# Dynamic right-sizing for power-proportional data centers Extended version

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2 ore than 15% cost savings even when the background work akes up 4% of the workload when the PMR is larger than 3. II. MODEL FORMULATION Right-sizing probles vary between data centers: soe systes have flexible quality of service (QoS) requireent, others have hard service level agreeents (SLAs); the electricity price ay be tie-varying or fixed; workloads can be hoogeneous or heterogeneous, etc. We first introduce a general odel which captures the ain issues in any rightsizing probles in different systes. We then give exaples and show how they fit into this general odel. A. General odel We now describe the general odel used to explore the cost savings possible fro dynaic right-sizing. An instance consists of a constant β >, a horizon 1 T >, a sequence of non-negative convex functions g t ( ) for t = 1,..., T 1; g t ( ) can take on the value but cannot be identically. For notational siplicity, let g T (x) and x = x T =. The odel is: iniize x 1,...,x T 1 g t ( ) + β ( 1 ) + (1) subject to where (x) + = ax(, x), and { } are non-negative scalar variables. The solution to optiization (1) ay not be unique. By Corollary 1 in Appendix A, there exists a solution that is the eleentwise axiu. Unless otherwise stated, the solution refers to this axiu solution. We discuss the extension to vector in Section VIII. To apply this to data center right-sizing, recall the ajor costs of right-sizing: (a) the cost associated with the increased delay fro using fewer servers and the energy cost of aintaining an active server with a particular load; (b) the cost incurred fro toggling a server into and out of a power-saving ode (including the delay, igration, and wearand-tear costs). We call the first part operating cost and the second part switching cost and consider a discretetie odel where the tieslot length atches the tiescale at which the data center can adjust its capacity. There is a (possibly long) tie-interval of interest t {, 1,..., T } and the capacity (i.e., nuber of active servers) at tie t is. The operating cost in each tieslot is odelled by g t ( ), which presents the cost of using servers ( has a vector value if servers are heterogeneous) to serve requests at tieslot t. Moreover, g t ( ) captures any other factors in tieslot t, including the arrival rate, the electricity price, the cap on available servers, the SLA constraints. For the switching cost, let β be the cost to transition a server fro the sleep state to the active state and back again. We dee this cost to be incurred only when the server wakes up. Thus the switching cost for changing the nuber of active 1 If paraeters such as β are known in advance, then the results in this paper can be extended to a odel in which each instance has a finite duration, but the cost is sued over an infinite horizon. servers fro 1 to is β( 1 ) +. The constant β includes the costs of (i) the energy used, (ii) the delay in igrating connections/data/etc. (e.g., by VM techniques), (iii) increased wear-and-tear on the servers, and (iv) the risk associated with server toggling. If only (i) and (ii) atter, then β is either on the order of the cost to run a server for a few seconds (waking fro suspend-to-ram or igrating network state [4] or storage state [5]), or several inutes (to igrate a large VM [3]). However, if (iii) is included, then β becoes on the order of the cost to run a server for an hour [9]. Finally, if (iv) is considered then our conversations with operators suggest that their perceived risk that servers will not turn on properly when toggled is high, so β ay be any hours server costs. Throughout, denote the operating cost of a vector X = (x 1,..., x T ) by cost o (X) = T g t( ), the switching cost by cost s (X) = β T ( 1 ) +, and cost(x) = cost o (X) + cost s (X). We have seen that Forulation (1) captures the operating cost and switching cost of right-sizing probles. However, Forulation (1) ay look not that general at first glance due to the lack of constraints. Actually constraints in rightsizing probles can be handled iplicitly in functions g t ( ) by extended-value extension, i.e., defining g t ( ) to be outside its doain. This extension akes our description/ notation ore clear without introducing particular constraints in different data centers. Forulation (1) akes a siplification that it does not enforce that be integer valued. This is acceptable since the nuber of servers in a typical data center is large; oreover, preliinary investigation suggests that this can be enforced by an appropriate choice of cost functions, g t ( ). This odel also ignores issues surrounding reliability and availability. In practice, a solution that toggles servers ust still aintain the reliability and availability guarantees. Such issues are beyond the scope of this paper, however previous work shows that solutions are possible [5]. This optiization proble would be easy to solve offline, i.e., given functions g t ( ) for all t. However, our goal is to find online algoriths for this optiization, i.e., algoriths that deterine x using only inforation up to tie + w, where the prediction window w is part of the proble instance. Here, we assue that the predictions [g,..., g +w ] are known perfectly at tie, but we show in Section VI that our algorith is robust to this assuption in practice. We evaluate the perforance of an online algorith A using the standard notion of copetitive ratio. The copetitive ratio of A is defined as the supreu, taken over all possible inputs, of cost(a)/cost(op T ), where cost(a) is the objective function of (1) under A and OP T is the optial offline algorith. The analytic results of Sections IV and V assue that the service has a finite duration, i.e. T <, but hold for arbitrary sequences of convex functions g t ( ). Thus, the analytic results provide worst-case guarantees. However, to provide realistic cost estiates, we consider case-studies in Section VI where g t ( ) is based on real-world traces. 2

3 B. Special cases Now let us consider two choices of g t ( ) to fit two different right-sizing probles into Forulation (1). The discrete-tie odel consists of finitely any ties t {, 1,..., T }, where T ay be a year, easured in tieslots of 1 inutes. Assue the workload affects the operating cost at tieslot t only by its ean arrival rate, denoted λ t. (This ay have a vector value if there are ultiple types of work). This assuption is reasonable for any services such as web search, social network or eail service since the request interarrival ties and the response ties are uch shorter than the tieslots so that the provisioning can be based on the arrival rate. We odel a data center as a collection of hoogeneous servers (except Section VIII where heterogeneous systes are discussed) and focus on deterining, the nuber of active servers during each tie slot t. For notational siplicity, we assue a load balancer assigns arriving jobs to active servers uniforly, at rate λ t / per server (which is widely deployed and turns out to be optial for any systes). Assue the power of an active server handling arrival rate λ is e(λ), and the perforance etric we care about is d(λ) (e.g., average response tie or response tie violation probability). For exaple, a coon odel of the power for typical servers is an affine function e(λ) = e + e 1 λ where e and e 1 are constants; e.g., see [1]. The average response tie can be odeled using standard queuing theory results. If the server happens to be odeled by an M/GI/1 Processor Sharing queue then the average response tie is d(λ) = 1/(µ λ), where the service rate of the server is µ [11]. Other exaples include, for instance, using the 99th percentile of delay instead of the ean. In fact, if the server happens to be odeled by an M/M/1 Processor Sharing queue then the 99th percentile is log(1)/(µ λ), and so the for of d(λ) does not change [11]. Note that, in practice, e(λ) and d(λ) can be epirically easured by observing the syste over tie without assuing atheatical odels for the. Our analytical results work for any e( ) and d( ) as long as they result in convex g t ( ). But for siplicity, we will use e(λ) = e +e 1 λ and d(λ) = 1/(µ λ) for our nuerical experients. 1) Exaple 1: Services with flexible QoS requireent: Many Internet services, such as web search, eail service and social network are flexible in response tie in certain range. For such applications, it is hard to find a threshold to distinguish good services and bad services. Instead, the revenue is lost gradually as the response tie increases and thus our goal is to balance the perforance and the power cost [8], [12] [14]. One natural odel for the lost revenue is d 1 λ(d(λ) d ) + where d is the iniu delay users can detect and d 1 is a constant. This easures the perceived delay weighted by the fraction of users experiencing that delay. For the energy cost, assue the electricity price is p t at tieslot t, then the energy cost is p t e(λ). There ay be a tie-varying cap M t on the nuber of active server available due to other activities/services in the data center (e.g., backup activities). The cobination together with the dispatching rule at the load balancer gives iniize + β ( ( d 1 λ t d ( λt ) d ) + + p t e ( λt ) ) ( 1 ) + (2) subject to λ t M t Copared to Forulation (1), we have g t ( ) = ( ( ) ) + ( ) d x 1λ t λ t (d ) t d + λ pt e t if λ t M t, otherwise. We can see that g t ( ) is in the for of the perspective function of f(z) = d 1 z(d(z) d ) + + p t e(z), which is convex under coon odels of d( ) and e( ), such as M/G/1 queueing odels with or without speed scaling [15] of individual servers. Thus g t ( ) is convex under coon odels. 2) Exaple 2: Services with hard QoS constraints: Soe data centers have to support services with hard constraints such as SLA requireent or delay constraints for ultiedia applications. For such systes, the goal is to iniize energy cost while satisfy the constraints, as shown in [16], [17]. The optiization becoes ( ) λt iniize p t e + β ( 1 ) + (3) subject to λ t and d ( λt ) D t where d( ) can be average delay, delay violation probability or other perforance etrics. This constraint usually defines a convex doain for. Actually if we assue that d( ) is an increasing function of the load, then it is equivalent to iposing an upper bound on the load per server (i.e., λ t / is less than certain threshold) and thus results in a convex optiization proble. The corresponding g t ( ) can be defined as follows: g t ( ) = { ( ) λ xt p t e t if λ t and d otherwise. III. RECEDING HORIZON CONTROL ( λ t ) D t, As otivation for deriving a new algorith, let us first consider the standard approach. An online policy that is coonly proposed to control data centers [7], [8] (and other dynaic systes) is Receding Horizon Control (RHC) algoriths, also known as Model Predictive Control. RHC has a long history in the control theory literature [18] [2] where the focus was on stability analysis. In this section, let us introduce briefly the copetitive analysis result of RHC for our data center proble (which is proven in [21]) and see why we ay need a better online algorith. Inforally, RHC(w) works by solving, at tie, the cost optiization over the window (, +w) given the starting state 3

4 x 1, and then using the first step of the solution, discarding the rest. Forally, define X (x 1 ; g... g +w ) as the vector in R w+1 indexed by t {,..., + w}, which is the solution to iniize +w t= subject to Then, RHC(w) works as follows. +w g t ( ) + β t= ( 1 ) + (4) Algorith 1. Receding Horizon Control, RHC(w). For all t, set the nuber of active servers to =. At each tieslot 1, set the nuber of active servers to x = X (x 1 ; g... g +w ) Note that (4) need not have a unique solution. We define RHC(w) to select the solution with the greatest first entry. Define e the iniu cost per tieslot for an active server, then we have the following theore [21]: β Theore 1. RHC(w) is (1 + (w+1)e )-copetitive for optiization (1) but not better than ( 1 w+2 + β (w+2)e )-copetitive. This highlights that, with enough lookahead, RHC(w) is guaranteed to perfor quite well. However, its copetitive ratio depends on the paraeters β, e and w. These paraeters vary widely in different data centers since they host different services or have different SLAs. Thus, RHC(w) ay have a poor copetitive ratio for a data center with big switching cost or sall prediction window. This poor perforance is to be expected since RHC(w) akes no use of the structure of the optiization we are trying to solve. We ay wonder if there exists better online algorith for this proble that perfors well for a wider range of paraeters, even when the prediction window w = (no workload prediction except current tieslot). IV. THE OPTIMAL OFFLINE SOLUTION In order to exploit the structure of the specific proble (1), the first natural task is to characterize the optial offline solution, i.e., the optial solution given g t ( ) for all t. The insight provided by the characterization of the offline optiu otivates the forulation of our online algorith. It turns out that there is a siple characterization of the optial offline solution X to the optiization proble (1), in ters of two bounds on the optial solution which correspond to charging cost β either when a server coes out of powersaving ode (as (1) states) or when it goes in. The optial x can be viewed as lazily staying within these bounds going backwards in tie. More forally, let us first describe lower and upper bounds on x, denoted x L and x U, respectively. Let (x L,1,..., x L, ) be the solution vector to the optiization proble iniize g t ( ) + β ( 1 ) + (5) subject to where x =. Then, define x L = x L,. Siilarly, let (x U,1,..., x U, ) be the solution vector to the optiization iniize g t ( ) + β subject to ( 1 ) + (6) Then, define x U = x U,. Notice that in each case, the optiization proble includes only ties 1 t, and so ignores the future inforation for t >. In the case of the lower bound, β cost is incurred for each server toggled on, while in the upper bound, β cost is incurred for each server toggled into power-saving ode. Lea 1. For all, x L x x U. Given Lea 1, we now characterize the optial solution x. Define (x) b a = ax(in(x, b), a) as the projection of x into [a, b]. Then, we have: Theore 2. The optial solution X = (x 1,..., x T ) of the data center optiization proble (1) satisfies the backward recurrence relation x =, > T ; ( x )x U +1 x L, T. Theore 2 and Lea 1 are proven in Appendix A. An exaple of the optial x t can be seen in Figure 1(a). Many ore nueric exaples of the perforance of the optial offline algorith are provided in Section VI. Theore 2 and Figure 1(a) highlight that the optial algorith can be interpreted as oving backwards in tie, starting with x T = and keeping x = x +1 unless the bounds prohibit this, in which case it akes the sallest possible change. This interpretation hightlights that it is ipossible for an online algorith to copute x since, without knowledge of the future, an online algorith cannot know whether to keep x constant or to follow the upper/lower bound. V. LAZY CAPACITY PROVISIONING A ajor contribution of this paper is the presentation and analysis of a novel online algorith, Lazy Capacity Provisioning (LCP(w)). At tie, LCP(w) knows only g t ( ) for t + w, for soe prediction window w. As entioned before, we assue that these are known perfectly, but we show in Section VI that the algorith is robust to this assuption in practice. The design of LCP(w) is otivated by the structure of the optial offline solution described in Section IV. Like the optial solution, it lazily stays within upper and lower bounds. However, it does this oving forward in tie instead of backwards in tie. Before defining LCP(w) forally, recall that the bounds x U and x L do not use knowledge about the loads in the prediction window of LCP(w). To use it, define refined bounds x U,w and x L,w such that x U,w = x U +w, in the solution of (6) and x L,w = x L +w, in that of (5). Note that w = is allowed (no future prediction) and x U, = x U and x L, = x L. The following generalization of Lea 1 is proven in Appendix B. (7) 4

5 nuber of servers nuber of servers bounds tie t (hours) (a) Offline optial LCP() bounds tie t (hours) (b) LCP() load load tie (hours) (a) Hotail tie (hours) (b) MSR Fig. 1. Illustrations of (a) the offline optial solution and (b) LCP() for the first day of the MSR workload described in Section VI with a sapling period of 1 inutes. The operating cost is defined in (2) with d = 1.5, d 1 = 1, µ = 1, p t = 1, e = 1 and e 1 = and the switching cost has β = 6 (corresponding to the energy consuption for one hour). Lea 2. x L x L,w x x U,w x U for all w. Now, we are ready to define LCP(w) using x U,w x = and x L,w. Algorith 2. Lazy Capacity Provisioning, LCP(w). Let X = (x,..., xt ) denote the vector of active servers under LCP(w). This vector can be calculated using the following forward recurrence relation, ; ( x 1 )x U,w x L,w, 1. Figure 1(b) illustrates the behavior of LCP(). Note its siilarity with Figure 1(a), but with the laziness in forward tie instead of reverse tie. The coputational deands of LCP(w) ay initially see prohibitive as grows, since calculating x U,w and x L,w requires solving convex optiizations of size + w. However, it is possible to calculate x U,w and x L,w without using the full history. Lea 1 in Appendix B iplies that it is enough to use only the history since the ost recent point when the solutions of (5) and (6) are either both increasing or both decreasing, if such a point exists. In practice, this period is typically less than a day due to diurnal traffic patterns, and so the convex optiization, and hence LCP(w), reains tractable even as grows. In Figure 1(b), each point is solved in under half a second. Next, consider the cost incurred by LCP(w). Section VI discusses the cost in realistic settings, while in this section we focus on worst-case bounds, i.e., characterize the copetitive ratio. The following theore is proven in Appendix B. Theore 3. cost(x ) cost(x ) + 2cost s (X ). Thus, LCP(w) is 3-copetitive for optiization (1). Further, (8) Fig. 2. Illustration of the traces used for nuerical experients. for any finite w and ɛ > there exists an instance, with g t of the for f(λ t / ), such that LCP(w) attains a cost greater than 3 ɛ ties the optial cost. Note that Theore 3 says that the copetitive ratio is independent of the prediction window size w, the switching cost β, and is uniforly bounded above regardless of the for of the operating cost functions g t ( ). Surprisingly, this eans that even the yopic LCP() is 3-copetitive, regardless of {g t ( )}, despite having no inforation beyond the current tieslot. It is also surprising that the copetitive ratio is tight regardless of w. Seeingly, for large w, LCP(w) should provide reduced costs. Indeed, for any particular workload, as w grows the cost decreases and eventually atches the optial, when w > T. However, for any fixed w, there is a worstcase sequence of cost functions such that the copetitive ratio is arbitrarily close to 3. Moreover, there is such a sequence of cost functions having the natural for f(λ t / ), which corresponds to a tie varying load λ being shared equally aong servers, which each has a tie-invariant (though pathological) operating cost f(λ) when serving load λ. Finally, though 3-copetitive ay see like a large gap, the fact that cost(x ) cost(x )+2cost s (X ) highlights that the gap will tend to be uch saller in practice, where the switching costs ake up a sall fraction of the total costs since dynaic right-sizing would tend to toggle servers once a day due to the diurnal traffic. VI. CASE STUDIES In this section the goal is two-fold: The first is to evaluate the cost incurred by LCP(w) relative to the optial solution and RHC(w) for realistic workloads. The second is ore generally to illustrate the cost savings and energy savings that coe fro dynaic right-sizing in data centers. To accoplish these goals, we experient using two real-world traces. 5

11 we need only to consider the case β ν t. Then the dual function becoes D(ν) ( T ) 1 = inf g t ( ) + (ν t ν t+1 ) + ν T x T ν 1 x x T 1 = gt (ν t+1 ν t ) gt ( ν T ) ν 1 x. where g t ( ) is the convex conjugate of function g t ( ) (enforcing g t ( ) = when < ). Therefore the corresponding dual proble of (9) is 1 ax gt (ν t+1 ν t ) gt ( ν T ) ν 1 x (1) subject to ν t β, where the copleentary slackness conditions are ν t ( 1 y t ) = (11) (β ν t )y t =, (12) and the feasibility condition is, y t, y t 1 and ν t β. Using the above, we now observe a relationship between the data center optiization in (9) and the upper and lower bounds, i.e., optiizations (6) and (5). Specifically, if ν +1 = in a solution of optiization (1), then ν 1,..., ν is a solution to: 1 ax gt (ν t+1 ν t ) g ( ν ) ν 1 x (13) subject to ν t β, which is identical to the dual proble of optiization (5). Thus, the corresponding x 1,..., x is a solution to optiization (5). On the other hand, if ν +1 = β in a solution of optiization (1), then ν 1,..., ν is a solution to: 1 ax gt (ν t+1 ν t ) g (β ν ) ν 1 x (14) subject to ν t β. Let ν t = β ν t, which akes (14) becoe 1 ax gt (ν t ν t+1) g (ν ) + ν 1x βx subject to ν t β. (15) It is easy to check that the corresponding x 1,..., x is a solution to optiization (6). We require soe notation and two technical leas before oving to the proofs of Lea 1 and Theore 2. Denote h i,j (x; x S ; x E ) = j t=i g t( )+β(x i x S ) + +β j t=i+1 ( 1 ) + + β(x E x j ) + for j i, and h i,j (x; x S; x E ) = j t=i g t( )+β(x S x i ) + +β j t=i+1 ( 1 ) + +β(x j x E ) + for j i. Then the objective of (5) is h 1, (x; x ; ). Siilarly, the objective of (6) is h 1, (x; x ; x M ) where x M = ax(ax,t The first lea is to connect h i,j and h i,j : x U,t, ax x t ). (16) t Lea 3. Given x S and x E, any solution iniizing h i,j (x; x S ; x E ) also iniizes h i,j (x; x S; x E ) and vice versa. Proof: h i,j (x; x S ; x E ) h i,j(x; x S ; x E ) j =β(x i x S ) + β ( 1 ) + β(x E x j ) =β(x E x S ), t=i+1 which is a constant. Thus any iniizer of h i,j (x; x S ; x E ) or h i,j (x; x S; x E ) is also a iniizer of the other. Lea 4. Given x S and x E ˆx E, let X = (x i,..., x j ) iniize h i,j (x; x S ; x E ), then there exists an ˆX = (ˆx i,..., ˆx j ) iniizing h i,j (x; x S ; ˆx E ) such that X ˆX. Proof: If there is ore than one solution iniizing h i,j (x; x S ; ˆx E ), let ˆX be a solution with greatest ˆxj. We will first argue that x j ˆx j. By definition, we have h i,j (X; x S ; x E ) h i,j ( ˆX; x S ; x E ) and h i,j ( ˆX; x S ; ˆx E ) h i,j (X; x S ; ˆx E ). Note that if the second inequality is an equality, then x j ˆx j. Otherwise, su the two inequalities, to obtain the strict inequality (x E x j ) + + (ˆx E ˆx j ) + < (ˆx E x j ) + + (x E ˆx j ) +. Since x E ˆx E, we can conclude that x j < ˆx j. Therefore, we always have x j ˆx j. Next, recursively consider the subproble h i,j 1 ( ) with x E = x j and ˆx E = ˆx j. The sae arguent as above yields that x j 1 ˆx j 1. This continues and we can conclude that (x i,..., x j ) (ˆx i,..., ˆx j ). Lea 4 shows that the optiizations in our paper have a unique axial solution, as follows. Corollary 1. If there is ore than one solution to optiization proble (1), (4), (5) or (6), there exists a axiu solution which is not less than others eleentwise. Proof: Let us consider proble (1) with ultiple solutions and prove the clai by induction. Assue that x is a solution with the first ( 1) entries not less than those in other solutions, but x +1 < ˆx +1 where ˆx is a solution with the greatest ( +1)th entry. Since (x 1,..., x ) iniizes h 1, (x; x ; x +1), Lea 4 iplies there exists a solution ( x 1,..., x ) iniizing h 1, (x; x ; ˆx +1) which is not less than (x 1,..., x ). Thus we can replace (ˆx 1,..., ˆx ) in ˆx by ( x 1,..., x ) to get a solution with the first + 1 entries not less than other solutions. The proof for optiization proble (4), (5) and (6) are siilar and thus oitted. We now coplete the proofs of Lea 1 and Theore 2. Proof of Lea 1: Let X L = (x L,1, x L,2,..., x L, ) be the solution of optiization (5) at tie and define X U syetrically for (6). Also, let X = (x 1,..., x ) be the 11

12 first entries of the offline solution to optiization (1). We know that X L iniizes h 1, (x; x ; ) and X iniizes h 1, (x; x ; x +1). Since x +1, Lea 4 iplies X X L, and thus, in particular, the last entry satisfies x x L,. Syetrically, X U iniizes h 1, (x; x ; x M ) where x M satisfies (16). By Lea 3, X U also iniizes h 1, (x; x ; x M ). Since x +1 x M, Lea 4 iplies that X X U, and thus x x U,. Proof of Theore 2: As a result of Lea 1, we know that x [x L, x U ] for all. Further, if x > x +1, by the copleentary slackness condition (11), we have that ν +1 =. Thus, in this case, x solves optiization (5) for the lower bound, i.e., x = x L. Syetrically, if x < x +1, we have that copleentary slackness condition (12) gives ν +1 = β and so x solves optiization (6) for the upper bound, i.e., x = x U. Thus, whenever x is increasing/decreasing it ust atch the upper/lower bound, respectively. APPENDIX B ANALYSIS OF LAZY CAPACITY PROVISIONING, LCP(w) In this section we will prove Lea 2 and Theore 3. Proof of Lea 2: First, we prove that x L,w x. By definition, x L,w = x L +w,, and so it belongs to a solution iniizing h 1,+w (x; x ; ). Further, we can view the optial x as an entry in a solution iniizing h 1,+w (x; x ; x +w+1). Fro these two representations, we can apply Lea 4, to conclude that x L,w x. Next, we prove that x L x L,w. To see this we notice that x L is the last entry in a solution iniizing h 1, (x; x ; ). And we can view x L,w as an entry in a solution iniizing h 1, (x; x ; x L +w,+1) where x L +w,+1. Based on Lea 4 we get x L x L,w. The proof that x x U,w x U is syetric. Fro the above lea, we iediately obtain an extension of the characterization of the offline optiu. Corollary 2. The optial solution of the data center optiization (1) satisfies the following backwards recurrence relation {, > T ; x = (17), T. (x +1) xu,w x L,w Moving to the proof of Theore 3, the first step is to use the above leas to characterize the relationship between x and x. Note that x = x = xt +1 = x T +1 =. Lea 5. Consider the tieslots = t < t 1 <... < t = T such that xt i = x t i. Then, during each segent (t i 1, t i ), either (i) xt > x t and both xt and x t are nonincreasing for all t (t i 1, t i ), or (ii) xt < x t and both xt and x t are nondecreasing for all t (t i 1, t i ). Proof: The result follows fro the characterization of the offline optial solution in Corollary 2 and the definition of LCP(w). Given that both the offline optial solution and LCP(w) are non-constant only for tieslots when they are equal to either x U,w t or x L,w t, we know that at any tie t i where xt i = x t i and xt i+1 x t i+1, we ust have that both xt i and x t i are equal to either x U,w t i or x L,w t i. Now we ust consider two cases. First, consider the case that xt i+1 > x t i+1. It is easy to see that xt i+1 doesn t atch the lower bound since x t i+1 is not less than the lower bound. Thus xt i xt i+1 since, by definition, LCP(w) will never choose to increase the nuber of servers it uses unless it atches the lower bound. Consequently, it ust be that x t i = xt i xt i+1 > x t i+1. Since x is decreasing, both xt i and x t i atch the lower bound. Further, the next tie, t i+1, when the optial solution and LCP(w) atch is the next tie either the nuber of servers in LCP(w) atches the lower bound x L,w t or the next tie the nuber of servers in the optial solution atches the upper bound x U,w t. Thus, until that point, LCP(w) cannot increase the nuber of servers (since this happens only when it atches the lower bound) and the optial solution cannot increase the nuber of servers (since this happens only when it atches the upper bound). This copletes the proof of part (i) of the Lea. The proof of part (ii) is syetric. Given Lea 5, we bound the switching cost of LCP(w). Lea 6. cost s (X ) = cost s (X ). Proof: Consider the sequence of ties = t < t 1 <... < t = T such that xt i = x t i identified in Lea 5. Then, each segent (t i 1, t i ) starts and ends with the sae nuber of servers being used under both LCP(w) and the optial solution. Additionally, the nuber of servers is onotone for both LCP(w) and the optial solution, thus the switching cost incurred by LCP(w) and the optial solution during each segent is the sae. Next, we bound the operating cost of LCP(w). Lea 7. cost o (X ) cost o (X )+β T x t x t 1. Proof: Consider the sequence of ties = t < t 1 <... < t = T such that xt i = x t i identified in Lea 5, and consider specifically one of these intervals (t i 1, t i ) such that xt i 1 = x t i 1, xt i = x t i. There are two cases in the proof: (i) xt > x for all (t i 1, t i ) and (ii) xt < x for all t (t i 1, t i ). We handle case (i) first. Our goal is to prove that t i t=t i 1 +1 g t(x t ) t i t=t i 1 +1 g t(x t ) + β x t i 1 x t i. (18) Define X = (x,1,..., x, 1, x ) where (x,1,..., x, 1 ) iniizes h 1, 1(x; x ; x ). Additionally, define X = (x,1,..., x, ) as the solution iniizing h 1, (x; x ; x ); by Lea 3 this also iniizes h 1, (x; x ; x ). We first argue that x, = x via a proof by contradiction. Note that if x, > x, based on siilar arguent in the proof of Theore 2, we have x, = x L, which contradicts the fact that x, > x x L. Second, 12

13 if x, < x, then we can follow a syetric arguent to arrive at a contradiction. Thus x, = x. Consequently, x,t = x,t for all t [1, ] and we get h 1, (X ; x ; x M ) = h 1, (X ; x ; x M ) (19) Next, let us consider X +1 = (x +1,1,..., x +1,, x+1 ) where (x +1,1,..., x +1, ) iniizes h 1, (x; x ; x+1 ). Recalling xt is non-increasing in case (i) by Lea 5, we have X +1 X by Lea 4. In particular, x +1, x. Thus h 1,+1(X +1 ; x ; x M ) (2) h 1, ((x +1,1,..., x +1, ); x ; x M ) + g +1 (x+1 ) =h 1, ((x +1,1,..., x +1, ); x ; x By definition of X, we get ) + g +1 (x +1 ) h 1, ((x +1,1,..., x +1, ); x ; x ) (21) h 1, (X ; x ; x ) h 1, (X ; x ; x M ) Cobining equations (19), (2) and (21), we obtain h 1,+1(X +1 ; x ; x M ) h 1, (X ; x ; x M )+g +1 (x+1 ). By suing this equality for [t i 1, t i ), we have t i t=t i 1+1 g t (xt ) h 1,t i (X ti ; x ; x M ) h 1,t i 1 (X ti 1 ; x ; x M ). Since xt i 1 = x t i 1, xt i = x t i, we know that both X ti 1 and X ti are prefixes 2 of the offline solution x, thus X ti 1 is the prefix of X ti. Expanding out h ( ) in the above inequality gives (18), which copletes case (i). In case (ii), i.e., segents where xt < x for all t (t i 1, t i ), a parallel arguent shows that (18) again holds. To coplete the proof we cobine the results fro case (i) and case (ii), suing equation (18) over all segents (and the additional ties when xt = x t ). We can now prove the copetitive ratio in Theore 3. Lea 8. cost(x ) cost(x ) + 2cost s (X ). Thus, LCP(w) is 3-copetitive for the data center optiization (1). Proof: Cobining Lea 7 and Lea 6 gives that cost(x LP C(w) ) cost(x ) + β x t x t 1. Note that, because both LCP(w) and the optial solution start and end with zero servers on, we have T x t x t 1 = 2 T (x t x t 1) +, which copletes the proof. All that reains for the proof of Theore 3 is to prove that the bound of 3 on the copetitive ratio is tight. Lea 9. The copetitive ratio of LCP(w) is at least 3. Proof: The following is a faily of instances paraeterized by n and 2, whose copetitive ratios approach the bound of 3. The operating cost for each server is defined as f(z) = z + f for z 1 and f(z) = otherwise, 2 That is, X ti 1 is the first t i 1 coponents of x and X ti is the first t i coponents of x. whence g t ( ) = f(λ t / ). The switching cost is β =.5. 1 Let δ (1, 1.5) be such that n = log δ δ 1. The arrival rate at tie i is λ i = δ i 1 for 1 i n, and λ i = for n < i T, where T > β/f + n with f = β(δ 1) n(δ n 1). For the offline optiization, denote the solution by vector x. First note that x i = x n for i [1, n] since x i is nondecreasing for i [1, n] and, for the above f, d dx [xf(λ i/x)] < for x [λ i, x n]. (22) Moreover, x i = for i [n + 1, T ]. Hence the iniu cost is n λ i i=1 (x + (nf + β)x n. Then by the first order n (stationarity) condition ) 1 we get (x n) = (nf + β)(δ 1) ( 1)(δ n 1), (23) which is saller than λ n because nf + β 1, 1 1 and δ > 1. Thus it is feasible (f(λ i /x n) is finite). The cost for the offline optial solution is then C = 1 (nf + β)x n. We consider with w = before considering the general case. Let C [i,j] denote the cost of LCP() on [i, j]. For the online algorith LCP(), denote the result by vector ˆx. We know ˆx i is actually atching x L i in [1, n] (x U i is not less than x i = x n), thus ˆx i is non-decreasing for i [1, n] and ˆx n = x n. By the sae arguent as for x n, we have =1 (ˆx ) = (f + β)(δ 1) ( 1)(δ 1) Thus the cost for LCP() in [1, n] is n ( ) λ C [1,n] = (ˆx ) 1 + f ˆx + βx n > > = Thus by (23) C [1,n] x n ) 1 n ( δ ( 1) (f + β)(δ 1) ( 1)(δ 1) =1 ( β(δ ) 1 1) n δ + βx n 1 =1 ( β(δ ) 1 1) δ n 1 1 (δ 1)δ 1 + βx n (24) + βx n δ n 1 > (δ n 1) β(δ 1) 1/ ( 1)(δ 1)δ 1 + β > δn 1 β(δ 1) δ n ( 1)(δ 1)δ 1 + β As n, both δ 1 and (δ n 1)/δ n 1, since n =. Since is independent of δ, L Hospital s Law gives log δ 1 δ 1 C [1,n] li n x li n δ 1 βδ 1 ( 1)(δ 1 ( 1)δ 2 ) + β = 1 β + β Now let us calculate the cost for LCP() in [n + 1, T ]. For > n, by LCP(), we know that x will stay constant 13

14 until it hits the upper bound. Let X = {x,t } be the solution of optiization (6) in [1, ] for any > n. We now prove that for such that ( n)f < β, we have x, x n, and thus ˆx = x n. Note that x,n x n since x n belongs to the lower bound. Given x,n, we know that x,n+1,..., x, is the solution to the following proble: iniize f X,t + β (X,t 1 X,t ) + t=n+1 subject to X,t t=n+1 Let x in = in{x,n,..., x, }. Then f x,t + β (x,t 1 x,t ) + t=n+1 t=n+1 t=n+1 f x in + β(x,n x in ) ( n)f x,n The last inequality is obtained by substituting β > ( n)f. We can see that x,i = x,n (i [n + 1, ]) is a solution, thus x, x n. (Recall that, if there are ultiple solutions, we take the axiu one). Therefore, we have C [n+1,] = ( n)f x n. Since f as δ 1, we can find an so that ( n)f β, and thus C [n+1,] βx n. (25) By cobining the cost for LCP() in [1, n] and [n + 1, T ], we have C [1,T ] /C 1 β + 2β 1 (nf + β) = 3 2/ 1 + nf /β. Using the relationship between n, f and δ, we can choose a large enough and n to ake this arbitrarily close to 3. This finishes the proof for LCP(). Now let us consider LCP(w) for w >. Denote the solution of LCP(w) by ˆx. At tie [1, n w], LCP(w) solves the sae optiization proble as LCP() does at + w. Thus ˆx = ˆx +w of LCP(). Thus C [1,n w] = = ( λ (ˆx ) 1 + f ˆx n ( 1 δ w n w =1 =1+w ) + βx n λ (ˆx ) 1 + f ˆx ) + βx n 1 > δ w C [1+w,n] By pushing n, and hence δ 1, we have C [1,n w] /C [1,n] 1. And for > n + 1, if f ( n) < β, then C [n+1, w] = ( w n)f x n. By pushing δ 1, we can find a such that C [n+1, w] βx n, the sae as (25). Therefore, as δ 1, we have C [1,T ] /C [1,T ] 1, thus the supreu over arbitrarily large 2 and arbitrarily sall δ > 1 of the copetitive ratio of LCP(w) on this faily is also 3. Finally, the following lea ensures that the optiizations solved by LCP(w) at each tieslot reain sall. Lea 1. If there exists an index [1, t 1] such that x U t,+1 < x U t, or x L t,+1 > x L t,, then (x U t,1,..., x U t, ) = (x L t,1,..., x L t, ). No atter what the future functions g i ( ) are, solving either (5) or (6) in [1, t ] for t > t is equivalent to solving two optiizations: one over [1, ] with initial condition x and final condition x U t, and the second over [ +1, t ] with initial condition x U t,. Proof: Consider the case x L t,+1 > x L t,. By the copleentary slackness conditions (12), the corresponding dual variable ν +1 = β. Based on the arguent in the proof of Theore 2, we know the dual proble up to tie is (14), which is identical to (15), the dual proble of (6). Since the optial x i depends only on ν i and ν i+1, the optial x i for i are copletely deterined by ν i for i [1, + 1], with ν +1 = β. Hence (x L t,1,..., x L t, ) = (x U t,1,..., x U t, ). The proof for the case x U t,+1 < x U t, is syetric. Notice that (x U t,1,..., x U t,t) iniizes h 1,t(x; x ; x M ) and thus h 1,t (x; x ; x M ) based on Lea 3, (x L t,1,..., x L t,t) iniizes h 1,t (x; x ; ). No atter what the future functions g i ( ) are, the first t entries of its solution ust iniize h 1,t (x; x ; +1 ) for soe +1 [, x M ] by the axiality of x M. Based on Lea 4, the first t entries are bounded by (x U t,1,..., x U t,t) and (x L t,1,..., x L t,t). However, we have seen that (x U t,1,..., x U t, ) = (x L t,1,..., x L t, ), thus the first entries of its solution are equal to (x U t,1,..., x U t, ) no atter what the future is. Minghong Lin received the B.Sc. degree in Coputer Science fro University of Science and Technology of China in 26, and the M.Phil degree in Coputer Science fro the Chinese University of Hong Kong in 28. Currently he is a Ph.D. student in Coputer Science at California Institute of Technology. His research interests include energy efficient coputing, online algoriths and nonlinear optiization. He has received best paper award at IEEE INFOCOM 11, IGCC 12 and best student paper award at ACM GREENMETRICS 11. Ada Wieran is an Assistant Professor in the Departent of Coputing and Matheatical Sciences at the California Institute of Technology, where he is a eber of the Rigorous Systes Research Group (RSRG). He received his Ph.D., M.Sc. and B.Sc. in Coputer Science fro Carnegie Mellon University in 27, 24, and 21, respectively. He received the ACM SIGMETRICS Rising Star award in 211, and has also received best paper awards at ACM SIGMETRICS, IFIP Perforance, IEEE INFOCOM, and ACM GREENMETRICS. He has also received ultiple teaching awards, including the Associated Students of the California Institute of Technology (ASCIT) Teaching Award. His research interests center around resource allocation and scheduling decisions in coputer systes and services. More specifically, his work focuses both on developing analytic techniques in stochastic odeling, queueing theory, scheduling theory, and gae theory, and applying these techniques to application doains such as energy-efficient coputing, data centers, social networks, and the electricity grid. 14

15 Lachlan Andrew (M 97-SM 5) received the B.Sc., B.E. and Ph.D. degrees in 1992, 1993, and 1997, fro the University of Melbourne, Australia. Since 28, he has been an associate professor at Swinburne University of Technology, Australia, and since 21 he has been an ARC Future Fellow. Fro 25 to 28, he was a senior research engineer in the Departent of Coputer Science at Caltech. Prior to that, he was a senior research fellow at the University of Melbourne and a lecturer at RMIT, Australia. His research interests include energy-efficient networking and perforance analysis of resource allocation algoriths. He was corecipient of the best paper award at IGCC212, IEEE INFOCOM 211 and IEEE MASS 27. He is a eber of the ACM. Eno Thereska received his PhD (ECE, 27), Masters (ECE, 23) and Bachelor (ECE + CS + Math inor, 22) degrees all fro Carnegie Mellon. He s been a Researcher with Microsoft Research since 27. He has broad interests in systes but also enjoys venturing into other areas occasionally, like achine learning and HCI. He was co-recipient of the best paper award at IEEE INFOCOM 211 and Usenix FAST 25, and the best student paper at Usenix FAST

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