Average Rate Speed Scaling

Transcription

1 Average Rate Speed Saling Nikhil Bansal David P. Bunde Ho-Leung Chan Kirk Pruhs May 2, 2008 Abstrat Speed saling is a power management tehnique that involves dynamially hanging the speed of a proessor. This gives rise to dual-objetive sheduling problems, where the operating system both wants to onserve energy and optimize some Quality of Servie (QoS measure of the resulting shedule. Yao, Demers, and Shenker [4] onsidered the problem where the QoS onstraint is deadline feasibility and the objetive is to minimize the energy used. They proposed an online speed saling algorithm Average Rate (AVR that runs eah job at a onstant speed between its release and its deadline. They showed that the ompetitive ratio of AVR is at most (2 /2 if a proessor running at speed s uses power s. We show the ompetitive ratio of AVR is at least ((2 δ /2, where δ is a funtion of that approahes zero as approahes infinity. This shows that the ompetitive analysis of AVR by Yao, Demers, and Shenker is essentially tight, at least for large. We also give an alternative proof that the ompetitive ratio of AVR is at most (2 /2 using a potential funtion argument. We believe that this analysis is signifiantly simpler and more elementary than the original analysis of AVR in [4]. Introdution Current proessors produed by Intel and AMD allow the speed of the proessor to be hanged dynamially. Intel s SpeedStep and AMD s PowerNOW tehnologies allow the Windows XP operating system to dynamially hange the speed of suh a proessor to onserve energy. In this setting, the operating system must not only have a job seletion poliy to determine whih job to run, but also a speed saling poliy to determine the speed at whih the job will be run. In urrent CMOS based proessors, the speed satisfies the well-known ube-root-rule, that the speed is approximately the ube root of the power. Energy onsumption is power integrated over time. The operating system is faed with a dual objetive optimization problem as it both wants to onserve energy, and optimize some Quality of Servie (QoS measure of the resulting shedule. The first theoretial worst-ase study of speed saling algorithms was in the seminal paper [4] by Yao, Demers, and Shenker. Their QoS objetive was deadline feasibility and the objetive was to minimize the energy used. More preisely, eah job i has a release time r i when it arrives in the system, a work requirement w i, and a deadline d i by whih the job must be finished. If job i runs at onstant speed s, then IBM T. J. Watson Researh Center, nikhil@us.ibm.om Computer Siene Department, Knox College, dbunde@knox.edu. Supported in part by Howard Hughes Medial Institute grant Computer Siene Department, University of Pittsburgh, hlhan@s.pitt.edu. Computer Siene Department, University of Pittsburgh, kirk@s.pitt.edu. Supported in part by NSF grants CNS , CCF and IIS

2 it ompletes in w i /s units of time. In this setting, an optimal job seletion poliy is Earliest Deadline First (EDF. They assumed a speed to power funtion P(s = s, where > is some onstant. If the ube-root rule holds, then = 3. Yao, Demers, and Shenker [4] showed that the optimal energy feasible shedule is found by a simple greedy algorithm that we all YDS. Yao, Demers, and Shenker [4] also proposed an online speed saling algorithm, Average Rate (AVR. Coneptually, AVR runs eah job i at speed w i /(d i r i throughout interval [r i, d i ], independent of other jobs. This spreads the work of eah job as evenly over time as possible. By the onvexity of the speed to power funtion, this even spreading is energy optimal if the instane onsists of only one job. The speed of the proessor at any time t is then just the sum of the speeds of the jobs ative at that time, that is i:t [r i,d i ] w i d i r i. AVR is an appealing speed saling algorithm beause in some sense it is perfetly fair to all jobs, and eah job runs as if it were the only job in the instane. Yao, Demers, and Shenker [4] showed that the ompetitive ratio, with respet to energy, of AVR is at least. They also showed that the ompetitive ratio of AVR, with respet to energy, is at most (2 /2. We now outline this upper bound ompetitive analysis of AVR. A job is defined to be of type A if the optimal shedule is always ahead of AVR on this job. A job is defined to be of type B if AVR is always ahead of the optimal shedule on this job. A shedule is bitoni if every job is of type A or type B. [4] observes that there is a worst-ase instane that is bitoni, and that the ompetitive ratio of AVR is at most 2 times the ompetitive ratio of AVR on instanes of jobs of just one type (A or B. [4] then onsiders instanes onsisting only of type-a jobs. [4] then introdues an auxiliary objetive funtion that is related to, but is not exatly, the energy used. In a somewhat involved redution, [4] shows that with respet to this auxiliary objetive, there is a worst-ase instane where the optimal shedule is non-preemptive, eah job starts when it is released, and the spans of the jobs are nested (where the span of job i is the interval [r i, d i ]. When = 2, [4] then shows that for suh instanes, optimizing the auxiliary objetive funtion an be represented in terms of the eigenvalues of a partiular tree-indued matrix, and shows how to bound the largest eigenvalue for suh tree-indued matries. [4] states that this argument an be readily generalized to an arbitrary, and using Hölder s inequality, give a bound on the l p norm of a ertain tree-indued matrix that would replae the eigenvalue argument used in the = 2 ase. So the natural question left open is, What is the exat ompetitive ratio of AVR? Based on simulation results, [4] onjetured that the ompetitive ratio of AVR is exatly. That is, that the lower bound in [4] is orret, and intuitively, that AVR an not simultaneously be losing badly on both type-a and type-b jobs. In the ase that the ube-root rule holds, = 3 3 = 27 is the best known ompetitive ratio for any online algorithm. If the onjeture from [4] was true, this would be evidene in favor of adopting the AVR speed saling poliy. Not only would AVR have the best known ompetitive ratio in the ase that the ube-root rule holds, but AVR is appealingly fair to all jobs. Unfortunately, in setion 4, we show that the upper bound on the ompetitive ratio from [4] is essentially tight, at least for larger. More preisely, we show that AVR has ompetitive ratio at least ((2 δ /2, where δ is a funtion of that approahes zero as approahes infinity. In the ase obeying the ube-root rule, we get a lower bound of approximately 48 on the ompetitive ratio of AVR. In setion 5, we give an alternative proof that the ompetitive ratio of AVR is at most (2 /2. Our analysis uses a potential funtion argument. We believe that this analysis is signifiantly simpler and more elementary than the original analysis of AVR in [4]. Our ompetitive analysis of AVR branhes off from the analysis in [4] outlined above after the observation that the ompetitive ratio of AVR is at most 2 times the ompetitive ratio of AVR on jobs of just one type. We give a potential funtion argument that AVR is -ompetitive on type-a jobs. We inlude a omplete analysis of AVR in this paper, inluding the elements of the analysis from [4] that we use. In priniple, verifying this analysis requires only basi 2

3 algebra, exept that some basi alulus is used to verify the positivity/negativity of ertain polynomials over partiular ranges. 2 Other Related Results There are now enough speed saling papers in the literature that it is not pratial to survey all suh papers here. We limit ourselves to those papers most related to the results presented here. Yao, Demers, and Shenker [4] also proposed another online speed saling algorithm, Optimal Available (OA. The algorithm OA runs at the optimal speed (whih an be omputed using the YDS algorithm assuming the urrent state and that no more jobs will be released in the future. [4] showed that the ompetitive ratio of OA is at least. Using a potential funtion analysis, Bansal, Kimbrel, and Pruhs [2] showed that OA is atually -ompetitive. Bansal, Kimbrel, and Pruhs [2] also introdued an online speed saling algorithm that we all BKP. Intuitively, BKP tries to mimi the offline YDS shedule in some way. Formally, at time t BKP runs at speed e v(t where v(t = max t >t w(t,et (e t,t e(t t and w(t, t, t 2 is the amount of work that has release time at least t, deadline at most t 2, and that has already arrived by time t. [2] showed that BKP is simultaneously O(-ompetitive for total energy, maximum temperature (assuming ooling obeys Newton s law, maximum power, and maximum speed. Speifially, [2] showed that the ompetitive ratio of BKP with respet to energy is at most 2(/( e. Albers, Müller, and Shmelzer [] onsider the problem of finding energy-effiient deadline-feasible shedules on multiproessors. [] showed that the offline problem is NP-hard, and gave O(-approximation algorithms. [] also gave online algorithms that are O(-ompetitive when job deadlines our in the same order as their release times. 3 Formal Problem Statement A problem instane onsists of n jobs. Job i has a release time r i, a deadline d i > r i, and work w i > 0. In the online version of the problem, the sheduler learns about a job only at its release time; at this time, the sheduler also learns the exat work requirement and the deadline of the job. We assume that time is ontinuous. A shedule speifies for eah time a job to be run and a speed at whih to run the job. The speed is the amount of work performed on the job per unit time. A job with work w run at a onstant speed s thus takes w s time to omplete. More generally, the work done on a job during a time period is the integral over that time period of the speed at whih the job is run. A shedule is feasible if for eah job i, work at least w i is done on job i during [r i, d i ]. Note that the times at whih work is performed on job i do not have to be ontiguous. If a job is run at speed s, then the power is P(s = s for some onstant >. The energy used during a time period is the integral of the power over that time period. Our objetive is to minimize the total energy used by the shedule. If A is a sheduling algorithm, then A(I denotes the shedule output by A on input I. A shedule is R-ompetitive for a partiular objetive funtion if the value of that objetive funtion on the shedule is at most R times the value of the objetive funtion on an optimal shedule. An online sheduling algorithm A is R-ompetitive, or has ompetitive ratio R, if A(I is R-ompetitive for all instanes I. For a shedule T, let s T,j (t denote the speed job j runs at time t in the shedule T, and let s T (t = j s T,j(t denote the speed of the proessor at time t in shedule T. If U is a subolletion of jobs, let s T,U (t denote the sum of the speeds of the jobs in U at time t in the shedule T. We will also substitute 3

4 an algorithm for a shedule in this notation. So for example, s AV R (t is the speed of the algorithm AVR at time t. We use OPT to denote a partiular optimal shedule. We say that job i is ative between its release time and its deadline. We all w i /(d i r i the density of job i sine this is the job s work divided by the length of the interval in whih it is ative. Algorithm AVR: At all times t, run the earliest-deadline job at speed s AV R (t = w i i d i r i, where the sum is over jobs i ative at time t. Consider a fixed optimum shedule OPT. A job is said to be of type A if t r j s OPT,j (tdt t r j w j d i r i dt for all r j t d j Intuitively, these are the jobs that OPT runs onsistently ahead of their density. Similarly, the jobs of type B are those that OPT runs onsistently behind their density, meaning they satisfy t r j s OPT,j (tdt t r j w j d i r i dt for all r j t d j. In general, a job need not be of either type (or it an also be of both types, in whih ase OPT exeutes exatly as in AVR. We say an instane is bitoni if every job is of type A, type B, or both (in whih ase it is arbitrarily assigned one of the types. A simple observation (Lemma 5 shows that if AVR is -ompetitive for bitoni instanes, then it is also -ompetitive in general. 4 The Lower Bound We give an instane on whih AVR uses up at least ((2 δ /2 times the energy used by an energy optimum solution, where δ is a funtion of that tends to zero as inreases. Instane Desription: For onveniene we will work with a ontinuous version of the job instane. We say that work arrives at rate a(t at time t to mean that a(tdt units of work arrive during the infinitesimally small interval [t, t + dt]. The instane onsists of two sets of jobs A and B. The work in A arrives during the time interval [0, ǫ], at rate a(t = ( t / and all the work in A has deadline. Here ǫ > 0 is an arbitrarily small but fixed onstant. The work in B arrives during the interval [ /, ǫ/] (where is a onstant that will be set to later at rate b(t = / ( t / and the work in B arriving at time t has deadline + ( t. Lemma On the instane above, the optimal algorithm uses total energy at most 2 ln(/ǫ. Proof: It suffies to give some feasible shedule that uses energy 2 ln(/ǫ. Consider the shedule that ompletes all jobs in A by running at speed a(t during [0, ǫ]. The energy usage is ǫ 0 (a(t dt = [ ln( t] ǫ 0 = ln(/ǫ 4

5 For jobs in B, note that they are released before time and have deadlines in [ + ǫ, 2]. Consider any time x [ + ǫ, 2]. The jobs with deadline in [ + ǫ, x] are released during [ x, ǫ ]. Their total amount of work is ǫ/ ǫ/ b(tdt = / ( t /dt (x / (x / Let y = + ( t. Then dy = dt, and the amount of work equals ǫ/ (x / +ǫ / ( t /dt = x x (y /dy = +ǫ (y /dy Therefore, onsider the shedule that proesses jobs in B at speed ˆb(y = ontinuously during (y / [ + ǫ, 2]. For any x [ + ǫ, 2], the amount of work done by time x equals the amount work with deadline by x. So the shedule ompletes eah job in B by its deadline. The energy usage to omplete all jobs in B is 2 (ˆb(t dt = [ln(y ] 2 +ǫ = ln(/ǫ +ǫ Sine the intervals of exeution of work in A and B do not overlap, the total energy used is 2 ln(/ǫ and the lemma follows. Lemma 2 On the instane above, AVR uses total energy at least ( + / (+ ln(/ǫ + K, where K is a onstant independent of ǫ. Proof: Consider the work in A. The work released at time t is sheduled by AVR uniformly during the interval [t, ]. Thus, at any time x [0, ], the density due to work in A is x x ( den a (x = a(t 0 t dt = 0 ( t / t dt = ( x / Now onsider the work in B. Note that for work released at time t, the duration between its release time and deadline is + ( t t = ( + ( t. Thus, at any time x [, ǫ ], the density due to work in B is den b (x = = x / / ( t / ( + ( t dt ( / ( + / ( x / During the interval [, ǫ], AVR runs at speed equal to the total density due to work in A and B. Therefore, the energy usage of AVR is at least ǫ / (den a (t + den b (t dt = ǫ / ( ( + / ( dt ( ( t / + 5

6 Let Y = + / (+. Note that for all t [, ǫ], we have that t / and hene ( t/ Y / / ( + / ( ( + + = Then, by fatoring Y, the right side of ( an be written as ( t / ( ǫ Y 2 + ( t/ dt / t + Y ( ǫ Y 2 + ( t/ dt as x ( x for x / t + Y ǫ ( = Y Z( t(/ dt where Z = (2+ t Y (+ / = Y [ ln( t + Z( t /] ǫ / = Y ( ln ǫ + Zǫ / + ln Z( / Y ln(/ǫ + Y ( ln Z( / Sine,, Y and Z are independent of ǫ the lemma follows. sine ǫ > 0 Theorem 3 The ompetitive ratio of AVR is at least ((2 δ /2, where δ is a funtion of that tends to zero as inreases. Proof: By Lemma and 2, when ǫ tends to zero, the ompetitive ratio of AVR is at least (( + / + /2. Putting =, the ompetitive ratio is at least (( + ( / /2, whih equals ((2 δ /2 where δ = ( /. Note that for large (in partiular for 2, we have that δ = ( / = e ( /ln( ( ( ln( ( using ex + x for x < 0 = ln( + Hene δ approahes zero as approahes infinity. ln( 2 (2 We remark that our bound ((2 δ /2 is asymptotially 2 /2 o( for large, and hene within /2+o( of the best known upper bound. To see this, by (2, we obtain that ( ( ln( lim δ lim + ln( =. ln ln ln ln 6

7 Similarly, and hene δ / = e (ln/ ln + = ln + ln, ( lim δ lim ln + ln =. Thus the expression (2 δ /2 = 2 ( δ/2 2 δ/(2ln = 2 /2 o(. 5 An Elementary Proof that AVR is 2 -ompetitive This setion gives a omplete elementary proof that AVR is 2 -ompetitive. This proof uses some elements of the analysis of AVR in [4] and some variations on elements of the analysis of OA in [2]. We start with the analysis of AVR on instanes onsisting of only type-a jobs. The analysis for general instanes then follows along the same lines as in [4], and is inluded here for ompleteness. Lemma 4 For instanes onsisting of only type-a jobs, AVR is -ompetitive with respet to energy. Proof: We use an amortized loal ompetitiveness argument. At any time t, either a task arrives or finishes, or else an infinitesimal interval of time dt elapses and AVR onsumes s AV R (t dt units of energy. We will define a potential funtion φ(t that satisfies the following properties: The potential funtion φ(t has value 0 before any jobs arrive and after the last deadline. The potential funtion φ(t does not inrease as a result of AVR ompleting a job, OPT ompleting a job, or the release of a job. At any time t, s AV R (t + dφ(t dt s OPT (t. (3 Integrating equation 3 over time and using the other two stated properties, we an onlude the desired result. For a more detailed treatment of amortized loal ompetitiveness arguments, see [3]. Before we an define the potential funtion we need to introdue some notation. Let t 0 denote the urrent time and t i denote the time of the i th deadline ourring after t 0. Then let I i denote the interval of time [t i, t i+. Let τ i = t i+ t i be the length of interval I i. Let s i denote the speed at whih AVR will work during interval I i if no new jobs arrive. This an be omputed by summing the densities of ative jobs whose deadline is at or after time t i+. Let w AV R,i = s i τ i denote the amount of work that AVR plans to omplete during interval I i. Let w OPT,i be the portion of the work AVR alloates to interval I i that OPT has not yet ompleted. Beause all jobs are of type A, all work that is unfinished by OPT is also unfinished by AVR. Without loss of generality, we assume that when OPT is working on a job j, work is removed from the term w OPT,i that ontains work from job j with the smallest index i. That is, OPT removes work from the earlier intervals first. We define the potential funtion φ(t as follows: φ(t = i 0 s i (w AV R,i w OPT,i (4 7

8 This potential funtion is a slight modifiation of the potential funtion used in [2] to analyze the algorithm OA. The differene is that the potential funtion in [2] uses w OPT,i to denote the work of jobs unfinished for OPT with deadline in I i. Now we show that φ has the laimed properties. This funtion is learly 0 when there are no ative jobs. The ompletion of a job by OPT also has no effet sine the potential is a ontinuous funtion of w OPT,i. The situation when AVR ompletes a job is slightly more ompliated. Observe that a job ompletes under AVR if and only if the size of the interval I 0 shrinks to 0, i.e. when the urrent time t 0 beomes equal to t, whih shifts all the indies. At the moment this happens AVR has ompleted all the work alloated to I 0 and hene w AV R,0 = 0. Beause all jobs are of type A, OPT has also ompleted the work alloated to I so w OPT,0 = 0. Thus, the potential is ontinuous even in this ase. (This is the only time we use that all the jobs are of type A. Arrival Case: The next ase to onsider is when a new job j arrives. First observe that adding a zero work job with deadline d j does not hange the value of the potential funtion φ. Thus, we may assume that the new job s deadline is t k for some k. Let y be the density of the new job. Then the release inreases the density of intervals I 0, I,..., I k by y, inreasing the weight of interval I i by yτ i for 0 i k. This hanges the potential funtion by k ( wav R,i + yτ i φ = ((w AV R,i + yτ i (w OPT,i + yτ i i=0 k i=0 τ i ( wav R,i This expression an be rearranged into k i=0 τ i (w AV R,i w OPT,i. (5 ( τi (w AV R,i + yτ i (w AV R,i w OPT,i ( yτ i w AV R,i (w AV R,i w OPT,i By making the substitutionsq = w AV R,i, δ = yτ i and r = w OPT,i eah term of this sum beomes a quantity shown to be at most 0 by Lemma 8. Working ase: We now onsider times when no job arrives, and no jobs omplete. Eah s i, inluding s 0, remains fixed during this time. We have to show or equivalently, s AV R (t 0 s OPT (t 0 + dφ(t dt s 0 s OPT (t 0 + d dt ( i 0 0 (6 s i (w AV R,i w OPT,i 0 (7 As AVR works, w AV R,0 is dereasing at rate s 0, and w AV R,i remains fixed for all i. Sine OPT takes work from a single interval I i, only one of the w OPT,i hanges; let it be w OPT,k. Then equation (7 is equivalent to s 0 s OPT (t 0 + ( s0 s s k s OPT (t 0 0 8

9 Sine a job ative during one interval is also ative in all earlier intervals, s k s 0 and it suffies to show that ( s s 0 s OPT (t 0 s OPT (t 0 0 Substituting z = s 0 /s OPT (t 0 gives ( z + 2 z 0 (8 Let u(z be the polynomial on the left hand side of inequality 8. Note that u(0 = and u(+ =. In addition, the derivative of u(z is 0 at only the point z =. Sine u( = 0, we onlude that u(z is non-positive for z 0, whih holds beause of the definition of z. This establishes inequality 6. Lemma 4 and the argument of Yao, Demers, and Shenker [4] proves the 2 -ompetitiveness of AVR. We now give their argument for ompleteness. Lemma 5 [4] Among those instanes on whih AVR has it worst-ase ompetitive ratio, there is a bitoni instane. Proof: Consider a worst-ase instane I that is not bitoni. We explain how to transform I into another worst-ase instane that is bitoni. There must be a job i that is of neither type A nor type B. By the definition of the types, there has to be some times s, u, with s < u, for whih one of AVR or OPT is ahead of the other on job i at time s, but behind at time u. By the intermediate value theorem, there must be a time t (s, u where AVR and OPT have ompleted an equal amount of work w on job i. We say that the lead hanges at suh a time t. We now reate a new instane I from I by replaing job i with two jobs: one with work w released at time r i with deadline t, and one with work w i w released at time t with deadline d i. It is easy to see that both AVR and OPT always run at the same speed in I that they did in I. This transformation however redues the number of lead hanges by one. Sine there an only be a bounded number of lead hanges between YDS = OPT and AVR, a bounded number of appliations of this transformation leads to a bitoni instane. Lemma 6 [4] AVR is 2 -ompetitive on bitoni instanes. Proof Sketh: Given a bitoni instane, let A be the set of type-a jobs and B be the others. Let AVR A and AVR B denote the energy attributable to A and B in the AVR shedule, respetively. Define OPT A and OPT B similarly with referene to the shedule OPT. Next observe that the roles of type-a jobs and type-b jobs an be swapped by reversing time and swapping the release time and deadline for eah job. Both YDS and AVR give the same shedule to the forward and bakwards versions so Lemma 4 implies that AVR is simultaneously -ompetitive with respet to energy attributable to type-a jobs and energy attributable to type-b jobs. The proof follows by ombining the shedules for the jobs of different types. The optimal ost is learly at least OPT A +OPT B. To bound the ost of AVR, define s AV R,A (t and s AV R,B (t as the speed of AVR on type-a and type-b jobs respetively. Then the ost of AVR is at most s AV R (t dt = (s AV R,A (t + s AV R,B (t dt 2 (s AV R,A (t + s AV R,B (t dt = 2 (AVR A + AVR B 2 (OPT A + OPT B, 9

10 whih gives the desired ratio. Thus we reah our final theorem, whih is an immediate onsequene of Lemma 4, Lemma 5, and Lemma 6. Theorem 7 AVR is 2 -ompetitive. The following lemma from [2] was used in the proof of Lemma 4: Lemma 8 [2] Let q, r, δ 0 and. Then (q + δ (q r ( δ q (q r 0. Proof: The lemma is equivalent to showing that (q r[(q + δ q ] (q + δ ( δ 0 Sine [(q + δ q ] 0, it suffies to show that q[(q + δ q ] (q + δ ( δ 0 Let δ = zq, whih implies z 0. The left hand side of the above beomes q [( + z ] q [( + z ( z] Fatoring out q and differentiating the rest with respet to z gives (( ( + z 2 [ ( z] + ( + z ( + = (( ( + z 2 [ ( z ( + z] = ( z( + z 2 This is non-positive sine > and z 0. Thus, the expression is maximized at z = 0, where it has value 0. This implies the result. 6 Conlusion Even though AVR is not optimally ompetitive, one ould imagine situations where a system designer might still adopt AVR beause AVR is in some sense fair to eah job. This is analogous to the reason that Proessor Sharing (Round Robin is adopted in some systems even though Proessor Sharing is known not to have the best ompetitive ratio for the standard QoS measures. Aknowledgments: We thank Don Coppersmith for helpful disussions. Referenes [] S. Albers, F. Müller, and S. Shmelzer. Speed saling on parallel proessors. In Pro. ACM Symposium on Parallel Algorithms and Arhitetures (SPAA, pages , [2] N. Bansal, T. Kimbrel, and K. Pruhs. Speed saling to manage energy and temperature. JACM, 54(,

11 [3] K. Pruhs. Competitive online sheduling for server systems. SIGMETRICS Performane Evaluation Review, 34(4:52 58, [4] F. Yao, A. Demers, and S. Shenker. A sheduling model for redued CPU energy. In Pro. IEEE Symp. Foundations of Computer Siene, pages , 995.