i the content migration overhead. A pare content placement optimization i performed every t minute under varying (predicted) demand, the placement olution may change from time to time, leading to the overhead of tranferring video copie. To mitigate migration overhead, we further propoe the following Iterative L 1 -Penalized Optimization, which not only yield a pare placement olution, but alo limit the tranfer or creation of video copie in each time period by attempting to generate a placement olution that i imilar to that of the preceding time period. Firt, we add a regularizing term to the original content placement problem (25) to yield minimize A + λ φ i (w i ) (43) W (,i):w pre i = ubject to A = µ T w + θ w T Σw, (44) µ T w + θ w T Σw C,, (45) w = 1, (46) w 1,, (47) where λ >, and w pre i denote the pare olution for the previou time period. The regularizer (,i):w pre i = φ i(w i ) repreent the number of video copie that need to be copied or tranferred in (48)

13 Bandwidth Auto-Scaling and Content Placement for Video-on-Demand in the Cloud 1:13 thi time period; a copy of video i need to be tranferred to erver only if erver doen t tore video i previouly, i.e., w pre i =, and w i > (φ i (w i ) = 1) a a reult of the current optimization. Note that in thi cae, we may remove the contraint w k, ince the regularizer itelf can already generate pare olution, a will be explained ubequently. To olve problem (43), we apply Algorithm 1 to (43) with iteratively reweighted L 1 -norm relaxation for the regularizer, leading to the following Iterative L 1 -Penalized algorithm. ALGORITHM 4: Iterative L 1-Penalized. b 1 for = 1,..., S do Solve the following ubproblem uing Algorithm 5 to obtain w : b b w. Exit if b. end maximize w µ T w λ i:w pre i = φ i(w i) (49) ubject to µ T w + θ w T Σw A C, (5) w b (51) ALGORITHM 5: Subroutine to olve the penalized problem (49) in Algorithm 4 for a particular erver. Initially, wi = 1 δ, for all i. for t = 1,..., maximum iteration do Given the olution w t 1 in the previou iteration t 1, define ˆφ t i a ˆφ t w i i(w i) = w t 1, i. (52) i + δ Solve the following modified problem to obtain w t : Break if w t and w t 1 end Return w w. t are approximately equal. maximize µ T w λ t ˆφ i(w i) (53) w i:w pre i = ubject to µ T w + θ w T Σw C, (54) w b (55) Now we explain the rationale behind Algorithm 4. Initially, for the firt t-minute time period, we et w pre i = for all and i. By doing o, the regularizer in problem (43) will drive the olution W to be pare, uch that the placement i pare. Therefore, by properly etting λ in the regularizer, we do not need to include the contraint w k. For each t-minute time period afterward, the optimization problem (43) i approximately olved by Algorithm 4 with penalty on the new 1 introduced in matrix W. In other word, if w pre i = in the content placement deciion for the previou time period, Algorithm 4 will penalize w pre i = 1 for the current time period, thu preventing new video

14 1:14 D. Niu, H. Xu and B. Li copie from being created or tranferred. Since the content placement for the firt time period i pare and each ubequent period penalize the difference from the previou period, the content placement for ubequent period alo tend to be pare. Thi fact will be demontrated in Sec. 6 through tracedriven imulation. 5. DEMAND PREDICTION The derivation of load direction deciion critically depend on parameter u and Σ, which are etimate of the expected demand and demand covariance for the hort-term future [t, t + t). In thi ection, we preent efficient time erie forecating method to make uch prediction baed on pat obervation. We aume that the bandwidth demand of channel i at any point in the period [t, t + t) can be repreented by the ame random variable D it. Thi i a reaonable aumption when t i mall. Similarly, let µ t = [µ 1t,..., µ Nt ] and Σ t = [σ ijt ] repreent the demand expectation vector and demand covariance matrix for all N channel in [t, t+ t). We aume that before time t, the ytem ha already collected enough demand hitory from cloud monitoring ervice with a ampling interval of t. The quetion i how to ue the available ampled bandwidth demand hitory {D iτ : τ =,..., t 1, i = 1,..., N} to etimate µ t and Σ t? In thi paper, we combine our previouly propoed eaonal ARIMA model [Niu et al. 211b] for conditional mean (expectation conditioned on the hitory) prediction with the GARCH model [Niu et al. 211a] for conditional variance prediction to obtain a multivariate GARCH model that can forecat the demand covariance matrix. The model extract the periodic evolution pattern from each channel demand time erie, and characterize the remaining innovation erie a autocorrelated GARCH procee. We briefly decribe thee tatitical model here. The difficulty in modeling the bandwidth demand of a channel i i that it exhibit diurnal periodicity, a downward trend a the video become le popular over time, and changing level of fluctuation a population goe up and down. Such non-tationarity in traffic render unbiaed linear predictor uele. We tackle thi problem by applying one-day-lagged difference (the lag i 144 if t = 1 minute) onto {D iτ } to remove daily periodicity to obtain the tranformed erie {D iτ := D iτ D iτ 144 }, which can be modeled a a low-order autoregreive moving-average (ARMA) proce: { D iτ φ i D iτ 1 = N iτ + γ i N iτ 1, D iτ = D iτ D i,τ 144, (56) where {N iτ } WN(, σ 2 ) denote the uncorrelated white noie with zero mean. Model (56) fall in the category of eaonal ARIMA model [Niu et al. 211b], [Box et al. 28]. Model parameter φ i and γ i in (56) can be trained baed on hitorical data uing a maximum likelihood etimator [Box et al. 28]. To predict the expected demand µ it of channel i, we firt predict µ it := E[D it D it 1, D it 2,...] for the tranformed erie {D iτ } to obtain the etimate ˆµ it, uing an unbiaed minimum mean quare error (MMSE) predictor. We then retranform ˆµ it into an etimate ˆµ it of the conditional mean µ it, with the invere of one-day-lagged differencing. Given the conditional mean {ˆµ iτ } of channel i over all time τ, we denote the innovation in {D iτ } by {Z iτ }, where Z iτ := D iτ ˆµ iτ. (57) Since the innovation term Z iτ repreent the fluctuation of D iτ relative to it projected expectation ˆµ iτ, and uch fluctuation may be changing over time, we model the innovation {Z iτ } uing a GARCH

15 proce: Bandwidth Auto-Scaling and Content Placement for Video-on-Demand in the Cloud 1:15 { Ziτ = h iτ e τ, {e τ } IIDN (, 1), h iτ = α i + α i1 Z 2 iτ 1 + β ih iτ 1, where {Z iτ } i modeled a a zero-mean Gauian proce yet with a time-varying conditional variance h iτ. Intead of auming a contant variance for {Z iτ }, (58) introduce autocorrelation into volatility evolution and forecat the conditional variance h it of Z it a a regreion of pat h iτ and Ziτ 2. The model parameter in (58) can be learned uing maximum likelihood etimation (pp. 417, [Box et al. 28]) baed on training data. Furthermore, the intantaneou hock to demand for different video can be correlated in a largecale ytem. An increae in one video demand may or may not affect the demand for other video depending on factor like video genre, releae time, etc. To incorporate demand correlation, intead of etimating volatility for each video eparately, we can etimate the time-varying conditional covariance matrix Σ t uing multivariate GARCH [Ender 21]. However, multivariate GARCH model are very difficult to etimate for large-cale problem. For the 2-video cae, the number of model parameter to etimate in GARCH(1, 1) i 21, and for the 3-video cae, uch a number ecalate to 78. To efficiently predict the covariance matrix Σ t, we introduce a contant conditional correlation (CCC) model [Ender 21], which i a popular multivariate GARCH pecification that retrict the correlation coefficient ρ ij to be contant. ρ ij can be etimated a the correlation coefficient between erie {Z iτ } and {Z jτ } in recent time period, and ρ ij = 1 if i = j. The covariance σ ijt between video i and j at time t i thu predicted a (58) ˆσ ijt = h ijt = ρ ij hit h jt, (59) with h it and h jt predicted uing (58) for channel i and j individually. The full tatitical model i a eaonal ARIMA conditional mean model (56) with a CCC multivariate GARCH innovation model given by (58) and (59). The above eemingly complex model i extremely efficient to train, a the five parameter φ i, γ i, α i, α i1 and β i are learned for each video i eparately following the procedure mentioned above, and ρ ij i calculated traightforwardly from recent hitory. 5.1 Model Validation via Real Trace We verify the effectivene of the propoed workload prediction model baed on the workload trace of UUSee video-on-demand ytem over a 21-day period during 28 Summer Olympic [Liu et al. 21]. A a commercial VoD company, UUSee tream on-demand video to million of Internet uer acro over 4 countrie through a downloadable client oftware. The dataet collected contain performance naphot taken at a 1-minute frequency of 1693 video channel, including port event, movie, TV epiode and other genre. The tatitic we ue in thi paper are the time-averaged total bandwidth demand in each video channel in each 1-minute period. There are 144 time period in a day. A an example, we make 1-minute-ahead (one-tep) prediction of the bandwidth demand of a popular video channel i = 121 releaed at time period t = 264 ( :47:39). The channel ha a maximum online population of The bandwidth conumption erie of the firt 1.25 day i ued a the training data tarting from time period 81. The initial 8 time period are excluded which may not conform to later evolution pattern. The prediction i teted on the data of 3 day following the training period. We fit the low-order model (56) and (58) to the training data and obtain model parameter through a maximum likelihood etimator [Box et al. 28]. A hown in Fig. 4, uch a low-order model merely trained baed on the data of 1.25 day can yield conditional mean prediction that are cloe to the actual demand. The reulted prediction error plotted in Fig. 4(b), with a mean of zero, have a varying conditional tandard deviation predicted by the GARCH model in Fig. 4(c).

16 1:16 D. Niu, H. Xu and B. Li Aggregate bandwidth (Mbp) Error (Mbp) Trace data 1 tep prediction Training data Time (unit: 1 minute) (a) One-tep conditional mean demand prediction Time (unit: 1 minute) Standard deviation (Mbp) (b) Prediction error Time (unit: 1 minute) (c) One-tep prediction for conditional tandard deviation Fig. 4: 1-minute-ahead (one-tep) prediction for the bandwidth demand of a popular video channel i = 121. Then, we verify that D it approximately follow a Gauian ditribution in each 1-minute period. Recall that for each channel i, given conditional mean prediction ˆµ it at time t, the innovation i Z it := D it ˆµ it. Fig. 5(a) how the QQ plot of Z it for a typical channel i = 121, which indicate {Z it } ampled at 1-minute interval i a Gauian proce. Thu, it i reaonable to aume D it follow a Gauian ditribution within the 1 minute following t, with mean ˆµ it. Fig. 5(b) how the QQ plot of i Z it, which indicate that the aggregated demand i D it tend to Gauian even if D jt i not for ome channel j. Since the load L of each erver i aggregated from many video, it i reaonable to aume L i Gauian.

17 Bandwidth Auto-Scaling and Content Placement for Video-on-Demand in the Cloud 1:17 Quantile of Input Sample Standard Normal Quantile (a) Z it of a typical channel Quantile of Input Sample Standard Normal Quantile (b) i Z it of all channel combined Fig. 5: QQ plot of innovation for t = v. normal ditribution. Aggregate bandwidth (Mbp) Video959F2 Video4EB2 Day 12 Day 13 Day 14 Day 1 Day 11 Day 12 Aggregate bandwidth (Mbp) Video9AB4A Video4E5A Day 16 Day 17 Day 18 Day 1 Day 11 Day 12 Fig. 6: Video releaed on different day but at the ame time of day exhibit imilar initial demand pattern. 5.2 A Channel Interleaving Scheme Although we have preented a complete framework for efficient forecat of expected future demand µ t and demand covariance matrix Σ t, the parameter learning for the eaonal ARIMA model (56) require a training data of more than 1 day (pecifically 1.25 day in our predictor) to incorporate daily periodicity into the model. A new video do not have enough hitorical obervation for model training, their demand can hardly be forecated from hitory. In thi ection, we propoe method to predict demand for newly releaed video that lack hitorical obervation and unpopular mall video channel. We tackle thi iue by intelligently interleaving traffic of new video to form virtualized video channel for demand prediction. We alo ue a imilar technique to combine mall channel to improve prediction accuracy. Let u conider new video that have been in the ytem for le than 1.25 day. Although thee video do not have ufficient hitorical obervation for model training, we oberve that their initial demand pattern are quite imilar to video that were releaed earlier around the ame time of day. For example, the left half of Fig. 6 how the initial demand of 2 video channel 959F2 and 4EB2

18 1:18 D. Niu, H. Xu and B. Li releaed at :31:56 and :23:2, repectively. A both video are releaed around the ame time of day, though on different day, they are aligned in Fig. 6 for comparion, with double line of x-label howing the firt 3 day of each video. ( i deemed a Day 1 and the firt time period of each day i 14:5.) We can ee the two video exhibit a imilar initial demand evolution pattern, though with different popularity. The major reaon for uch imilarity i that mot uer watch VoD channel around everal peak time in a day: both video are releaed between 21: and 22: and will expect the firt peak demand at midnight, followed by a econd peak at noon on the next day. Similarly, the right half of Fig. 6 compare the initial demand of video 9AB4A releaed at :11:54 and video 4E5A releaed at :2:38. They alo exhibit imilar initial demand pattern, with the firt peak around 18:, which i the tart of off-work hour, before the econd peak around midnight. Different video, however, may attract different ize of population depending on their popularity. From the above analyi, we can predict the demand for a new video baed on other video releaed on an earlier date but at the ame time of day. To implement thi idea, we define virtual new channel k a a combination of all video channel with an age le than 1.25 day and releaed in hour k {1,..., 24} on any date. Upon releae, a new video join virtual new channel k baed on it releae hour k, and quit thi virtual new channel when it ha been in the ytem for 1.25 day and accumulated enough obervation for eparate model training. A a reult, each virtual new channel k contain a dynamic et of video channel releaed in hour k yet poibly on different day. For example, Fig. 7 how the aggregate bandwidth demand of virtual new channel 11 from time 433 to 18, and Fig. 9 how the number of video contained in virtual new channel 11 from time 1 to 18. We can ee that although virtual new channel 11 repreent a dynamic group of video, it aggregate bandwidth demand exhibit repetition of a imilar pattern becaue the video in thi virtual channel are all releaed in hour 11, poibly on different date. Similarly, we aggregate mall video channel and et up 24 virtual mall channel. When a video reache the age of 1.25 day, it quit it virtual new channel. If it demand never exceeded a threhold (e.g., 4 Mbp) in the firt 1.25 day, it will join one of the virtual mall channel in a round robin fahion. Otherwie, it become a mature channel. Each mature or virtual channel i deemed a an entity to which prediction and optimization are applied. For example, we make 1-minute-ahead prediction of bandwidth demand for virtual new channel 11, and plot the conditional mean prediction in Fig. 7 and the conditional tandard deviation prediction in Fig. 8 for a tet period of 1.5 day. Satifactory prediction performance i oberved. Although conditional mean prediction i ubject to error, the GARCH model can predict the conditional error tandard deviation, a hown in Fig. 8, which contribute to the rik factor (11) in the bandwidth reervation minimization. Furthermore, the combination of everal real video channel into a virtual channel uppree random hock, making prediction more accurate. 6. PERFORMANCE EVALUATION We conduct a erie of imulation to evaluate the performance of our auto-caling reervation cheme for video torage ytem. The imulation are driven by the replay of the workload trace of UUSee video-on-demand ytem over a 21-day period during 28 Summer Olympic [Liu et al. 21]. We ak the quetion what the performance would have been if UUSee had all it workload in thi period erved by cloud ervice through our auto-caled bandwidth reervation ytem? We conduct performance evaluation for 4 typical time pan which are near the beginning, middle and end of the 21-day duration. We implement tatitical learning and demand prediction technique preented in Sec. 5 to forecat the expected demand µ t and demand covariance matrix Σ t every 1 minute. The model parameter are retrained daily, with training data being the bandwidth demand

19 Bandwidth Auto-Scaling and Content Placement for Video-on-Demand in the Cloud 1:19 Aggregate bandwidth (Mbp) Trace data 1 tep prediction Training data Time (unit: 1 minute) Fig. 7: The conditional mean prediction for S t in virtual new channel 11, with a tet period of 1.5 day from time 1585 to 18. Only a part of the entire training data i plotted. Mbp Prediction error Error tandard deviation Time (unit: 1 minute) Fig. 8: The prediction error and predicted error tandard deviation for S t in virtual new channel 11. Number of Video Training Period Tet Period Time (unit: 1 minute) Fig. 9: The number of video in virtual new channel 11 during the entire training and tet period. erie {D iτ } in the recent 1.25 day of each channel i. Once trained, the model will be ued for the next 24 hour. Although video uer may join or quit a channel unexpectedly, our prediction i till effective, ince it deal with the aggregate demand in the channel which feature diurnal pattern. We aume that there i a pool of erver from which UUSee can reerve bandwidth. To pread the load acro erver, we et C = 3 Mbp for each. The QoS parameter θ := F 1 (1 ɛ) i et to θ = 2.5 to confine the under-proviion probability to ɛ = 2%. 6.1 Algorithm for Comparion We compare our optimal load direction (14) under full replication, and Algorithm 1, Algorithm 2, Algorithm 3 and Algorithm 4 with pare content placement, againt the following baeline algorithm: Reactive without Prediction. Initially, replicate each video to K randomly choen erver, which limit the initial content replication degree to K. Each client requeting channel i i randomly directed to a erver that ha video i and idle bandwidth capacity. A requet i dropped if there i no uch erver. In thi cae, the algorithm react by replicating video i to an additional erver choen randomly that ha idle capacity. Replicating content i not intant: we aume that the replication involve a delay of one period of time.

20 1:2 D. Niu, H. Xu and B. Li Table I. : The performance of different cheme averaged over each tet period, in term of QoS, reource utilization, and replication. Period Time period (91 mature and virtual channel) Peak demand 6.56 Gbp, mean demand 5.19 Gbp Time period (161 mature and virtual channel) Peak demand 6.81 Gbp, mean demand 4.91 Gbp Short Drop Util Rep Booked Over-prov Short Drop Util Rep Booked Over-prov Optimal Ch.66% 92.9% Gbp 18.5% Ch.25% 91.1% Gbp 11.3% Per-Server Opt 1. Ch.37% 9.% Gbp 112.2% 1.2 Ch.13% 88.6% Gbp 113.4% Per-Server Lim.3 Ch.6% 85.7% Gbp 117.8%.2 Ch.3% 84.6% Gbp 118.8% Random 5.9 Ch.2% 83.3% Gbp 121.2% 7.6 Ch.% 82.2% Gbp 122.4% Reactive 7.9 Ch.47% 77.2% Gbp 132.4% 7.2 Ch.34% 7.4% Gbp 146.% Itr L 1-Contr Ch.18% 88.2% Gbp 114.3% Itr L 1-Penal.1 Ch.6% 85.1% Gbp 118.7%.1 Ch % 84.7% Gbp 118.8% Period Time period (176 mature and virtual channel) Peak demand 7.55 Gbp, mean demand 5.62 Gbp Time period (199 mature and virtual channel) Peak demand 9.19 Gbp, mean demand 7.62 Gbp Short Drop Util Rep Booked Over-prov Short Drop Util Rep Booked Over-prov Optimal Ch.31% 91.1% Gbp 11.4% Ch.11% 85.4% Gbp 118.1% Per-Server Opt.7 Ch.16% 88.3% Gbp 114.% 1. Ch.9% 82.7% Gbp 121.8% Per-Server Lim 1.4 Ch.% 83.9% Gbp 119.9% 2.7 Ch.17% 82.3% Gbp 122.6% Random 6.2 Ch.% 8.4% Gbp 125.4% 33.4 Ch.2% 77.9% Gbp 129.3% Reactive 5.9 Ch.27% 72.7% Gbp 14.4% 15.8 Ch.43% 74.6% Gbp 14.3% Itr L 1-Penal 1.1 Ch.3% 84.5% Gbp 119.1% 1. Ch.1% 81.1% Gbp 124.5% Short: Average # channel with dropped requet; Drop: average requet drop rate; Util: average utilization of allocated reource; Rep: average replication degree; Booked: average booked bandwidth; Over-prov: average over-proviioning ratio. Note: Iterative L 1 -Contrained i only evaluated for time period 72-78, ince it cannot efficiently complete within 1 minute for more than 91 channel, which i the cae for other time pan. Random with Prediction. Initially, let = 1 and b = 1. Second, randomly generate w in (, b) and recale it o that the QoS contraint (11) i achieved with equality for. Update b to b w and update to +1. Go to the econd tep unle b = or = S +1, in which cae the program terminate. The reactive cheme repreent proviioning for peak demand in Fig. 1 in ome way, with limited replication. It doe not leverage prediction or bandwidth reervation. We aume in Reactive, the total cloud capacity allocated i alway the minimum capacity needed to meet the peak demand in the ytem. The random cheme leverage prediction and make bandwidth reervation, but randomly direct workload intead of uing anti-correlation and optimization technique to minimize bandwidth reervation. We implement all of the ix cheme dicued above, and ummarize their performance comparion in Table I for each of the four time pan. Iterative L 1 -Contrained i only evaluated for time period 72-78, a it cannot converge within 1 minute for more than 91 channel. Note that the channel in the table include mature channel, virtual new channel and virtual mall channel. The number of video in each virtual channel can vary over time. A new video are introduced, more channel are preent in later tet period. We evaluate the algorithm performance with regard to QoS, bandwidth reource occupied, and replication cot. 6.2 The Benefit of Predictive Proviioning over Reactive Proviioning Table I how that Reactive generally ha a more alient QoS problem than all five predictive cheme in term of both the number of unatified channel and requet drop rate (percentage of unatified requet), demontrating the benefit of demand prediction. Fig. 1 preent a more detailed comparion for a typical peak period from time 72 to 78. Without urprie, Reactive ha many unfulfilled requet at the beginning. Since the video are randomly replicated to K = 2 erver (hown in Fig. 1(d) at t = 72) and requet are randomly directed, it i likely that a channel doe not acquire enough ca-