Consolidating Virtual Machines with Dynamic Bandwidth Demand in Data Centers

Transcription

1 Consolidating Virtual Maines wit Dynami Bandwidt Demand in Data Centers Meng Wang, Xiaoqiao Meng, and Li Zang Sool of ECE, Cornell University, Itaa, NY 4853, USA. IBM T.J. Watson Resear Center. 9 Skyline Drive, Hawtorne, NY 0532, USA. {xmeng,zangli}@us.ibm.om Abstrat Reent advanes in virtualization tenology ave made it a ommon pratie to onsolidate virtual maines(vms) into a fewer number of servers. An effiient onsolidation seme requires tat VMs are paked tigtly, yet reeiving resoures ommensurate wit teir demands. On te oter and, measurements from prodution data enters sow tat te network bandwidt demands of VMs are dynami, making it diffiult to araterize te demands by a fixed value and to apply traditional onsolidation semes. In tis work, we apture VM bandwidt demand by random variables following probabilisti distributions. We study ow VMs sould be onsolidated wit bandwidt limit imposed by network devies su as Eternet adapters and edge swites. We formulate a Stoasti Bin Paking problem and propose an online paking algoritm by wi te number of servers required is witin (+ɛ)( 2+) of te optimum for any ɛ > 0. In a speial ase tat tere are finite number of possibilities for te means of te random variables, te number of servers required by our algoritm is witin ( 2 + ) of te optimum. In addition, we use numerial experiments to evaluate te proposed onsolidation algoritm and observe 30% server redution ompared to several benmark algoritms. I. INTRODUCTION Virtualization tenologies promise opportunities for modern data enters to ost appliations on sared infrastruture. Now data enter operators an reate a large number of virtual maines (VMs) for different workload requests. Ea VM is provisioned wit ertain amount of omputing resoures ommensurate wit workload requirements. Te operators an ten onsolidate all te VMs into a smaller number of pysial servers, wit te goal of minimizing te total required server number. Su a VM onsolidation proedure is te key fator for a data enter to aieve eonomy of sale. Traditional VM onsolidation semes are onerned wit CPU, memory and disk I/O [3][23][26][29][3][32]. In tese semes, te operators need to estimate a deterministi amount of resoures required by te future workload; su an estimate is usually made from eiter an understanding of te underlying appliations or a foreast using istorial workload patterns. Te atual amount of alloated resoures to VMs are based on te estimate. In te onsolidation pase, all su VMs wit fixed size must be paked on servers wit known apaity limit. In tis way, te onsolidation pase beomes te lassial bin paking problem and an be solved by many euristis. Compared to te apaity limit of CPU, memory and disk I/O, network bandwidt limit as not been well studied. In fat, wit an explosive growt of data enter traffi, network bandwidt limit beomes more and more ritial, and su limit exists at multiple onsolidation levels. E.g., at te server level, due to te sared Eternet adapters, te aggregate traffi rate for VMs plaed on te same server must not exeed te adapter apaity. At te assis/rak level, te aggregate rate must not exeed te apaity of te end-of-row/top-of-rak swit. Tus it is imperative to find effiient onsolidation metods tat respet te limit imposed by various network devies. One an easily apply te traditional VM onsolidation semes to andle network bandwidt limit. Neverteless, tis migt raise a fundamental drawbak as reent measurement studies sow tat, in prodution data enters, network traffi are igly volatile and bursty[2][9][27]. It tus beomes diffiult for te traditional semes to make a reliable, deterministi estimate of bandwidt demand. Altoug one an take a onservative strategy by making an estimate mu iger tan te atual traffi rate, it leads to over-provisioning and resoure waste. In tis work, instead of using deterministi values, we use random variables to araterize te future bandwidt usage. Te random variables follow ertain distributions tat are estimated from eiter istorial traffi rates or foreasting algoritms. Su a probabilisti araterization an better represent te unertainty of te future bandwidt demand. Wit VM size being a random variable, te VM onsolidation issue is formulated into a NP-ard Stoasti Bin Paking problem (SBP), wi states tat items wit sizes following a probabilisti distribution must be paked into bins su tat te number of bins used is minimal and te ane for violating any bin size is below a tresold. Similar problems are studied in [][22], assuming te random variables all ave Bernoulli-type distributions, Poisson distributions, or exponential distributions. In our work, all random variables ave Gaussian distributions, and we propose an approximation algoritm and prove its asymptotial worst-ase performane bound. In our algoritm, random variables are divided into groups aording to teir means and varianes. Te random variables are ten assigned to different bins based on wi groups tey belong to. We prove te worst-ase performane ratio of our algoritm is witin (+ɛ)( 2+) for any ɛ > 0. In a speial ase were tere are finite number of possibilities of te mean value, te worst-ase performane ratio is witin 2+. We run numerial experiments to evaluate performane of te proposed algoritm. Te results sow tat our algoritm saves up to 30% of required servers ompared wit several

2 2 benmark semes. Te rest of te paper is organized as follows. Setion II disusses te related work. Setion III elaborates on problem formulation. Setion IV and V present algoritms and bound analysis for a speial ase and generi ases respetively. Setion VI onsiders a more general ase wit variable-sized bins. Setion VII uses numerial experiments to evaluate te proposed algoritms. Setion VIII onludes te paper. II. RELATED WORK A. Virtual Maine Consolidation VM onsolidation in Compute Clouds as been takled in bot ommerial produts and resear work. Representative produts inlude VMWare s Distributed Resoure Seduler (DRS) [33] and IBM Server Planning Tool (SPT) [2]. Tese tools use server utilization data to dynamially assign resoures to servers based on various poliies. In te resear literature, most work [][8][26][28][34] formulate VM onsolidation as an optimization problem wit te number of required servers as te objetive. Su an optimization problem is often assoiated wit onstraints imposed by server apaity, servie-level-agreement, legal, et. To solve te optimization problem, tese work use various algoritms wit a nature similar to tese bin paking euristis su as First Fit Dereasing (FFD) []. Wile tese produts and resear work onsider CPU, memory or disk, tey do not take into aount te network apaity limit imposed by network devies. On te metodology aspet, tese work denote te resoure demand of VMs by a deterministi value. In ontrast, in tis work we model te VM resoure demand as a random variable wi better aptures te dynamis of bandwidt usage in data enters. B. Bin Paking In te lassial bin paking problem, a list L wit real numbers between 0 and must be assigned to bins wit unit apaity su tat te sum of te numbers in ea bin is no more tan and te number of bins used is minimal. Let OP T denote te minimum. For a paking algoritm B, let B(L) denote te number of bins used for L, ten te worst ase performane ratio of tis algoritm is defined as lim OP T sup L [B(L)/OP T ]. Sine finding te optimal paking involves solving an NP-omplete partition problem [0][20], various euristi algoritms [4][4][5][6][35] wit performane guarantee ave been proposed. One simple algoritm First Fit Dereasing (FFD) as a worst ase performane ratio of 9 [0][36], and is later sown to be tigt [7]. MFFD [7], a variane of FFD an ave a ratio of Tese algoritms require to first sort te items, and are not appliable for an online bin paking problem. An online algoritm is required to proess L in te same order as given. First Fit [0] is an simple online algoritm wit a performane ratio of 7 0, and a refined version [35] as a ratio of 5 3. Next Fit [4] is anoter popular online algoritm wi as a worst ase performane ratio of 2 and takes O(n) time and O() spae. First Fit (FF), on te oter and, takes O(n log n) time and O(n) spae. [24] proposes a HARMONIC algoritm wi as a worst ase performane ratio of.692. [35] sows tat no online algoritm as a performane ratio less tan 3 2, and tis lower bound is later improved to [6][25]. Our problem formulation is te same as te Stoasti Bin Problem in [][22]. Given a positive onstant α, te goal is to use a minimum number of bins to pak te items su tat te violation probability of ea bin is at most α. [22] onsider te ase tat item sizes ave a Bernoullitype distribution, and te worst-ase performane ratio of log p teir algoritm is O( log log p ). [] provide a polynomial time approximation algoritm for te ases tat item sizes ave Poisson or exponential distributions, a quasi-polynomial approximation seme wit running time polynomially on n and log /p for Bernoulli-distributed items. III. PROBLEM FORMULATION In tis setion, we present a model of VM onsolidation wit network bandwidt onstraint imposed by network devies. We first introdee te Stoasti Bin Paking (SBP) in Setion III-A. We ten ompare SBP to te lassial bin paking problem in Setion III-B and illustrate its speial properties in Setion III-C. A. Stoasti Bin Paking We onsider a senario in wi n VMs wit known bandwidt demands must be onsolidated onto a number of servers (or assis, raks). We assume an online onsolidation senario tat ea VM sould be onsolidated one a request is made. We assume servers ave idential apaity limit. Tis orresponds to te ommon ase tat server onfigurations are omogeneous in te same data enter. Heterogeneous ases will be disussed in Setion VI. Let x i denote te normalized bandwidt demand of VM i, te n-vm onsolidation problem is to pak a list of items L = (x, x 2,..., x n ) into bins wit unit apaity. Instead of assuming x i a fixed value in te lassial bin paking problem, ere we assume x i follows a probabilisti distribution. Te distributions of te sizes of different items i and j are independent, and te probability distribution of teir total size x i + x j is te onvolution of te two probability distributions of x i and x j. In te lassial bin paking problem, a paking strategy is feasible if for ea bin, te total item size in tat bin does not exeed one. Here we define a paking strategy feasible if for ea bin, te probability tat te unit apaity is exeeded is no greater tan a given onstant α (0, ). Our goal is to find a feasible paking strategy tat uses te least number of bins. We formally desribe te Stoasti Bin Paking problem as following: Given a list of items L = (x, x 2,..., x n ), were x i s are independent random variables, wat is te minimum number of unit apaity bins needed to pak all te items, su tat for ea bin, te probability tat its apaity is exeeded is no greater tan a given onstant α (0, )? Different from previous assumptions tat x i follows a Bernoulli distribution, a Poisson distribution or an exponential

3 3 distribution [][22], in tis paper we assume tat x i independently follows a normal distribution N (µ i, σi 2 ). In reality server onsolidation usually ours at weekly or montly timesale. At su a large timesale, we believe te VM bandwidt usage an be approximated by a Gaussian distribution. We assume te mean µ i is positive and te standard deviation σ i is small enoug ompared wit µ i, tus te probability of x i being negative is very small and negligible. If σ i = 0 for all i, ten x i = µ i and te problem is redued to te lassial NP-ard one-dimensional bin paking problem. Clearly SBP is NP-ard. For a paking strategy, let U j denote te set of indies of te items tat are paked in bin j. Sine x i N (µ i, σi 2) and x i and x j are independent if i j, ten te total size of te items in a bin follows normal distribution wit mean i U µ j i and variane i U j σ2 i. For te normal distribution, we ave te following fat, Lemma. Given x N (µ, σ 2 ) wit µ (0, ) and α (0, ), ten Prob[x > ] α olds if and only if µ + Φ ( α)σ, were te a quantile funtion Φ is te inverse of te df Φ of N (0, ). Proof: Sine x N (µ, σ 2 ), ten Prob[x > µ + Φ ( α)σ] = α from definition. Tus Prob[x > ] α if and only if µ + Φ ( α)σ. Given te violation probability α, let β := Φ ( α). Tus, from Lemma, we say a paking strategy is feasible for a given α if and only if i U µ j i + β i U j σ2 i for all bin j. In tis paper, we only onsider te ase tat α 0.5 and β 0. In order to make sure tat tere exists a feasible paking for a given α, trougout tis paper we assume tat ea item sould be able to be paked into one bin, i.e. µ i + βσ i, i. B. SBP v.s. Deterministi Paking Beause i U j µ i + β i U j σ2 i i U j (µ i + βσ i ), if we use onstant µ i + βσ i as te size of item i, and redue te problem to te lassial bin paking, obviously any feasible paking strategy for te lassial bin paking is also feasible for SBP. Terefore, Observation. Te optimal number of bins used in SBP is no greater tan te optimum of a lassial bin paking problem wit µ i + βσ i as te size for item i. Sine SBP only requires µ i + µ j + β σi 2 + σ2 j wen item i and item j are paked in one bin, te required bins by simply fitting into te lassial bin paking problem may be mu larger tan te optimum of SBP. We provide an example to illustrate tis differene. i.i.d. Example Let n = 60 and x i N ( 2 4, ( 3 64 )2 ) for all i. α = 0.003, and β = Φ ( α) = 3. Sine µ i = 2 4, σ i = 3 64 for all i, and =, ten ea bin an pak exatly 6 items. For 60 items, te optimal strategy is to pak tem into 0 bins. If we use µ i + βσ i as te size of item i and treat it as a lassial bin paking problem, sine = 7 64, ten ea bin an pak at most 9 items. Tus, te best strategy is to pak 9 items in ea bin, wi results in using 8 bins. In general, if we diretly fit into te lassial bin paking problem wit sizes µ i + βσ i, even if we ould solve te NPard problem and find te minimum bin number, tis value migt still be mu larger tan te number of bins tat are indeed needed for SBP. C. Equivalent Sizes Vary under Different Paking In fat, for items tat an be paked in one bin wit U as te set of indies, we ave µ i + β σi 2 = (βσ i ) µ 2 i +. () i U i U i U β i U σ2 i Tus, we an view µ i + (βσi)2 β i U σ2 i as te equivalent size of item i, wi in general depends on oter items paked togeter. In a speial ase tat σ i = 0 or β = 0, te equivalent size redues to µ i, wi does not depend on te size of oter items and oinides wit te lassial ase. However, wen σ i 0 and β 0, te equivalent size anges if te items tat are paked in te same bin anges. For example, if item i is te only one in a bin, ten its equivalent size aieves te maximum, wi is µ i + βσ i. If item i is paked in te same bin wit some item aving a non-zero variane, ten te equivalent size is stritly less tan µ i + βσ i. Note tat in te lassial bin paking problem, te number of bins used is indeed te sum of te item sizes plus te sum of te residual apaity of ea bin. Sine te item size is fixed, we only need to pak ea bin in a ompat way so as to redue te residual apaity of ea bin. However, in our problem setup, reduing te residual apaity does not neessarily redue te number of bins used sine te equivalent size an ange. Let us first onsider a simple example. Example 2. Let n = 35, α = 0.003, β = 3. Tere are i.i.d. tree types of items:() x i N (/6, /324), i =,..., 4, i.i.d. (2) x i N (/2, /296), i = 5,..., 3, and (3) x i = /3 3/6, i = 4,..., 35. Sine =, ten te first four items an fit into one bin. One an ek tat te next nine items an exatly fit into one bin. Sine 22 ( ) <, ten te rest twenty-two items an be paked in one bin. Tis strategy uses tree bins and tere is no residual apaity in two bins. One an also ek tat tis strategy is also optimal sine tat tere is no way to pak all te items into two bins. If we pak two items in te first type and four items in te seond type and one item in te tird type togeter, one an ek tat tey an just fit into one bin wit no residual apaity. If two bins are paked in tis way, ten we ave one item in te seond type and twenty items in te tird type left, wi annot fit into one bin. Terefore, even toug te first two bins are fully paked, we need four bins in tis strategy.

4 4 Algoritm Group paking algoritm for items wit means oosing from a finite set Initial: t k = 0 for all k, Ea item i (i =,..., n) belongs to a group G k for some k {,..., m}, {,..., W k } 2 If item i fits into te urrent G k bin, te pak it. 3 Oterwise, t k = t k +, open a new G k bin to pak item i, and make it te urrent G k bin. As sown in te example, it is te variation of equivalent size tat leads to different paking results. Sine te equivalent size is larger in te seond strategy tan te first one, even toug tere is no residual apaity, we annot fit tem into tree bins. Terefore, in order to redue te number of bins, we need to () redue te residual apaity and (2) redue te equivalent size at te same time wile paking items. Sine te equivalent size of item i depends on oter items, we provide its upper bound and lower bound tat only depend on µ i and σ i. From (), we ave β i U σ2 i i U µ i µ i, terefore, µ i + (βσi)2 µ i + β i U (βσi)2 σ2 µ i, i were te rigtand side only depends on µ i and σ i. Let ten we ave g(µ, σ) := µ + (βσ) 2 /( µ), (2) g(µ i, σ i ) µ i + (βσ i ) 2 β i U σ2 i µ i + βσ i. (3) In later disussion, we will use te two bounds in (3) to araterize te equivalent size of item i. IV. A SPECIAL CASE: FINITE POSSIBILITIES OF MEANS In tis setion we onsider a speial ase tat te means of all items are from m distint values e, e 2,...,e m wit e k (η, ) (η > 0). Wile m is a onstant, n, te number of items, an grow to infinity. We will provide an online paking algoritm, and prove tat it an aieve a worse ase performane ratio of 2+. Te result is sligtly better tan in te general senario in Setion V. Besides, te algoritm and te analysis tenique used ere are te basis for te general senario. We first desribe te algoritm (see Algoritm ). In te algoritm, we divide items into different groups aording to teir means and varianes, we ten pak te items in te same group by Next Fit [4]. Speifially, we keep one ative bin for ea group. If te urrent item annot fit into te ative bin, we open a new bin and make it ative. For items wit te same mean e k (k =,..., m), te standard deviation is no greater tan ( e k )/β from previous assumptions. We divide tem into W k (= /e k /η ) groups aording to teir varianes. If µ i = e k, and σ i ( (+)e k β, e k + β ] ( =,..., W k ), ten we say item i belongs to group G k. If µ i = e k, and σ i [0, W ke k β ], ten item i belongs to group W k G kwk. Let t k be te number of bins used by Algoritm to pak te items in group G k, ten te total number of bins used is B := m Wk k= = t k. Sine te item lassifiation an be done in O(log(m /η )) time, Algoritm runs in O(n log(m /η )) time. For given m and η, it takes O(n) time. Sine tere are at most m /η ative bins at any time, te algoritm takes m /η storage spae to store ative bins. Now we derive te worst ase performane ratio of Algoritm. If W k 2, note tat sine for ea item i tat belongs to G k ( =,..., W k ) we ave µ i = e k, and σ i ( (+)e k β, e k + β ], ten te total size of te items in one bin for group G k follows normal distribution wit mean µ = de k and variane σ ( d( (+)ek ), d( ek ) ], were β + β d is te number of items in a bin. From Lemma, for a feasible paking wit violation probability α, we want µ + βσ. If d =, we ave µ + βσ e k + ( e k ) =, and if d = +, we ave µ+βσ > (+)e k +( (+)e k ) =. Terefore tere are exatly items in ea bin exept te last one for group G k ( =,..., W k ). For group G kwk, sine ea item as mean e k and variane no greater tan W ke k β, W k ten tere are at least W k items in ea bin exept te last one. Tus, we ave te following key observation, Observation 2. In te resulting paking strategy of Algoritm, tere are exatly items in ea bin exept te last one of group G k ( =,..., W k ), and at least W k items in ea bin exept te last one of group G kwk for K =,..., m. From Observation 2, in te resulting paking strategy, for group G k (k =,..., m, =,..., W k ), ea item (exept tose in te last bin) takes up exatly / of a bin. And for group G kwk, ea item (exept tose in te last bin) takes up at most /W k of a bin. Tus, like in [24] we an define an effetive oupation f(µ, σ) as follows f(µ, σ) = if ( + )µ β < σ µ + β for some {,..., /µ } (4) f(µ i, σ i ) an be rougly interpreted as te fration of a bin tat is oupied by ea item i in te paking strategy of Algoritm. In fat, te effetive oupation f(µ, σ) is very important for te performane analysis of te algoritms in tis paper. Given violation probability α, let OP T denote te number of bins used by te optimal strategy. Define S to be te set of all te possible item subsets tat an fit into one bin su tat te violation probability for te bin apaity is at most α, i.e., S := {U {, 2,..., n} µ i + β }. (5) i U i U Now we an state a general lemma wi will be used bot ere and in Setion V for algoritm analysis. Lemma 2. Given α, a list of items L = (x, x 2,..., x n ) wit x i N (µ i, σ 2 i ) are paked into B bins. Let U j be te set of indies of items in bin j. If tere exists funtion f su tat σ 2 i

5 5 for every bin j exept Q bins (Q does not depend on n), i U j f(µ i, σ i ), (6) and let r := max U S i U f(µ i, σ i ), were S is defined in (5), ten we ave Moreover, wen OP T go to infinity, B r OP T + Q. (7) lim B/OP T OP T r (8) Proof: Sine (6) olds for B Q bins, ten B Q n i= f(µ i, σ i ). It follows from te definition of r tat n i= f(µ i, σ i ) r OP T. Ten we ave (7) by ombing te above two inequalities. (8) follows if we let bot sides of (7) divide OP T and let OP T go to infinity. Sine it is ard to diretly ompute r, we will next give an upper bound wi is also an upper bound for te worstase performane ratio of Algoritm. Sine i U µ i + β i U σ2 i r max U S for every U in S, we know i U f(µ i, σ i ) i U µ i + β i U σ2 i max U S i U f(µ i, σ i ) i U g(µ i, σ i ) max max f(µ i, σ i ) U S i U g(µ i, σ i ) = max f(µ i, σ i ) i g(µ i, σ i ), (9) were g is defined in (2), te seond inequality follows from (3) and te tird olds sine f and g are always positive. Now onsider te effetive oupation f defined in (4). For group G k (k =,..., m, =,..., W k ), Algoritm uses t k bins, and for ea bin j exept te last one, one an easily ek tat i U f(µ i, σ j i ) =. In group G kwk, we ave tat i U f(µ i, σ j i ) for ea bin j {,.., t kwk }. Tus, te assumption of Lemma 2 is satisfied wit Q m k= W k, were m k= W k aounts for te total number of possibly unfilled bins in all te groups. Terefore, from Lemma 2 and (9), and e k η k, we ave m B r OP T + k= W k max i f(µ i, σ i ) g(µ i, σ i ) OP T +m. (0) η f(µ Now te question is ow to estimate max i,σ i) i g(µ i,σ i). We state an important lemma for f(µ,σ) g(µ,σ). Lemma 3. For all =,..., /µ, we ave f(µ, σ) max µ (0,),σ [0, µ β ] g(µ, σ) < λ() := ( 2 2 ( + )+), () were f and g are defined in (4) and (2), and λ() stritly dereases as inreases. Speially, max µ+βσ ( f(µ, σ)/g(µ, σ) ) < 2 +. (2) Proof: We study f(µ, σ)/g(µ, σ) in tree different ases. (i) If µ > /2, ten f(µ, σ) =, and g(µ, σ) = µ + σ 2 /( µ) > /2. Terefore, f(µ, σ)/g(µ, σ) < 2. (ii) If µ /2 and σ ( (+)µ β, µ + β ] for some belongs to {, 2,..., /µ }, ten f(µ, σ) = /. Terefore, f(µ, σ) g(µ, σ) < ( (µ + = ( 2 ( µ) + ( ( + )µ)2 ( + )( µ) )) 3 ( + )( µ) + 22) (2 2 ( /( + ) ) + ), (3) were te last inequality olds from te fat tat 2 ( µ) + 22, and te equality olds wen µ = 3 (+)( µ) + /( + ). Terefore, for = 2,..., /µ, we ave ( ) max max f(µ, σ)/g(µ, σ) < λ(). µ σ ( (+)µ β +, µ β ] One an ek tat λ() > 0 and λ () < 0 for all. Ten λ() stritly dereases as inreases. Wen =, λ() aieves te maximum value 2+. Wen goes to infinity, λ() goes to 4 3. (iii) If µ /2, and σ /µ µ β /µ, ten f(µ, σ) = / /µ, and g(µ, σ) µ. Terefore, f(µ,σ) g(µ,σ) µ /µ. Let D := /µ, ten D 2. Besides, µ < Dµ. Tus, /D µ, and 0 Dµ < µ. We laim tat Dµ > (D ) 2 (D )3 ( µ)+ D( µ) +(D ) 2(D )2. (4) Sine te rigtand side of (4) minus its leftand side is ((D )µ 2 ( Dµ) 2 )/( µ) + (/D µ) > ((D )µ 2 µ 2 )/( µ) + (/D µ) = (D 2)µ 2 /( µ) + (/D µ) 0, were te first inequality olds from Dµ < µ, and te seond inequality olds from D 2, µ and /D µ. Terefore (4) olds. From (4) we ave Dµ > 2 Tus, (D ) 2 ( µ)(d ) 3 D( µ) + (D ) 2(D ) 2 = 2(D ) 2 ( (D )/D ) + D. f(µ, σ) g(µ, σ) µ /µ < ( 2(D ) 2 ( (D )/D )+D ). Combing ases (i), (ii) and (iii), we onlude tat max µ (0,),σ [0, µ β ] (f(µ, σ)/g(µ, σ)) < λ(), were =,..., /µ. Wen =, we get (2). Now we are ready to present our first main result. Teorem. Given a finite set {e,..., e m } wit e k (η, ) (η > 0) for all k =,..., m and a violation probability α 0.5, a list L = (x,..., x n ) wit x i N (µ i, σ 2 i ) and µ i {e,..., e m } for all i =,..., n needs to be paked into te

6 6 least number of bins wile for ea bin, te probability tat its apaity is exeeded does not exeed α. Ten te number of bins B used by Algoritm to pak L satisfies B < ( 2 + )OP T + m /η, (5) were OP T is te minimum number of bins needed. And te worst-ase performane ratio satisfies lim B/OP T 2 +. (6) n Proof: (5) follows by ombing (7), (0) and (2). Wen n goes to infinity, sine te number of items tat ea bin ould pak is at most /η, ten OP T also goes to infinity. Sine m /η is also a onstant, ten (6) olds. Note tat in Teorem, we only assume tat σ i 0 and µ i + βσ i for all i. In fat, te upper bound of te worstase performane ratio is aieved in te group tat µ 2 and σ ( 2µ 2β, µ]. If no item belongs to tis group, te upper bound an be redued. Terefore, provided wit furter onstraints on µ and σ su tat no item belongs to groups wit a large ratio f(µ,σ) g(µ,σ), we an indeed obtain a stronger result tan tat in Teorem. Teorem 2. Same assumption as in Teorem. If it furter olds tat for some Z +, µ i + β σ i for all i, ten B < λ()op T + m /η, and were λ() = (2 2 ( /( + ) ) + ). lim B/OP T λ(), n Proof: It follows by ombing (7), (9) and (). Te worst-ase performane ratio λ() stritly dereases as inreases, and lim λ() = 4/3. In pratie, sine µ i > η for all item i, ten is no greater tan /η. V. GENERIC SCENARIOS Setion IV disussed te speial ase tat te means of te items form a finite set wose ardinality does not ange as te number of items inreases. Here we onsider te general ase tat every two items an ave different means and different varianes. In oter words, we only assume tat µ i (η, ), σ i 0, and µ i + βσ i for all i. For any ɛ > 0, we provide an online paking algoritm wose worst ase performane ratio is ( + ɛ)( 2 + ). We first introdue te algoritm. Same as Algoritm, te key idea is to divide items into different groups and pak items in te same group by Next Fit. In Algoritm, sine tere are only m possibilities for te mean, ten we lassify items wit te same mean aording to te variane su tat for different groups, te number of items tat one bin an pak is different. Here we ome aross te diffiulty tat ea item ould ave a unique mean. Terefore we need to divide te items in a different and more autious way. We will define a stritly dereasing sequene { k } to divide te means into intervals (, ] and ( k+, k ] (k 2), and for items wit means belong to ( k+, k ], we will use {d k, =,..., m k = / k } to divide teir standard deviations into intervals [d k m k, d k m k ] and (d k, dk ]( =,..., m k ). Ten items wit means in te same interval and standard deviations in te same interval belong to te same group. To ensure tat te number of groups is finite, (η, ) sould be overed by finite number of intervals we defined. Moreover, for interval ( k+, k ], te union of [d k m k, d k m k ] and (d k, dk ]( =,..., m k ) sould over [0, ( k+ )/β], sine µ i + βσ i for every item i. We define a sequene k togeter wit d k ( = 0,,..., / k ) as follows. Given ɛ > 0, let r := (+ɛ)( 2+ ), ten = /r, m = /, and d = ( + ) β, = 0,,..., m. (7) + For k Z + and k 2, p k = ( 2 r + ( r )2 + 4(βd k ) ), 2 (8) were =,..., m k, and p k m k = /rm k, k = max {,...,m k } pk, m k = / k, (9) d k = ( + ) k β, = 0,,..., m k. (20) + We also define m 0 =, and d k m k = 0 for all k 0. To sow tat tis partition is valid, note tat from (20), orresponding to an interval ( k+, k ], te union of [d k m k, d k m k ] and (d k, dk ]( =,..., m k ) is exatly [0, ( k )/β]. Terefore, we only need to sow tat (η, ) ould be overed by a finite number of intervals indued by { k }. Formally, we state te following lemma, Lemma 4. For a given ɛ > 0, te sequene { k } defined in (7) (20) satisfies k k /( + ɛ), k 2. Tus, { k } is stritly dereasing, and tere exists M log ( ( 2 + )η ) /(log( + ɛ)) su tat M η, were M = max(, M). Proof: If k 2, from te definition we know p k < for all =,..., m k. Ten from (9) we ave k <. Note tat given d k ( =,..., m k ), te funtion π(x) = x + (βd k ) 2 /( x) stritly inreases on [, ). One an also ek tat π(p k ) = /(r) wit pk defined in (8). Tus, we ave x + (βd k ) 2 /( x) /(r) for all x [p k, ), and x + (βd k ) 2 /( x) < /r for all x [0, p k ). From te definition of k in (9), we ave for all x [ k, ), x + (βd k ) 2 /( x) /(r), =,..., m k, (2) and for every y [0, k ), tere exists some su tat y + (βd k ) 2 /( y) < /(r). Let γ k = k /( + ɛ), ten in order to prove k γ k olds, we only need to prove tat γ k +(βd k ) 2 /( γ k ) /(r), =,..., m k. (22) If k > /2, ten m k =, and d k = 0. Terefore, γ k + (βd k ) 2 /( γ k ) = k /( + ɛ) > /r. (23)

7 7 G β Algoritm 2 General group paking algoritm Keep one ative bin for ea group G k. 2 For ea item, deide wi group it belongs to. If it fits into te urrent bin for tat group, ten pak it. Oterwise, open a new bin and make it te urrent bin for tat group. µ 2 k k M Fig.. G Gkm k 22 d k. k d d. m k.... Gk G k d σ d k 2 β G k k β Grouping items aording to µ and σ If k /2, ten γ k = k /( + ɛ) = ( + ɛ)( k )+ɛ( 2+ɛ +ɛ k ) (+ɛ)( k ). Terefore, for all =,..., m k, γ k + (βdk ) 2 ( k + (βdk ) 2 ). (24) γ k + ɛ k From (3) and te arguments following it, we know tat ( k + (βdk ) 2 ) 2 2 ( k + )+ 2, (25) for all k (0, /2] and =,..., m k. Combing (24) and (25), we ave for =,..., m k, γ k + (βd k ) 2 /( γ k ) ( 2 )/(( + ɛ)) = r. (26) Note tat sine k /2, ten k m k k (/ k ) /2 > 2, tus, γ k k ɛ > = ( + ɛ)m k m k r. (27) Combing (23), (26) and (27), we ave tat (22) olds, terefore k k /( + ɛ).ten, k (/( + ɛ)) k = ( 2 )(+ɛ) k. If η 2, ten η. If η < 2, ten C M η. Terefore, M η. If we furter let 0 =, ten (η, ) M k= ( k, k ] wit M defined in Lemma 4. Item i belongs to group G kmk if µ i ( k, k ](k =,..., M), and σ i [0, d k m k ]. It belongs to group G k if µ i ( k, k ] and σ i (d k, d k ] ( =,..., m k ). Sine k M, and m k /η for all k, ten tere are at most M /η groups. Figure illustrates ow to divide items aording to te mean and te standard deviation. We briefly summarize te paking strategy in Algoritm 2. Sine te lassifiation takes O(log( M /η )) time for ea item, Algoritm 2 runs in O(n log( M /η )) time. Sine tere are at most M /η ative bins at any time, te algoritm takes M /η storage spae to store ative bins. Let t k denote te number of bins used to pak items in group G k, ten te total number of bins used is B := M mk t k= = k. We ave one main result as follows, Teorem 3. A list L = (x,..., x n ) wit x i N (µ i, σi 2) (µ i (η, )) needs to be paked into te minimum of bins, su tat for ea bin, te probability tat its size is exeeded does not exeed a given probability α. For any ɛ > 0, Algoritm 2 wit { k } and {d k } defined in (7) (20) produes a feasible paking strategy wit te number of bins B satisfying B ( + ɛ)( 2 + )OP T + M /η, (28) were OP T is te minimum number of bins needed. And te worst-ase performane ratio satisfies lim B/OP T ( + ɛ)( 2 + ). (29) n Proof: Sine for any item i in group G k, µ i k and σ i d k, ten te total size of items in group G k follows N (µ, σ 2 ) wit µ k and σ d k. Sine µ + βσ k + β d k =, ten tere are at least items in one bin exept te last bin for group G k. We define te effetive oupation f(µ, σ) su tat f(µi, σ i ) = if item i belongs to G k (k =,..., M, =,..., m k ). For ea bin j of group G k exept te last bin, let U j denote te set of indies of te items in tis bin, ten i U f(µ i, σ j i ). Ten from Lemma 2, we ave B r OP T + M /η, (30) were M /η aounts for te total number of possibility unfilled bins in all te groups, and r = max U S i U f(µ i, σ i ) wit S defined in (5). From (9), we ave r f(µi,σ max i) i g(µ, i,σ i) were g(µ, σ) is defined in (2). For any item i in group G k, (k =,..., M, =,..., m k ), we ave µ i > k and σ i d k, tus, g(µ i, σ i ) = µ i + (βσ i ) 2 /( µ i ) k + (βd k ) 2 /( k ) = k + ( ( + ) k ) 2/ ( ( + )( k ) ) /(r), were te last inequality olds from (2). Ten f(µ i, σ i )/g(µ i, σ i ) /r = ( + ɛ)( 2 + ). (3) Sine for every item i, (3) olds, ten r ( max f(µi, σ i )/g(µ i, σ i ) ) ( + ɛ)( 2 + ). (32) i Terefore, (28) follows from (30) and (32). Sine given ɛ, M /η is a onstant, and OP T goes to infinity as n goes to infinity, ten (29) follows.

8 8 We remark tat if we define { k } as k = k /( + ɛ) instead of (9), and modify Algoritm 2 aordingly, Teorem 3 still olds. However, ompared wit te urrent algoritm wit (9), te items are divided into more groups if we make tis ange, wi ould lead to more unfilled bins. Tus, Algoritm 2 wit { k } and {d k } defined in (7) (20) as a better pratial performane. In te analysis of Algoritm and Algoritm 2, sine it is ard to ompute r and r diretly, we use teir upper bounds f(µ max i,σ i) i g(µ i,σ i) (for r f(µi,σ ) and max i) i g(µ i,σ i) (for r ) instead. However, sine tese upper bounds are in general loose, te worst-ase performane ratio we obtained are also not tigt. VI. EXTENSION TO VARIABLE-SIZED BINS Here we extend to a more general ase tat tere are q > kinds of bins wit different apaities 0 < α < α 2 <... < α q =, and tere are an inexaustible supply of bins of ea size. Ten for a given list L of items, and a given violation probability, we want to minimize te total apaities of te bins used to pak L. Let OP T v denote tis minimum total apaity. We also onsider paking L using only unit-apaity bins like wat we disussed previously and let OP T denote te minimum number of unit-apaity bins needed. Ten one an easily ek tat OP T v OP T. Teorem establised te relation of te number bins B used by Algoritm and OP T in (5) and (6). Teorem 3 establised te relation of te number of bins B used by Algoritm 2 and OP T in (28) and (29). Here we laim tat (5-6) and (28-29) still old if we replae OP T wit OP T v. In oter words, even if te optimal strategy an oose bin size for ea bin, te worst-ase performane ratio of implementing Algoritm and Algoritm 2 wit unit-apaity bins do not ange. Tis result follows from te following lemma, Lemma 5. Given α, a list of items L = (x, x 2,..., x n ) wit x i N (µ i, σi 2) are paked into B bins. Let U j be te set of indies of item in bin j. If tere exists funtion f su tat for every bin j exept Q bins (Q does not depend on n), i U f(µ i, σ j i ), and g is defined in (2), ten ( B max f(µi, σ i )/g(µ i, σ i ) ) OP T v + Q, i Proof: Note tat B Q n i= f(µ i, σ i ) from assumption. Sine te maximum possible bin size is, ten g(µ i, σ i ) is still a lower bound of te equivalent size of item i, i.e. (3) still olds. Tus, n i= g(µ i, σ i ) OP T v. Ten, B n i= f(µ f(µ i, σ i ) + Q max i,σ i) i g(µ ( n i,σ i) i= g(µ i, σ i )) + Q ( f(µ i, σ i )/g(µ i, σ i ) ) OP T v + Q. Ten using te same analysis in Setion IV and ombining Lemma 3 and Lemma 5, we onlude tat B < ( 2 + )OP T v + m /η, were B is te number of bins used by Algoritm. One an ompare tis result wit (5). Similarly, for te number of bins B used by Algoritm 2, following te analysis in Setion V and Lemma 5, we ave B ( + ɛ)( 2 + )OP T v + M /η. Fig. 2. Perentage of Servers Algoritm 2 HARMONIC (µ+β σ) HARMONIC (µ+σ) Violation Probability (α) Violation probability on pysial maines in different strategies Number of Bins Used Fig. 3. HARMONIC FF FFD Algoritm 2 Lower Bound Number of Items (n) x 0 4 Number of bins used by different algoritms Terefore, te worst ase performane bounds we obtained for Algoritm and Algoritm 2 still old for te general problem of paking random variables wit variable-sized bins. VII. NUMERICAL RESULTS We use traffi dataset from global operational data enters. Details of tis dataset are provided in [27]. We ompute te mean and variane of te traffi rates for about 9K VMs in a six-our period. In our simulations, we assume tese VMs are onsolidated onto servers equipped wit Gbps Eternet ard. Te apaity violation probability is α = 0.0, tus we ave β = Te number of servers used by te proposed group paking algoritm, i.e., Algoritm 2, is 42. For omparison purposes, we also implement te HARMONIC algoritm [24] (wit M=2), a popular online algoritm for te lassial bin paking problem. To guarantee te atual violation probability does not exeed α, we use µ i +βσ i as te bandwidt requirement for VM i wen HARMONIC is tested. In tis test, te number of servers used is 609. Comparing te proposed algoritm and HARMONIC, te proposed one redues te required servers by 30%. If we are less onservative and use µ i +.2σ i as te bandwidt requirement for VM i, ten te number of servers used is 402, wi is omparable to te number used by Algoritm 2. However, te violation probability in tis ase exeeds α for some servers. We ten draw 0000 samples from te obtained distributions for all

9 9 te VMs and alulate te empirial violation probability for ea server in differen semes. As sown in Figure 2, bot our algoritm and HARMONIC wit µ + βσ as VM size an guarantee tat for ea server, te violation probability does not exeed 0.0, wile in te HARMONIC ase wit µ+.2σ as VM size, te violation probability exeeds 0.0 in about 7 perents of te servers. Next, we generate random samples and pak tem wit four different algoritms: Algoritm 2, HARMONIC, FFD and FF. We inrease te number of items to pak from 2000 to Given α = and β = 2, we let te distribution of te size of item i follow N (µ i, σi 2) wit µ i and σ i satisfying µ i + βσ i. Algoritm 2 paks te distributions, wile HARMONIC, FFD and FF take µ i +βσ i as te size of item i. We also find a lower bound of te minimum number of bins needed (OPT) as follows. We use FFD to pak items wit g(µ i, σ i ) as te size of item i and let B be te number of bins used. Sine te minimum number of bins needed g(µ i, σ i ) (i =,..., n), denoted by OP T g, is less tan OP T, and te FFD algoritm guarantees tat B 9 OP T g + ([36]), ten 9 (B ) OP T g OP T, and 9 (B ) is a lower bound of OP T. Figure 3 ompares te number of bins used by te four algoritms. Te result for ea n is averaged over 000 runs. All te algoritms an guarantee tat te violation probability does not exeed Te result sows tat te number of bins used varies signifiantly: Algoritm 2 uses te least among te four, and is witin.6 times optimum. VIII. CONCLUSION Most traditional VM onsolidation semes only onsider CPU, memory and disk onstraints, and solve a bin paking type of problem by making a deterministi estimate of resoure demands. Tis paper studies VM onsolidation problem wen network devies in data enters impose bandwidt onstraints. Beause of te dynami nature in data enter traffi, we formulate te VM onsolidation as a novel Random Variable Paking problem wi models te bandwidt demands of VMs as probabilisti distributions. Su a probabilisti approa better aptures te unertainty in network bandwidt demand tan te traditional deterministi model. We propose an approximation algoritm and prove its worst-ase performane ratio of (+ɛ)( 2+) for any ɛ > 0. Te ratio is furter improved to 2 + in speial ases. We demonstrate wit numerial experiments tat te proposed algoritm saves many servers witout violating te server apaity onstraints. Due to te generality of te defined RVP problem and te algoritm, te results in tis work an apply to te VM onsolidation problem for oter resoure types and beyond. SBP wit general item size distributions is interesting to explore. REFERENCES [] Y. Ajiro and A. 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