Stochastic Analysis of Power-Aware Scheduling

Transcription

1 Stochatic Analyi of Power-Aware Scheduling Adam Wierman Computer Science Department California Intitute of Technology Lachlan L.H. Andrew Computer Science Department California Intitute of Technology Ao Tang School of ECE Cornell Univerity Abtract Energy conumption in a computer ytem can be reduced by dynamic peed caling, which adapt the proceing peed to the current load. Thi paper tudie the way to adjut peed to balance mean repone time and mean energy conumption, when job arrive a a Poion proce and proceor haring cheduling i ued. Both bound and aymptotic for the peed are provided. Interetingly, a imple cheme that halt when the ytem i idle and ue a tatic rate while the ytem i buy provide nearly the ame performance a the dynamic peed caling. However, dynamic peed caling which allocate a higher peed when more job are preent ignificantly improve robutne to burty traffic and mi-etimation of workload parameter. I. INTRODUCTION Two threat to the growth of the internet have their root in power conumption. The mot preing i that Moore law ha increaed the thermal denity of electronic to uch an extent that cooling i a major concern, and ha halted the previouly inexorable increae in clock peed. The longer term threat i that the need to reduce foil fuel conumption require all apect of ociety to conerve energy, while aggregate internet energy conumption i a ignificant and growing fraction of the energy conumption of developed countrie [1]. A a reult, all modern ytem deign mut conider the tradeoff between energy ue and other performance metric. Power can often be aved imply by running device more lowly. Dynamic peed caling, which electively reduce the peed when the load i light, reduce energy conumption with minimal impact on performance. It i widely implemented in current proceor, in the form of Intel SpeedStep and AMD PowerNow. Much of the theory of dynamic peed caling [2] [6] conider wort-cae bound. While uch wort-cae reult are important for temperature management [7], or when the energy to a pecific computer i contrained [8], [9], global energy conumption i affected by the average cae, rather than the wort cae. Conequently, thi paper tudie the average performance in a tochatic etting. In particular, thi paper eek to minimize a weighted um of the mean repone time and the energy ue per job. Thi performance metric ha been tudied both theoretically [1] [12] and in implementation [13]. Algorithm are known [12] for finding the peed which optimize thi objective in a tochatic M/M/1/FCFS etting. However, thee are highly recurive, and provide little inight. The goal of thi paper i to identify imple tructural propertie of the olution, and to ue them to compare the gain of uncontrained peed caling with that of an optimized tatic deign. The paper make three main contribution. Firt, the paper provide bound on the performance of dynamic peed caling (Section IV-A). Surpriingly, thee bound how that even an idealized verion of dynamic peed caling improve performance only marginally compared to a imple cheme where the erver ue a tatic peed when buy and run at peed when idle at mot a factor of 2 for typical parameter and often le (ee Section V). Second, the paper provide bound and aymptotic for the peed ued by the dynamic peed caling cheme (Section IV-B and IV-C). Thee reult provide inight into how the peed cale with the arriving load, the queue length, and the relative cot of energy. Third, the paper illutrate through analytic reult and numerical experiment that, though dynamic peed caling provide limited performance gain, it dramatically improve robutne to mi-etimation of workload parameter and burty traffic (Section VI). Note that many proof are omitted from thi document; all proof can be found in [14]. II. MODEL AND NOTATION In order to tudy the performance of dynamic peed caling, we focu on a imple model: an M/GI/1 PS queue with controllable ervice rate, dependent on the queue length. In thi model, job arrive to the erver a a Poion proce with rate λ, have intrinic ize with mean 1/µ, and depart at rate n µ when there are n job in the ytem. Under tatic cheme, the (contant) ervice rate i denoted by. Define the load a ρ = λ/µ, and note that the ρ i not the fraction of time the erver i buy. The performance metric we conider i E[T ] E[E]/β, where T i the repone time of a job, E i the energy expended on a job, and β repreent how delay-avere the deign i. It i often convenient to work with the expected cot per unit time, intead of per job, which by Little law can be written a z = E[N] λe[f()]/β, where N i the number of job in the ytem and f() determine the power ued when running at peed. The remaining piece of the model i to define the form of f(). The dynamic power of a circuit i typically a low-order polynomial in the peed [15]. A a reult, we will model the power ued by running at peed by λ f() β = α β where α > 1 and β take the role of β, but ha dimenion (1)

2 2 (time) α. The cot per unit time then become z = E[N] α β. (2) We will often focu on the cae of α = 2 to provide intuition. The impact of the workload parameter ρ, β, and α can often be captured by = ρ/β 1/α, which i a dimenionle meaure. Alo, it will often be convenient to ue the a natural dimenionle unit of peed /β 1/α. III. POWER-AWARE SPEED SELECTION Thi paper conider two natural form of peed caling: (i) Gated tatic peed: The erver gate it clock (etting = ) if no job are preent, and if job are preent it work at a contant rate choen to balance energy uage and repone time. (ii) Dynamic peed caling: The erver adapt it peed to the current number of requet preent in the ytem. The goal of thi paper i to undertand how to chooe peed in each of thee cenario and to contrat the relative merit of each cheme. Clearly the expected cot i reduced each time the erver i allowed to adjut it peed more dynamically. Thi mut be traded againt the cot of witching, uch a a delay of up to ten of microecond to change peed [16]. The important quetion i What i the magnitude of improvement at each level? For our comparion, we will ue idealized verion of each cheme. In particular, in each cae we will aume that the erver can be run at any deired peed in [, ) and ignore witching cot. In thi ection, we will derive expreion for the peed in cae (i). For cae (ii), we will decribe a numerical approach for calculating the peed which i due to George and Harrion [12]. Though thi numerical approach i efficient, it provide little inight into the tructure of the dynamic peed or the overall performance. Providing uch reult will be the focu of Section IV. A. The tatic peed for a gated ytem In the implet dynamic peed caling, a erver either run at a contant rate, or ha it clock gated uing zero dynamic power when the ytem i empty. We call thi policy the gated-tatic policy, and denote it cot z g. Since the erver can gate it clock, the energy cot i only incurred ρ/ of the time, when the erver i buy. Thu z = ρ ρα 1 ρ β. The optimum occur when > ρ and (α 1) α 2 ( ρ) 2 = β. (3) When α = 2, g = ρ β. In general, define G(; α) = σ.t. σ > (α 1)σ α (1 /σ) 2 = 1. (4) The gated-tatic peed i g = β 1/α G(; α). The following lemma bound G. Lemma 1. For α 2, 2 α α 1 G(; α) (α 1) 1/α 2 α (5) and the inequalitie are revered for α 2. B. Optimal dynamic peed caling A popular alternative to tatic power management i to allow the peed to adjut dynamically to the number of requet in the ytem. The tak of deigning an dynamic peed caling cheme in our model can be viewed a a tochatic control problem. We tart the analyi by noting that we can implify the problem dramatically with the following obervation. An M/GI/1 PS ytem i well-known to be inenitive to the job ize ditribution. Thi till hold when the ervice rate i queue-length dependent ince the policy till fall into the cla of ymmetric policie introduced by Kelly [17]. A a reult, the mean repone time and entire queue length ditribution are affected by the ervice ditribution through only it mean. Thu, we can conider an M/M/1 PS ytem. Further, the mean repone time and entire queue length ditribution are equivalent under all non-ize baed ervice ditribution in the M/M/1 queue [17]. Thu, to determine the dynamic peed caling cheme for an M/GI/1 PS queue we need only conider an M/M/1 FCFS queue. The ervice rate control problem in the M/M/1 FCFS queue ha been tudied extenively [12], [18], [19]. In particular, George and Harrion [12] provide an elegant olution to the problem of electing the tate-dependent proceing peed to minimize a weighted um of an arbitrary holding cot with a proceing peed cot. Specifically, the tatedependent proceing peed can be framed a the olution to a tochatic dynamic program, to which [12] provide an efficient numerical olution. In the remainder of thi ection, we will provide an overview of thi numerical approach. The core of thi approach will form the bai of our derivation of bound on the peed in Section IV. We will decribe the algorithm of [12] pecialized to the cae conidered in thi paper, where the holding cot in tate n i imply n. Further, we will generalize the decription to allow arbitrary arrival rate, λ. The olution tart with an etimate z of the minimal cot per unit time, including both the occupancy cot and the energy cot. A in [12], [19], [2], the minimum cot of returning from tate n to the empty ytem i given by the dynamic program v n = inf A { 1 λ µ [λ αβ n z ] µ λ µ v n 1 λ } λ µ v n1 where A i the et of available peed. We will uually aume A = R {}. With the ubtitution u n = λ(v n v n 1 ), thi can be written a [12], [2] u n1 = up A { z n λ α β u n ρ }. (6)

3 3 Two additional function are defined. Firt, ( ) α/(α 1) u φ(u) = up{ux/ρ λx α /β} = (α 1). (7) x A α Second, the minimum value of x which achieve thi upremum, normalized to be dimenionle, i ψ(u) = 1 { min x : ux } ( ) 1/(α 1) β1/α ρ λxα u β = φ(u) =. α (8) Given the etimate of z, u n atify u 1 = z u n1 = φ(u n ) n z. (9a) (9b) The value of z can be found a the minimum value uch that (u n ) n=1 i an increaing equence. Thi allow z to be found by an efficient binary earch, after which u n can in principle be found recurively. The peed in tate n i then given by n β 1/α = ψ(u n). (1) Thi highlight the fact that = ρ/β 1/α provide the appropriate caling of the workload information becaue the cot z, normalized peed β 1/α and variable u n depend on λ, µ and β only through. IV. BOUNDS ON OPTIMAL DYNAMIC SPEED SCALING In the prior ection, we preented the deign for gated-tatic and dynamic peed caling. In the firt cae, the peed wa preented more-or-le explicitly, however in the third cae we preented only a recurive numerical algorithm for determining the dynamic peed caling. In thi ection, we provide reult exhibiting the tructure of the dynamic peed and the performance they achieve. The main reult of thi ection are ummarized in Table I. The bound on z for arbitrary α are eentially tight (i.e., agree to leading order) in the limit of mall or large. Due to the complicated form of the general reult, we illutrate the bound for the pecific cae of α = 2 to provide inight. In particular, it i eay to ee the behavior of n and z a a function of and n in the cae of α = 2. Thi lead to intereting obervation. For example, it illutrate a connection between the tochatic policy and policie analyzed in the wort-cae model. In particular, Banal, Pruh and Stein [11] howed that, when nothing i known about future arrival, a policy that give peed of the form n = (n/(α 1)) 1/α i contant-competitive, i.e., in the wort cae the total cot i within a contant of. Thi matche the aymptotic behavior of the bound for α = 2 for large n. Thi behavior can alo be oberved for general α (Lemma 7 and Theorem 4). A. Bound on cot We tart the analyi by providing bound on z in thi ubection, and then uing the bound on z to bound n above and below (Section IV-B and IV-C). Recall that z g i the total cot under gated-tatic. Theorem 2. max ( α, α(α 1) (1/α) 1) z z g = G(; α)α 1 G(; α) Proof: The cot z i bounded above by the cot of the gated-tatic policy, which i imply z g = G(; α) G(; α)α 1. (15) Two lower bound can be obtained a follow. In order to maintain tability, the time-average peed mut atify E[] ρ. But z > E[ α ]/β (E[]) α /β by Jenen inequality and the convexity of ( ) α. Thu z > E[α ] β ρα β = α. (16) For mall load, thi bound i quite looe. Another bound come from conidering the minimum cot of proceing a ingle job of ize X, with no waiting time or proceor haring. It i to erve the job at a contant rate [2]. Thu z λ E X [ min ( X α β X )]. The right hand ide i minimized for = (β/(α 1)) 1/α independent of X, giving z ρβ 1/α α(α 1) (1/α) 1. Thu ( z max α, α(α 1) (1/α) 1). (17) The form of the bound on z are complicated, o it i ueful to look at the particular cae of α = 2. Corollary 3. For α = 2, gated-tatic ha cot within a factor of 2 of. Specifically, max( 2, 2) z z g = 2 2. (18) It i perhap urpriing that uch an idealized verion of dynamic peed caling provide uch a mall magnitude of improvement over a implitic policy uch a gated-tatic. In fact, the bound of 2 i very looe when i large or mall. Further, empirically, the maximum ratio for typical α are below 1.1 (ee Figure 2). Thu there i little to be gained by dynamic caling in term of mean cot. However, Section VI how that dynamic caling dramatically improve robutne. A econd intereting obervation about Corollary 3 i that the expected repone time under thee power aware cheme remain bounded a the arrival rate λ grow. Specifically, by (16), E[T ] = z λ E[2 /β] λ 2 µ β. Thi i a marked contrat to the tandard M/GI/1 queue.

4 4 For any α, TABLE I BOUNDS ON TOTAL COSTS AND SPEED AS A FUNCTION OF THE NUMBER n 1 OF JOBS IN THE SYSTEM. ( max α, α(α 1) (1/α) 1) z σ n G(; α) G(; α)α 1 Theorem 2 (11) ( ( 1 n σ α α min α )) 1/(α 1) σ> (σ ) (σ ) 2 Theorem 8 and 4 (12) n β 1/α where σ n atifie σn α 1 ((α 1)σ n α) n (/(G(; α) ) G(; α) α 1 For α = 2, max ( 2, 2 ) z 2 2 Corollary 3 (13) n 2 n ( n min β 2n, 3 ( ) ) 1/3 Corollarie 9 and 5 (14) 2 4 For α = 2 and n < 2, a lower bound on n reult from linear interpolation between max(/2, 1) at n = 1 and at n = 2. B. Upper bound on the dynamic peed We now move to providing upper bound on the dynamic peed caling cheme. Theorem 4. For all n and α, u n n σα α σ 2 (σ ) 2 (19) for all σ >, whence ( ( n 1 n σ α β 1/α α min α )) 1/(α 1) σ> (σ ) (σ ) 2. (2) In particular, for σ = n 1/α, which i concave in n. u n n (α 1)/α (1 (1 ) α ) 2 (21) Proof: A explained in [2], (6) can be rewritten a [ ] α u n = ρ min n /β n u n1 z. (22) n Unrolling the dynamic program (22) give a joint minimization over all n [ 1 u n = ρ min α n n/β n z n ] 1 [ ] ρ min α n1 n1 /β (n 1) z u n2 n1 i = min ρ ( α i /β i z). (23) i,i n i=n j=n j An upper bound can be found by taking any (poibly ub) choice of ni for i 1, and bounding the z. Taking i = σβ 1/α > for all i n give ( ) j u n min (σ α (n j) z) σ> σ σ j= [ n σ α ] z = min σ> σ (σ ) 2. n Since z α from (17), equation (19) follow. With (1), thi etablihe (2). For n =, (21) hold ince u =. Otherwie, it follow from the inequality σ α = n(1 n 1/α ) α n(1 ) α and the fact that n 2/α 1. By pecializing to the cae when α = 2, we can provide ome intuition for the upper bound on the peed. Factoring the difference of quare in the firt term of (19) yeild one increaing term and two decreaing term. Minimizing pair of thee term give the following upper bound on u n. Corollary 5. For α = 2, n β ( n min 1/α 2n, 3 ( ) ) 1/3. (24) 2 4 C. Lower bound on the dynamic peed Finally, we prove lower bound on the dynamic peed caling cheme. We begin by bounding the peed ued when there i one job in the ytem. The following reult i an immediate conequence of Corollary 3 and (9a). Corollary 6. For α = 2, ( ) max 2, (25) β 2 Oberve that the bound in (25), like thoe in Corollary 3, are eentially tight for both large and mall, but looe for near 1, epecially the lower bound. In conjunction with (21) and (1), the following lemma how that peed choen to perform well in the wort-cae are aymptotically (for large n) in the tochatic model. Lemma 7. For ufficiently large n, ( ) 1/α n n β >. (26) 1/α α 1 The following tighter bound on the peed i obtained by uing u n u n1 and (15). Theorem 8. The caled peed σ n = n/β 1/α atifie ( (α 1)σn α ) n σ α 1 n G(; α) G(; α)α 1.

5 5 rate/qrt(β) tatic leep 1 tatic bound occupancy, n (a) = 1 rate/qrt(β) tatic leep tatic bound occupancy, n (b) = 1 Fig. 1. Rate v n, for α = 2 and different energy-aware-load,. For α = 2, thi become: Corollary 9. For α = 2 and any n 2, n β 1/α n 2. (27) Thi prove that the mode n = min n { n ρ} atifie n 2. By the following lemma, linear interpolation between max(/2, 1) and give a lower bound on n for n < 2. Lemma 1. The equence u n i trictly concave increaing. V. COMPARING STATIC AND DYNAMIC SCHEMES To thi point, we have only provided analytic reult. We now ue numerical experiment to contrat tatic and dynamic cheme. In addition, thee experiment will illutrate the tightne of the bound proven in Section IV on the dynamic peed caling cheme. We will tart by contrating the peed under each of the cheme. Figure 1 compare the dynamic peed with the tatic peed. Note that the bound on the dynamic peed are quite tight, epecially when the number of job in the ytem, n, i large. For reference, the mode of the occupancy ditribution are about 1 and 5, cloe to the point at which the peed matche the tatic peed. Note alo that the rate grow only lowly for n much larger than the typical occupancy. Thi i important ince the range over which DVS i poible i limited [15]. Although the peed of the cheme differ ignificantly from that of gated-tatic, the actual cot are very imilar, a predicted by the remark after Corollary 3. Thi i hown in Figure 2. The bound on the peed are alo very tight, both for large and mall. Part (a) how that the lower bound i looet for intermediate, where the weight given to power and repone time are comparable. Part (b) how that gated-tatic (i.e., the upper bound) ha very cloe to the cot. In addition to comparing the total cot of the cheme, it i important to contrat the mean repone time and mean energy uage. Figure 3 how the breakdown. A reference load of ρ = 3 with delay-averion β = 1 and power caling α = 2 wa compared againt changing ρ for fixed, changing β for fixed ρ and changing α. Note = 3 wa choen to maximize the ratio of z g /z. The econd cenario how that when i held fixed, but the load ρ i reduced and delay-averion cot per job, z/λ tatic leep tatic bound (a) Abolute cot, α = 2 z / z α=1.6 α=2 α= (b) Ratio of cot for gated-tatic to, z g/z. Fig. 2. Cot z v energy-aware-load. i reduced commenurately, the energy conumption become negligible. VI. ROBUST POWER-AWARE DESIGN We have een both analytically and numerically that (idealized) dynamic peed caling only marginally reduce the cot compared to the imple gated-tatic. Thi raie the quetion of whether dynamic caling i worth the complexity. Thi ection illutrate one reaon: robutne. Specifically, dynamic cheme provide ignificantly better performance in the face of burty traffic and mi-etimation of workload. We focu on robutne with repect to the load, ρ. The peed are enitive to ρ, but in reality thi parameter mut be etimated, and will be time-varying. It i eay to ee the problem mi-etimation of ρ caue for tatic peed deign. If the load i not known, then the elected peed mut be atifactory for all poible anticipated load. Conider the cae that it i only known that ρ [ρ, ρ]. Let z(ρ 1 ρ 2 ) denote the expected cot per unit time if the arrival rate i ρ 1, but the peed wa optimized for ρ 2. Then, the robut deign problem i to elect the peed ρ uch that min ρ max ρ [ρ, ρ] z(ρ ρ ). The deign i to proviion for the highet foreeen load, i.e., max ρ [ρ, ρ] z(ρ ρ ) = z( ρ ρ ). However, thi i wateful in the typical cae that the load i le than ρ. The fragility of tatic peed deign i illutrated in Figure 4, which how that when peed i underproviioned, the erver i untable, and when it i overproviioned the deign i wateful. Optimal dynamic caling i not immune to mi-etimation of ρ, ince n i highly dependent on ρ. However, becaue the peed adapt to the queue length, dynamic caling i more robut. Figure 4 how thi improvement. Though the dynamic cheme i more robut than a tatic cheme, robutne can be improved further. Specifically, conider the following peed caling cheme that we term linear. It cale the erver peed in proportion to the queue length, i.e., n /β 1/α = n. Figure 4 how that the linear caling provide ignificantly improved robutne when compared with the dynamic cheme; indeed, the cheme i only for deign with ρ [7, 14]. Further, when ρ i in thi region, the linear caling provide only lightly higher cot than the caling. The price that linear caling pay i that it require very high proceing peed

6 6 Fig. 3. delay or energy (normalized unit) energy repone time = 3 = 3 α = 2 α = 2 ρ = 3 ρ =.3 β = 1 β=.1 Optimal Static leep Static = 3 = 3 α = 2 α = 2 ρ = 3 ρ = 3 β=.1 β = 1 Breakdown of E[T ] and E[ α ], for everal cenario. when the occupancy i high, which may not be upported by the hardware. In addition to the numerical illutration above, we can compare robutne analytically in the cae of α = 2. Theorem 11 how that the cot of the linear cheme i exactly the ame a the cot of the gated-tatic cheme when ρ i known exactly. Thu, the cot of the linear cheme i within a factor of 2 of, even without uing information about ρ. Theorem 11. When α = 2, z g = z lin. Thu, z lin 2z. Theorem 12. Conider a ytem deigned for target load ρ that i operating at load ρ. When α = 2, z lin = ρ2 β 2 ρ (28) β z = z lin ρ ( ) ɛ 2. (29) β β ɛ VII. CONCLUDING REMARKS Speed caling i an important method for reducing energy conumption in computer communication ytem. Intrinically, it trade off the mean repone time and the mean energy conumption, and thi paper provide inight into thi tradeoff uing a tochatic analyi. Specifically, in the M/GI/1 PS model, both bound and aymptotic for the peed caling cheme are provided. Thee bound are tight for mall and large and provide a number of inight, e.g., that the mean repone time i bounded a the load grow under the dynamic peed caling and that the dynamic peed in the tochatic model match (for large n) dynamic peed caling that have been hown to have good wort-cae performance. Surpriingly, the bound alo illutrate that a imple cheme which run at peed when the ytem i idle and ue a tatic rate while the ytem i buy provide performance within a factor of 2 of the dynamic peed caling. However, the value of dynamic peed caling i alo illutrated dynamic peed caling cheme provide ignificantly improved robutne to burty traffic and mi-etimation of workload parameter. Interetingly, the dynamic cheme that optimize the mean cot i no longer when robutne i conidered. In particular, a cheme that cale peed linearly with n can provide ignificantly improved robutne while increaing cot only lightly. cot, z tatic leep linear peed deign ρ Fig. 4. Cot at load ρ = 1, when peed are deigned for deign ρ, uing β = 1, α = 2. REFERENCES [1] J. Baliga, R. Ayre, W. Sorin, K. Hinton, and R. Tucker, Energy conumption in acce network, in IEEE Conf. Optical Fiber Communication (OFC), Feb. 28, pp [2] F. Yao, A. Demer, and S. Shenker, A cheduling model for reduced CPU energy, in Proc. IEEE Symp. Foundation of Computer Science (FOCS), 1995, pp [3] N. Banal, T. Kimbrel, and K. Pruh, Speed caling to manage energy and temperature, J. ACM, vol. 54, no. 1, pp. 1 39, Mar. 27. [4] Y. Zhu and F. Mueller, Feedback EDF cheduling of real-time tak exploiting dynamic voltage caling, Real Time Sytem, vol. 31, pp , Dec. 25. [5] L. Yuan and G. 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