Speed Scaling with an Arbitrary Power Function

Transcription

1 Spee Scaling with an Arbitrary Power Function Nikhil Bansal Ho-Leung Chan Kirk Pruhs What matters most to the computer esigners at Google is not spee, but power, low power, because ata centers can consume as much electricity as a city. Dr. Eric Schmit, CEO of Google [12]. Abstract All of the theoretical spee scaling research to ate has assume that the power function, which expresses the power consumption P as a function of the processor spee s, is of the form P = s α, where α > 1 is some constant. Motivate in part by technological avances, we initiate a stuy of spee scaling with arbitrary power functions. We consier the problem of minimizing the total flow plus energy. Our main result is a (3+ǫ)-competitive algorithm for this problem, that hols for essentially any power function. We also give a (2+ǫ)-competitive algorithm for the objective of fractional weighte flow plus energy. Even for power functions of the form s α, it was not previously known how to obtain competitiveness inepenent of α for these problems. We also introuce a moel of allowable spees that generalizes all known moels in the literature. 1 Introuction Energy consumption has become a key issue in the esign of microprocessors. Major chip manufacturers, such as Intel, AMD an IBM, now prouce chips with ynamically scalable spees, an prouce associate software that enables an operating system to manage power by scaling processor spee. Within the last few years there has been a significant amount of research on the scheuling problems that arise in this setting. Generally these problems have ual objectives as one wants both to optimize some scheule quality of service objective (for example, total flow) an some power relate objective (for example, the total energy use). IBM T.J. Watson Research, P.O. Box 218, Yorktown Heights, NY. nikhil@us.ibm.com Max-Planck-Institut für Informatik. hlchan@mpi-inf.mpg.e. This paper was one when the author was in University of Pittsburgh. Computer Science Department, University of Pittsburgh. kirk@cs.pitt.eu. Supporte in part by an IBM faculty awar, an by NSF grants CNS , CCF-51458, IIS , an CCF Scheuling algorithms for these problems have two components: A job selection policy that etermines which job to run an a spee scaling policy to etermine the spee at which the processor is run. All of the theoretical spee scaling research to ate has assume that the power function, which expresses the power consumption P as a function of the processor spee s, is of the form P = s α, where α > 1 is some constant. Let us call this the traitional moel. The traitional moel was motive by the fact that in CMOS base processors, the well known cube-root rule states that the spee is approximately the cube root of the power. So historically P = s 3 was a reasonable assumption. In the literature one fins ifferent variations on this traitional moel base on which spees are allowable. Most of the literature assumes the unboune spee moel, in which a processor can be run at any real spee in the range [, ). Some of the literature assumes the boune spee moel in which the allowable spees lie in some real interval [,T]. Some of the literature on offline algorithms, assumes the iscrete spees moel in which there are a finite number of allowable spees. Our main contribution in this paper is to initiate theoretical investigations into spee scaling problems with more general power functions, an evelop algorithmic analysis techniques for this setting. For an explanation of the historical technological motivation for the traitional moel, an the current technological motivations for consiering more general power functions, see section 1.3. A seconary contribution is to introuce a moel for allowable spees that generalizes all of various moels foun in the literature. We will consier the objective of minimizing a linear combination of total (possibly weighte) flow an total energy use. Our thir contribution is to improve on the known results for this important funamental problem. Optimizing a linear combination of energy an total flow has the following natural interpretation. Suppose that the user specifies how much improvement in flow, call this amount ρ, is necessary to justify spening one unit of energy. For example, the user might specify that he is willing to spen 1 erg of energy from the battery for a ecrease of 4 micro-secons in flow. Then the optimal scheule, from this user s perspective, is the scheule 693 Copyright by SIAM. Unauthorize reprouction of this article is prohibite.

2 that optimizes ρ = 4 times the energy use plus the total flow. By changing the units of either energy or time, one may assume without loss of generality that ρ = 1. Weighte flow generalizes both total flow, an total/average stretch, which is another common QoS measure. The stretch/slowown of a job is the flow ivie by the work of the job. When the user is aware of the size of a job (say if the user knows that he/she is ownloaing a vieo file instea of a text file) then perhaps slowown is a more appropriate measure of the happiness of a user than flow. Many server systems, such as operating systems an atabases, have mechanisms that allow the user or the system to give ifferent priorities to ifferent jobs. For example, Unix has the nice comman. In a spee scaling setting, the weight of a job is inicative of the flow versus energy trae-off for this job. The user may be willing to spen more energy to reuce the flow of a higher priority job, than for a lower priority job. 1.1 The Literature on Flow + Energy in the Traitional Moel Let us start with results in the unboune spee moel. Pruhs, Uthaisombut an Woeginger [15] gave an efficient offline algorithm to fin the scheule that minimizes average flow subject to a constraint on the amount of energy use, in the case that jobs have unit work. This algorithm can also be use to fin optimal scheules when the objective is a linear combination of total flow an energy use. They observe that in any locally-optimal scheule, essentially each job i is run at a power proportional to the number of jobs that woul be elaye if job i was elaye. Albers an Fujiwara [1] propose the natural online spee scaling algorithm that always runs at a power equal to the number of unfinishe jobs (which is lower boun to the number of jobs that woul be elaye if the selecte job was elaye). They i not actually analyze this natural algorithm, but rather analyze a batche variation, in which jobs that are release while the current batch is running are ignore until the current batch finishes. They showe that (( for unit 3+ work jobs that this batche algorithm is O ) α ) competitive by reasoning irectly about the optimal scheule. This gave a competitive ratio of about 4 when the cube-root rule hols. They also gave an efficient offline ynamic programming algorithm. Bansal, Pruhs an Stein [4] consiere the algorithm that runs at a power equal to the unfinishe work (which is in general a bit less than the number of unfinishe jobs for unit work jobs). They showe that for unit work jobs, this algorithm is 2-competitive with respect to the objective of fractional flow plus energy using an amortize local competitiveness argument. A job that is say 2/3 complete at time t, only contributes 1/3 to the increase in fractional flow at time t. They then showe that the natural algorithm propose in [1] is 4-competitive for total flow plus energy for unit work jobs. For the more general setting where jobs have arbitrary sizes an arbitrary weights an the objective is weighte flow plus energy, Bansal, Pruhs an Stein [4] consiere the algorithm that uses Highest Density First (HDF) for job selection, an always runs at a power equal to the fractional weight of the unfinishe jobs. They showe that this algorithm is O( α log α )-competitive for fractional weighte flow plus energy using an amortize local competitiveness argument. Using the known resource augmentation analysis of HDF [5], they then showe how to obtain an O( α2 )-competitive algorithm for (integral) weighte flow plus energy. The com- log 2 α petitive ratio was a bit less than 8 when the cube-root rule hols. Recently, Lam et al. [11] improve the competitive ratio for total flow plus energy for arbitrary work an unit weight jobs. They consiere the job selection algorithm Shortest Remaining Processing Time (SRPT) an the spee scaling algorithm of running at a power proportional to the number of unfinishe jobs, an prove that this is O( α log α )-competitive for flow time plus energy. When the cube-root rule hols, this competitive ratio was about Their improvement came by reasoning irectly about integral flow time instea of arguing first about fractional flow time. Spee scaling papers prior to [11] use potential functions relate to the fractional amount of unfinishe work or weight. The main reason for this was that such a potential function varies continuously with time which substantially simplifies the proofs as one only nees to consier instantaneous states of the online an offline algorithms. The main technical contribution of [11] was the introuction of a ifferent form of continuously varying potential function that epens on the integral number of unfinishe jobs. Bansal et al. [2] extene the results of [4] for the unboune spee moel to the boune spee moel. The spee scaling algorithm was to run at the minimum of the spee recommene by the spee scaling algorithm in the unboune spee moel an the maximum spee of the processor. The contribution there was to evelop the algorithmic analysis techniques necessary to analyze this algorithm. The results for the boune spee moel in [2] were improve in [11], again by reasoning irectly about integral flow. Again [11] showe competitive ratios of O( α log α ). When the cuberoot rule hols, the obtaine competitive ratio was Copyright by SIAM. Unauthorize reprouction of this article is prohibite.

3 1.2 Our Results We assume that the allowable spees are a countable collection of isjoint subintervals of [, ). We assume that all the intervals, except possibly the rightmost interval, are close on both ens. The rightmost interval may be open on the right if the power P(s) approaches infinity as the spee s approaches the rightmost enpoint of that interval. We assume that P is non-negative, an P is continuous an ifferentiable on all but countably many points. We assume that either there is a maximum allowable spee T, or that the limit inferior of P(s)/s as s approaches infinity is not zero (if this conition oesn t hol then, then the optimal spee scaling policy is to run at infinite spee). Let us call this the general moel. We give two main results in the general moel. Theorem 1.1. Consier the scheuling algorithm that uses Shortest Remaining Processing Time (SRPT) for job selection an power equal to one more than the number of unfinishe jobs for spee scaling. In the general moel, this scheuling algorithm is (3 + ǫ)- competitive for the objective of total flow plus energy on arbitrary-work unit-weight jobs. Theorem 1.2. Consier the scheuling algorithm that uses Highest Density First (HDF) for job selection an power equal to the fractional weight of the unfinishe jobs for spee scaling. In the general moel, this scheuling algorithm is (2 + ǫ)-competitive for the objective of fractional weighte flow plus energy on arbitrary-work arbitrary-weight jobs. We establish these results through an amortize local competitiveness argument. As in [2], our potential function is base on the integral number of unfinishe jobs. However, our potential function is quite ifferent an more general than the potential function consiere in [2]. While Theorem 1.1 eals with integral flow, Theorem 1.2 only hols for fractional weighte flow. Obtaining a competitive ratio inepenent of α for the objective of (integral) weighte flow plus energy is rule out since resource augmentation is require to achieve O(1)-competitiveness for the objective of weighte flow on a fixe spee processor [3]. Let us consier what these theorems say in the traitional moel. Theorem 1.1 slightly improves the best known competitive ratios, when the cube root rule hols, in both the unboune an boune spee moels. The potential functions use in all previous papers are specifically tailore towar the function P = s α. Moreover, these potential functions can not be use to show competitive ratios of o( α ln α ) for arbitrary work jobs. So while the competitive ratios were O(1)-competitive for a fixe α, they were not O(1)- competitive for a general power function of the form s α. Our new potential function not only allows us to α break the barrier of ln α, an it allows us to obtain O(1)- competitiveness for general power functions. 1.3 Technological Motivations The power use by a processor can be partitione into ynamic power, the power use by switching when oing computations, an static/leakage power, which is the power lost in absence of any switching. Historically (say up until 5 years ago) the static power use by a processor was negligible compare to the ynamic power. Dynamic power is roughly proportional to sv 2, where V is the voltage. However as the minimum voltage require to rive the microprocessor at a esire spee is approximately proportional to the frequency [7], this leas to the well known cuberoot rule that the spee s is roughly proportional to the cube-root of the power P, or equivalently, P = s 3, the power is proportional to the spee cube [7]. However, currently the static power use by the common processors manufacture by Intel an AMD is now comparable to the ynamic power. The reason for this is that to increase processor spees, transistor sizes an esire transistor switching elays must ecrease. To obtain a smaller switching elay, the threshol voltage for the transistor must be ecrease. Unfortunately, subthreshol leakage current increases exponentially as threshol voltage ecreases [13, 8]. This suggests moeling the spee to power function as something like P(s) = s α + c, where there is some range of spees [s min,s max ] that the processor can run at. Here α 3 comes from the cube-root rule for ynamic power, an c is a constant specifying the static power loss. There is however, goo motivation for consiering more general spee to power functions. We will state three more examples here. Example 1: In general, the higher spee that one can scale a core or a processor, the greater the static power (because the threshol voltage must be less). It has been propose that it might be avantageous to buil a processor consisting of ifferent circuitry for running jobs at ifferent spees spee ranges [6, 8]. Each circuitry woul have ifferent spee ranges, ifferent static powers ue to the ifferent threshol voltages, an ifferent power functions. Thus the spee scaling algorithm might be able to choose among a collection of power functions of the form P i (s) = a i s α + c i, where each P i (s) is applicable over a collection S i = {s i,1,...,s i,ki } of spees. (Note that in this context that we can not scale our units so that the multiplicative constant is always 1). In this setting, the power function of might be of the form: P(s) = min i:s S i a i s α + c i 695 Copyright by SIAM. Unauthorize reprouction of this article is prohibite.

4 Example 2: Subthreshol leakage current/power is not inepenent of temperature. The temperature in turn epens on the spee that the processor is run. Some iscussion of how leakage current/power is relate to temperature can be foun in [8], but it seems that this issue is perhaps not so well unerstoo. But in any case, there appears to be no reason to presume that the resulting power function woul be of the form s α. Example 3: Another motivation for consiering general spee to power functions is at the ata center level. The opening quote inicates the importance of power management at the ata center level. A nice investigation, by Google researchers, into the energy that a ata center coul save from spee scaling can be foun in [9]. Ultimately what a ata center operator might care about is the cost of the energy use, not the actual amount of energy use. These can be quite ifferent because normally the contract that the ata center has with the electrical suppliers can have rastic increases in cost if various power threshols are exceee. So here the relevant function that one woul care about is the spee to cost (say measure in ollars per secon) function, not the spee to power function. There is no reason why these contracts nee to have a cost that rises as s α. 2 Preliminaries An instance consists of n jobs, where job i has a release time r i, an a positive work y i. In some cases each job may have a positive integer weight w i. An online scheuler is not aware of job i until time r i, an, at time r i, learns y i an weight w i. For each time, a scheuler specifies a job to be run an a spee at which the processor is run. We assume that preemption is allowe, that is, a job may be suspene an later restarte from the point of suspension. A job i completes once y i units of work have been performe on i. The spee is the rate at which work is complete; a job with work y run at a constant spee s completes in y s secons. The flow F i of a job i is its completion time minus its release time. The total flow is n i=1 F i. The weighte flow is n i=1 w i F i. Let us quickly review amortize local competitiveness analysis. Consier an objective G. Let G A (t) be the increase in the objective in the scheule for algorithm A at time t. So when G is total flow plus energy, G A (t) is P(s t a) + n t a, where s t a is the spee for A at time t an n t a is the number of unfinishe jobs for A at time t. This is because energy is power integrate over time, an total flow is the number of unfinishe jobs integrate over time. Let OPT be the offline aversary that optimizes G an A be some algorithm. To prove A is c-competitive it suffices to give a potential function Φ(t) such that the following two conitions hol. Bounary conition: Φ is zero before any job is release an Φ is non-negative after all jobs are finishe. General conition: At any time t, (2.1) G A (t) + Φ(t) t c G OPT (t) To prove the general conition, it suffices to show that Φ oes not increase when a job arrives or is complete by A or OPT, an Equation 2.1 is true uring any perio of time without job arrival or completion. By integrating Equation 2.1, we can see that A is c-competitive for H. For more information see [14]. 3 Flow Plus Energy Our goal in this section is to prove Theorem 1.1, that in the general moel SRPT plus the natural spee scaling algorithm is (3 + ǫ)-competitive for the objective of total flow plus energy. We start by showing that this algorithm is 3-competitive if P also satisfies the following aitional special conitions: P is efine, an continuous an ifferentiable at all spees in [, ). P() =. P is strictly increasing. P is strictly convex. That is, for a < b an for all p (,1), is the case that pp(a) + (1 p)p(b) > P(pa + (1 p)b). P is unboune. That is, for all c, the exists a spee s such that P(s) > c. Then in section 3.1 we show how to exten the analysis to the case when these special conitions o not hol, at the cost of an arbitrarily small increase in the competitive ratio. Definition of online algorithm A: Algorithm A scheules the unfinishe job with the least remaining unfinishe work, an runs at spee s t a where { s t P a = 1 (n t a + 1) if n t a 1 if n t a = where n t a is the number of unfinishe jobs for A at time t. As P() =, an P is strictly increasing, continuous an unboune, P 1 (i) exists an is unique for all integers i 1. Thus, A is well efine. Definition of potential function Φ: Let OPT be the offline aversary that minimizes total flow plus energy. We can assume without loss of generality that 696 Copyright by SIAM. Unauthorize reprouction of this article is prohibite.

5 OPT runs SRPT. At any time t, let n t o be the number of unfinishe jobs in OPT. Let n t a(q) an n t o(q) be the number of unfinishe jobs with remaining size at least q in A an OPT, respectively, at time t. Let n t (q) = max{,n t a(q) n t o(q)}. We efine the the potential function Φ(t) = 3 where f is efine by i 1, f(i) f(i 1) f(n t (q))q f() = = P (P 1 (i)) an P (x) is the erivative of P(x). Note that P (P 1 (i)) simply means substituting x = P 1 (i) into P (x). Since P(x) is ifferentiable, P (x) is well-efine. Hence, f is well-efine for all integer i. As both P (x) an P 1 (i) are increasing, we have f(i) f(i 1) f(j) f(j 1) if i > j. We enote (i) = f(i) f(i 1) for simplicity. We now wish to establish a crucial technical lemma, Lemma 3.1, which relies on the well known Young s inequality. Theorem 3.1. (Young s Inequality[1]) Let g be a real-value, continuous, an strictly increasing function on [,c] with c >. If g(), an a,b such that a [,c], an b [g(),g(c)], then a g(x)x + b g() g 1 (x)x ab where g 1 is the inverse function of g. Lemma 3.1. Let P be a strictly increasing, strictly convex, continuous an ifferentiable function. Let i,s a,s o be any real. Then P (P 1 (i))( s a + s o ) ( s a + P 1 (i))p (P 1 (i)) + P(s o ) i Proof. Since P is strictly increasing an strictly convex, P () an P (x) is strictly increasing. Thus, by Young s Inequality with g(x) = P (x), a = s o an b = P (P 1 (i)), we have s o P (P 1 (i)) so g(x)x + P (P 1 (i)) P () = P(s o ) + [ xg 1 (x) ] g(p 1 (i)) g() = P(s o ) + g(p 1 (w))p 1 (i) = P(s o ) + g(p 1 (i))p 1 (i) i g 1 (x)x g(p 1 (i)) g() P 1 (i) Case n o < n a : We show that either Φ 3 (n a n o )( s a + s o )t or Φ 3 (n a n o + 1)( s a + s o )t, as follows. Consier 3 subcases epening on whether x(g 1 (x)) q a is smaller than, equal to, or bigger than q o. g(y)y The first equality is obtaine by integration by parts an the secon equality is obtaine by letting y = g 1 (x). The lemma follows by aing s a P (P 1 (i)) to both sies. We are now reay to procee with our amortize local competitiveness argument. For the bounary conition, we observe that before any job is release an after all jobs are finishe, n t (q) = for all q, so Φ =. For the general conition, we observe that when a job is release, n t (q) is not change for all q, so Φ is unchange. When a job is complete by A or OPT, n t (q) is change only at the single point of q =, which oes not affect the integration, so Φ is also unchange. It remains to show at any time t uring a time interval without job arrival or completion, (3.2) n t a + P(s t a) + t Φ(t) 3(nt o + P(s t o)) The rest of this proof consiers any such time t an proves Equation 3.2. We omit t from the superscript an the parameter for convenience. First observe that if n a =, then n(q) = for all q, an tφ =. Thus equation 3.2 trivially hols. Henceforth, we assume n a 1. Let q a be the remaining size of the job with the shortest remaining size in A (q a exists as n a 1). Note that A is processing a job of size q a. Define q o similarly for OPT (q o = if n o = ). Consier an infinitesimal interval of time [t,t + t]. The processing of A causes n a (q) to ecrease by 1 for q [q a s a t,q a ]. Similarly, the processing of OPT causes n o (q) to ecrease by 1 for q [q o s o t,q o ]. Denote the change in Φ uring this time interval as Φ. We consier three cases epening on whether n o > n a, n o < n a, or n o = n a. Case n o > n a : We show that Φ, as follows. At time t, n o (q) is greater than n a (q) for q in [q o s o t,q o ]. Thus, at time t + t, n o (q) is still at least n a (q) for q in [q o s o t,q o ]. It means that n(q) remains zero in this interval an Φ is not increase ue to the processing of OPT. The processing of A only ecreases Φ or leaves Φ unchange. Hence, Φ. Thus implies that t Φ an n a + P(s a ) + t Φ n a + n a + 1 3n o, so (3.2) hols. 1. Assume q a < q o. As n a (q) ecreases by 1 for q in [q a s a t,q a ], by the efinition of Φ, the change in Φ ue to A is 3(f(n a (q a ) n o (q a ) 1) f(n a (q a ) n o (q a )))s a t = 3 (n a (q a ) n o (q a ))s a t. Since 697 Copyright by SIAM. Unauthorize reprouction of this article is prohibite.

6 q a < q o, we have n o (q a ) = n o (q o ) = n o. So the change in Φ ue to A is 3 (n a n o )s a t. As n o (q) ecreases by 1 for q in [q o s o t,q o ], by the efinition of Φ, the change in Φ ue to OPT is at most 3(f(n a (q o ) n o (q o ) + 1) f(n a (q o ) n o (q o )))s o t = 3 (n a (q o ) n o (q o ) + 1)s o t. Since q a < q o, we have n a (q o ) n a (q a ) 1. Thus, n a (q o ) n o (q o ) + 1 n a (q a ) n o (q o ). As (i) (j) for i j, the change in Φ ue to OPT is at most 3 (n a (q a ) n o (q o ))s o t = 3 (n a n o )s o t. Aing up the change in Φ ue to A an OPT, we obtain that Φ 3 (n a n o )( s a + s o )t. 2. Assume q a = q o. If s a s o, n(q) = n a (q) n o (q) ecreases by 1 for q in [q a s a t,q a s o t]. Hence, the change in Φ is 3(f(n a (q a ) n o (q a ) 1) f(n a (q a ) n o (q a )))(s a s o )t = 3 (n a n o )( s a + s o )t. Else if s a < s o, n(q) = n a (q) n o (q) increases by 1 for q in [q a s o t,q a s a t]. Hence, the change in Φ is 3(f(n a (q a ) n o (q a )+1) f(n a (q a ) n o (q a )))(s o s a )t = 3 (n a n +1)( s a +s o )t. 3. Assume q a > q o. As n a (q) ecreases by 1 for q in [q a s a t,q a ], the change in Φ ue to A is 3(f(n a (q a ) n o (q a ) 1) f(n a (q a ) n o (q a )))s a t = 3 (n a (q a ) n o (q a ))s a t. Since q a > q o, we have n o (q a ) n o (q o ) 1. Thus, n a (q a ) n o (q a ) n a (q a ) n o (q o ) + 1. As (i) (j) for i j, The change in Φ ue to A is at most 3 (n a (q a ) n o (q o ) + 1)s a t = 3 (n a n o + 1)s a t. On the other han, n o (q) ecreases by 1 for q in [q o s o t,q o ], so the change in Φ ue to OPT is at most 3(f(n a (q o ) n o (q o ) + 1) f(n a (q o ) n o (q o )))s o t = 3 (n a (q o ) n o (q o ) + 1)s o t. Since q a > q o, n a (q o ) = n a (q a ) = n a, an the change in Φ ue to OPT is at most 3 (n a n o + 1)s o t. Summing the change in Φ ue to A an OPT, Φ 3 (n a n o + 1)( s a + s o )t. Combining the above 3 subcases, it implies that if n o < n a, then either t Φ 3 (n a n )( s a + s o ) or t Φ 3 (n a n + 1)( s a + s o ). Note that (n a n o )( s a + s o ) = P (P 1 (n a n o ))( s a + s o ). Setting i = n a n o in Lemma 3.1, we have that P (P 1 (n a n o ))( s a + s o ) ( s a + P 1 (n a n o ))P (P 1 (n a n o )) + +P(s o ) n a + n o P(s o ) n a + n o, where the last inequality follows from s a = P 1 (n a + 1) P 1 (n a n o ). Similarly, (n a n o +1)( s a +s o ) = P (P 1 (n a n o +1))( s a +s o ). Setting i = n a n o +1 in Lemma 3.1, we have that P (P 1 (n a n o + 1))( s a + s o ) ( s a + P 1 (n a n o + 1))P (P 1 (n a n o + 1)) +P(s o ) n a + n o 1 P(s o ) n a + n o, where the last inequality follows from s a = P 1 (n a + 1) P 1 (n a n o +1)). Thus, t Φ 3P(s o) 3n a +3n o in both cases. It follows that n a + P(s a ) + t Φ n a + n a P(s o ) 3n a + 3n o 3(P(s o ) + n o ). So Equation 3.2 is true if n o < n a. Case n o = n a : We show that Φ or Φ 3 (n a n o + 1)( s a + s o )t, as follows. Again, we consier 3 subcases epening whether q a < q o, q a = q o, or q a > q o. 1. Assume q a < q o. Then n(q) remains zero for q [q a s a t,q a ] an q [q o s o t,q o ]. Thus, n(q) is not change for any q an t Φ =. 2. Assume q a = q o. If s a s o, n(q) remains zero for q [q a s a t,q a ] (which also contains [q o s o t,q o ]). Thus, n(q) is unchange for any q an t Φ = Else if s a < s o, n(q) increases by 1 for q [q a s o t,q a s a t]. The change in Φ is 3(f(n a (q a ) n o (q o ) + 1) f(n a (q a ) n o (q o )))( s a + s o )t = 3 (n a n o + 1)( s a + s o )t 3. If q a > q o, it is ientical to subcase 3 of the case n o < n a, so Φ 3 (n a n o + 1)( s a + s o )t. It implies that t Φ or t Φ 3 (n a n o + 1)( s a + s o )t. Similar argument as before shows that Equation 3.2 is true if n o = n a. This completes the analysis when P satisfies the special conitions that we impose at the start of this section. 3.1 Removing the special conitions on P Consier an arbitrary power function P in the general moel. We explain how to moify P so that it meets the special conitions without significantly raising the competitive ratio. If P is not increasing, one can make P unefine on those spees where there is a greater spee 698 Copyright by SIAM. Unauthorize reprouction of this article is prohibite.

7 that consumes less power. Similarly, we can make P convex by eliminating any points in a subinterval (a,b) of spees where the line segment between (a,p(a)) an (b,p(b)) lies below the curve P. We now argue that the online algorithm A with the new power function can simulate running a spee s (a,b) in the ol power function. Algorithm A can simulate any spee s (a,b) by alternately running at spees a an b. Specifically, assume s = pa + (1 p)b for some < p < 1, we can run at spee a for a p fraction of the time an run at spee b for a (1 p) fraction of the time. It gives an effective spee of s an the energy usage is pp(a)+(1 p)p(b). If the slowest spee s on which P is efine is not, then can shift P(s) own by reefining P as P(s) = P(s) P(s ), an P(s) = for s [,s ]. Then it is easy to see that if A is c-competitive with respect to this new P then A is also be c-competitive with respect to the original P. If P is efine at a an b but not at any point in the interval (a,b), then interpolate P linearly between a an b. Once again the online algorithm can simulate running a spee s (a,b) in the new power function by alternately running at spees a an b. P stays convex by the convexity of the original P. One can fin a power function between P an P +max{ǫ,ǫp } that is efine on the same omain of spees, is continuous, ifferentiable, strictly increasing, an strictly convex. The competitive ratio of A for the ol power function will be within ǫ of the competitive ratio of A for the new power function. If P is boune, where T is the maximum spee, then the natural interpretation of the algorithm is to run at spee min(p 1 (n a + 1),T). We can follow the line of reasoning from [11]. We first note that at all times it must be the case that n a + 1 n o P(T). Then the analysis essentially then goes through as in the unboune case. In particular, in the case that n o < n a, we arrive at the same inequality of Φ t 3( s a + P 1 (n a n o + 1))P (P 1 (n a n o +1)) n a +n o 1. If s a = P 1 (n a +1), then we have that s a P 1 (n a n o +1) as before. Else if s a = T, we still have that s a P 1 (n a n o + 1). 4 Fractional Weighte Flow Plus Energy Our goal in this section is to prove Theorem 1.2, that in the general moel, HDF plus the natural spee scaling algorithm is (2 + ǫ)-competitive for the objective of fractional weighte flow plus energy. We start by showing, using an amortize local competitiveness argument, that this algorithm is 2-competitive if P also satisfies the special conitions from the last section. Definition of Online Algorithm A: The algorithm always runs the job of highest ensity. The ensity of a job is its weight ivie by its work. The spee s t a at time t is s t a = P 1 (w t a) where w t a is the total fractional weight of all unfinishe jobs for A at time t. Definition of the potential function Γ. Let OPT be the offline aversary that minimizes fractional weighte flow plus energy. At any time t, let wo t be the total fractional weight of all unfinishe jobs in OPT. For a job j, its inverse ensity is efine to be the ratio of its original size ivie by its original weight. At any time t, let wa(m) t enote the total fractional weight of all unfinishe jobs with inverse ensity at least m in A at time t. Define wo(m) t similarly for that of OPT. Let w t (m) = max{,wa(m) t wo(m)}. t We efine Γ(t) = 2 h(w t (m))m where h is the function such that for any real w, w h(w) = P (P 1 (w)) Since both P (x) an P 1 are increasing, P (P 1 (w))) is an increasing of function in w. Furthermore, h(w) can be written as h(w) = w P (P 1 (y))y. We now start the amortize local competitiveness analysis. For the bounary conition, we observe that before any job is release an after all jobs are complete, w t (m) = for all m, so Γ =. For the general conition, when a job is release, w t (m) is unchange for all m, so Γ is unchange. When a job is processe by A or OPT, the fractional weight of the job ecreases continuously to zero, so Γ is continuous an oes not ecrease ue to the completion of a job. It remains to show that at any time t uring a time interval without job arrival or completion, (4.3) w t a + P(s t a) + t Γ(t) 2(wt o + P(s t o)) The rest of this proof consiers any such time t an proves Equation 4.3. We omit t from the superscript an the parameter for convenience. Then, t Γ = 2 = 2 = 2 t h(w(m))m (w(m)) h(w(m))( t w(m))m P (P 1 (w(m))( t w(m))m, where the secon equality follows from chain rule. Let m a an m o enote the minimum inverse ensity of an unfinishe job in A an OPT, respectively. (Let 699 Copyright by SIAM. Unauthorize reprouction of this article is prohibite.

8 m a (resp., m o ) be if A (resp., OPT) has no unfinishe job.) Note that 1/m a an 1/m o are the ensity of the highest ensity job in A an OPT, respectively. Since A is running HDF at spee s a, w a (m) is ecreasing at the rate of (s a /m a ) for all m [,m a ] an w a (m) remains unchange for m > m a. Similarly, w o (m) is ecreasing at the rate of (s o /m o ) for m [,m o ] an remains unchange for m > m o. We consier three cases epening on w o > w a, w o < w a or w o = w a. Case w o > w a : We show that tγ in this case. Note that for any m [,m o ], w o (m) = w o > w a w a (m). Thus, for any m [,m o ], w(m) = max{,w a (m) w o (m)} remains zero an oes not change. It means that tw(m) = for m m o. For any m > m o, w o (m) remains unchange, so t w(m) for m > m o. Therefore, t Γ = 2 P (P 1 (w(m))( t w(m))m. We have w a + P(s a ) + t Γ 2w a < 2(w o + P(s o )), so Equation 4.3 is true. Case w o < w a : The ecrease in w a (m) causes a ecrease in Γ an the ecrease in w o (m) causes an increase in Γ. We analyze these two effects separately an boun them appropriately. For any m [,m a ], w a (m) is ecreasing at a rate of (s a /m a ), which causes Γ to ecrease at the rate of 2 m a P (P 1 (w(m)))( sa m a )m. For m [,m a ], w(m) = w a (m) w o (m) w a w o, so P (P 1 (w(m))) P (P 1 (w a w o )). Thus, 2 2 ma ma P (P 1 (w(m)))( s a m a )m P (P 1 (w a w o )))( s a m a )m = 2P (P 1 (w a w o )))s a For m [,m o ], w o (m) is ecreasing at a rate of (s o /m o ), which causes Γ to increase at a rate at most 2 m o P (P 1 (w(m)))( so m o )m. For m [,m o ], w(m) = max{,w a (m) w o (m)} w a w o, so P (P 1 (w(m))) P (P 1 (w a w o )). Thus, 2 2 mo mo P (P 1 (w(m)))( s o m o )m P (P 1 (w a w o )))( s o m o )m = 2P (P 1 (w a w o )))s o Summing the above two terms, we have t Γ 2P (P 1 (w a w o )( s a + s o ). By Lemma 3.1, it is at most 2( s a + P 1 (w a w o ))P (P 1 (w a w o )) + 2(P(s o ) (w a w o )). Since s a = P 1 (w a ) P 1 (w a w o ), t Γ 2(P(s o) w a + w o ). We have w a + P(s a ) + t Γ 2w a +2(P(s o ) w a +w o ) 2(P(s o )+w o ), hence proving Equation 4.3. Case w o = w a : For m [,m o ], w o (m) is ecreasing at a rate of (s o /m o ), which causes Γ to increase at a rate at most 2 m o P (P 1 (w(m)))( so m o )m. For m [,m o ], w(m) = max{,w a (m) w o (m)} w a w o =, so P (P 1 (w(m))) = P (P 1 ()) an mo P (P 1 (w(m)))( so m o )m = P (P 1 ()))s o. By Lemma 3.1 with i =, s a =, we obtain that t Γ 2P (P 1 ()))s o 2P(s o ). We have w a +P(s a )+ t Γ 2w a + 2P(s o ) = 2(w o + P(s o )), hence proving Equation 4.3. This completes the analysis when the special conitions hol. The proof can be extene to the general moel in the same way as was one in the proof of Theorem 1.1. References [1] Susanne Albers an Hiroshi Fujiwara. Energy-efficient algorithms for flow time minimization. In Lecture Notes in Computer Science (STACS), volume 3884, pages , 26. [2] N. Bansal, H.L. Chan, T.W. Lam, an L.K. Lee. Scheuling for boune spee processors. In International Colloquium on Automata, Languages an Programming, ICALP, 28, to appear. [3] Nikhil Bansal an Ho-Leung Chan. Weighte flow time oes not amit o(1)-competitive algorithms. submitte to SODA 29. [4] Nikhil Bansal, Kirk Pruhs, an Cliff Stein. Spee scaling for weighte flow time. In SODA 7: Proceeings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, pages , 27. [5] Luca Becchetti, Stefano Leonari, Alberto Marchetti- Spaccamela, an Kirk R. Pruhs. Online weighte flow time an ealine scheuling. In Workshop on Approximation Algorithms for Combinatorial Optimization, pages 36 47, 21. [6] Fre Bower, Daniel Sorin, an Lanon Cox. The impact of ynamically heterogeneous multicore processors on threa scheuling. Unpublishe. [7] Davi M. Brooks, Praip Bose, Stanley E. Schuster, Hans Jacobson, Prabhakar N. Kuva, Alper Buyuktosunoglu, John-Davi Wellman, Victor Zyuban, Manish Gupta, an Peter W. Cook. Power-aware microarchitecture: Design an moeling challenges for nextgeneration microprocessors. IEEE Micro, 2(6):26 44, 2. [8] J. Aam Butts an Gurinar S. Sohi. A static power moel for architects. In MICRO 33: Proceeings of the 33r annual ACM/IEEE international symposium on Microarchitecture, pages , 2. [9] Xiaobo Fan, Wolf-Dietrich Weber, an Luiz Anre Barroso. Power provisioning for a warehouse-size computer. In ISCA 7: Proceeings of the 34th annual international symposium on Computer architecture, pages 13 23, Copyright by SIAM. Unauthorize reprouction of this article is prohibite.

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