Ben Juurlink, Cor Meenderinck Amdahl's law for predicting the future of multicores considered harmful

Transcription

1 Poweed by TCPDF ( Ben Juulink, Co Meendeinck Amdahl's law fo pedicting the futue of multicoes consideed hamful Aticle, Postpint vesion This vesion is available at Suggested Citation Juulink, B.; Meendeinck, C.: Amdahl's law fo pedicting the futue of multicoes consideed hamful. - In: ACM SIGARCH Compute Achitectue News. - ISSN: (online), (pint). - (1),. - pp DOI: 1.15/ (Postpint vesion is cited.) Tems of Use ACM, 1. This is the autho's vesion of the wok. It is posted hee by pemission of ACM fo you pesonal use. Not fo edistibution. The definitive vesion was published in ACM SIGARCH Compute Achitectue News, {VOL, ISS, (1)} doid=

2 Amdahl s Law fo Pedicting the Futue of Multicoes Consideed Hamful B.H.H. Juulink Belin Univesity of Technology Belin, Gemany b.juulink@tu-belin.de C.H. Meendeinck IntelliMagic Leiden, The Nethelands co.meendeinck@intellimagic.net ABSTRACT Seveal ecent woks pedict the futue of multicoe systems o identify scalability bottlenecks based on Amdahl s law. Amdahl s law implicitly assumes, howeve, that the poblem size stays constant, but in most cases moe coes ae used to solve lage and moe complex poblems. Thee is a elated law known as Gustafson s law which assumes that untime, not the poblem size, is constant. In othe wods, it is assumedthattheuntimeonpcoesisthesameastheuntime on 1 coe and that the paallel pat of an application scales linealy with the numbe of coes. We apply Gustafson s law to symmetic, asymmetic, and dynamic multicoes and show that this leads to fundamentally diffeent esults than when Amdahl s law is applied. We also genealize Amdahl s and Gustafson s law and study how this quantitatively effects the dimensioning of futue multicoe systems. 1. INTRODUCTION Seveal ecent woks pedict the futue of multicoe systems o identify scalability bottlenecks based on Amdahl s well-known law [1]. Fo example, Hill and Maty [11] extended Amdahl s law with an aea-pefomance model and applied it to symmetic, asymmetic, and dynamic multicoe chips. Thei esults show that obtaining optimal multicoe pefomance equies extacting moe paallelism as well as making sequential coes faste. Basically, sequential coes need to be made elatively lage and hence faste to execute the seial pat of an application faste, since the seial pat will eventually dominate when the numbe of coes inceases. Based on Amdahl s law they also showed that dynamic multicoes that can dynamically combine all esouces to fom one lage, sequentially coes povide the optimimal solution. Based on Hill and Maty s findings, two dynamic multicoe designs wee pesented at ISCA 1: WiDGET [1] and Fowadflow [7]. Othe examples of the use of Amdahl s law include the wok of Eyeman and Eeckhout [], who show that paallel speedup is not only limited by the seial faction but also by synchonization though citical sections, and the wok of Cho and Melhem [], who use Amdahl s law to detemine the optimal pocesso fequencies in the seial and paallel egions with the goal of minimizing the total enegy consumption. TheimplicitassumptioninAmdahl slawaswellastheextensions mentioned above is, howeve, that the poblem size emains constant. As obseved by Gustafson [1], this is vituallynevethecase. Moecoesaeusedtosolvelageand moe complex poblems. Fo example, moe coes ae used to pefom weathe foecasting on a lage aea (e.g., [1]) o fo video decoding with highe esolutions [3]. Futhemoe, fo many applications, when the poblem size scales the paallel pat scales faste than the seial pat. Gustafson theefoe poposed an altenative to Amdahl s law which is now known as Gustafson s law but which he himself attibuted to E. Basis. Gustafson s law is much moe optimistic than Amdahl s law. While Hill and Maty biefly mention Gustafson s law, they state that in thei view multicoe designs should also opeate well unde Amdahl s moe pessimistic assumptions and do not conside it futhe. The main contibution of this wok is two-fold. Fist we genealize Amdahl s and Gustafson s laws by assuming that the paallel faction does not stay constant as in Amdahl s law, no that it gows linealy with the numbe of coes as in Gustafson s law, but something in between (e.g., it is popotional to n, whee n is the numbe of coes). We efe to this equation as the genealized scaled speedup equation (GSSE), since it encompasses both Amdahl s and Gustafson s law by substituting the appopiate application scaling function. Second, we apply Gustafson s law and the GSSE to symmetic, asymmetic, and dynamic multicoes and show that they poduce esults that ae fundamentally diffeent fom the esults obtained by Hill and Maty based on Amdahl s law. Ou esults have seveal impotant implications of which we mention thee. Fist, while Amdahl s law indicates that symmetic multicoe pocessos should consist of fewe but lage and moe poweful coes, Gustafson s law suggests that many tiny coes yield the highest pefomance. The GSSE indicates that fewe, moe poweful coes can delive a pefomance impovement, but the impovement is much smalle and only occus fo smalle paallel factions f than when Amdahl s law is assumed. Second, when the seial faction is only 1%, Amdahl s law implies that one-eighth of the aea of a lage asymmetic multicoe chip should be devoted to a lage, high-pefomance coe, while the emaining aea is devoted to many small coes. Gustafson s law, on the othe hand, indicates that in this case an asymmetic design baely pefoms bette than a symmetic design, while the GSSE indicates that only 3.1% of the aea should be devoted to the lage coe and, even then, the asymmetic design is only 7.% faste (in theoy) than a symmetic design. Thid, unde Amdahl s law the speedup of a dynamic multicoe that can contain up to 5 simple coes is limited to 3.1. Gustafson s law, on the othe hand, shows that a speedup of can be achieved, while the GSSE which assumes

3 T s = 1 T s = +f p 1 coe f 1 coe f p p coes f/p p coes f T p = +f/p T p = 1 Figue 1: Illustation of Amdahl s law. Amdahl s assumes that the nomalized untime on one coe is 1. Figue : Illustation of Gustafson s law. Gustafson assumes that the nomalized untime on p coes is 1. less than pefect application scaling, indicates that still a speedup of 13 can be achieved. This pape is oganized as follows. Section eviews Amdahl s and Gustafson s law and explains thei undelying assumptions. In addition, it pesents the genealized scaled speedup equation that subsumes both Amdahl s and Gustafson s law. Section 3 applies Amdahl s and Gustafson s law and the genealized scaled speedup equation to symmetic, asymmetic, and dynamic multicoe chips using the aea-pefomance model poposed by Hill and Maty, and pesents and discusses the analytical speedup esults. Conclusions ae dawn in Section.. AMDAHL S AND RELATED LAWS Amdahl s law assumes that a faction f of a seial pogam s execution time is pefectly paallelizable with no communication and synchonization ovehead, while the emaining faction s = 1 f is totally sequential. Consequently, if T s is the execution time of the seial pogam, the paallel execution time on p coes, T(p), is given by T(p) = () T s +f T s/p, since only a faction f of the seial pogam s execution time is paallelizable. Speedup is the atio of sequential execution time to paallel execution time, giving S Amdahl = Ts T(p) = T s () T s + f Ts p 1 =. (1) ()+ f p The last expession in this equation goes to 1/() when p goes to infinity. So, fo example, when the seial faction s = 1 f is %, the speedup is limited to 5, no matte how many coes ae employed. Amdahl used this equation to ague fo the validity of the single-pocesso appoach. Amdahl s law is illustated in Figue 1. Amdahl s equation assumes, howeve, that the poblem size does not change when using moe coes to execute the application. In othe wods, the paallelizable faction emains constant, no matte how many coes ae employed. As obseved by Gustafson, this is vey ae. One does not take a fixed-sized poblem and un it on as many coes as possible. In vitually all application domains, moe coes ae used to solve lage and moe complex poblems. Gustafson theefoe agued that it is moe ealistic to assume that untime, not poblem size, is constant. Gustafson s law is illustated in Figue. Gustafson assumes that the nomalized untime on p coes is ()+ f = 1. If ()+f is the untime on p coes, the untime on one coe will be ()+p f. Consequently, the speedup accoding to Gustafson (which he efeed to as scaled speedup) is given by S Gustafson = ()+f p +f = ()+f p. () In this equation, if the seial faction (1 f) is %, the speedup will be. on 1 coes, which is much moe optimistic than the speedup of. pedicted by Amdahl s law. Figue 3, based on [17], illustates the diffeences between Amdahl s and Gustafson s law. Amdahl assumes that the amount of wok that can be paallelized,, is constant and independent of the numbe of coes p. This can be consideed ovely pessimistic. Gustafson assumes that the amount of wok that can be paallelized gows linealy with the numbe of coes p. This, on the othe hand, can be consideed ovely optimistic. Fo example, although video coding is a domain that can take benefit fom multi- and many-coes [3], the esolution and computational equiements will not gow indefinitely. To addess the limitations of Amdahl s and Gustafson s law, we popose an equation that is somewhee in between. In this equation the amount of wok that can be paallelized is not constant as in Amdahl s law, no does it gow linealy with the numbe of coes as in Gustafson s law. Instead, the amount of paallel wok is popotional to a function scale(p) that is sub-linea in p (e.g., scale(p) = p). Consequently, the nomalized execution time on p coes is given by ()+ f scale(p). p If this equation gives the nomalized untime on p coes, the untime on one coe will be ()+f scale(p). The speedup is theefoe given by S Geneal = ()+f scale(p). (3) ()+ f scale(p) p Note that when scale(p) = 1, this equation is identical to Amdahl s law, and when scale(p) = p, it is identical to Gustafson s law. The pecise scaling function is, of couse,

4 Amount of wok Amount of wok W s W s W s W s W s Numbe of coes p Numbe of coes p Execution time T p T s T p T s (a) Amdahl s assumption. W s Ws W s W s W s Execution time T p T s T p T p Ts T s T s (b) Gustafson s assumption. Numbe of coes p T p T p T p T p T p Ts Ts Ts Ts Numbe of coes p Figue 3: Amdahl s and Gustafson s assumption. Amdahl assumes the input size (o amount of wok) to be constant, while Gustafson assumes it to be dependent on N. 1 coe p coes T s = +f scale(p) f scale(p) p T p = + f scale(p) p f scale(p) Figue : Illustation of the genealized scaled speedup equation. application dependent. We efe to this equation as the genealized scaled speedup equation (GSSE). The GSSE is illustated in Figue. Figue 5 plots the speedups given by Amdahl s law, Gustafson s law, and the GSSE fo f =.5 and f =.9 (whee we assume that scale(p) = p, which will be done thoughout this aticle). While Amdahl s law indicates that the maximum speedups that can be achieved ae and 1 fo f =.5 and f =.9, espectively, the speedups on 1 coes obtained using Gustafson s law ae 5.5 and 9.1, and the speedups on 1 coes calculated using the GSSE ae 1 and 7.9. We note, howeve, that it is misleading to plot these functions in a single figue, since the undelying assumptions diffe. Fo example, a value of f of.5 on 1 coes in Gustafson s law implies that the paallel faction on a single coe is 1 times as lage as the seial faction. This coesponds to a paallel faction f of 1/11 = 99% in Amdahl s law. Speedup Numbe of coes n Amdahl, f =.5 Amdahl, f =.9 Gustafson, f =.5 Gustafson, f =.9 GSSE, f =.5 GSSE, f =.9 Figue 5: Speedup as a function of the numbe of coes fo f =.5 and f =.9 assuming Amdahl s law, Gustafson s law, and the GSSE. 3. IMPLICATIONS FOR MULTICORE DE- SIGN Hill and Maty [11] used Amdahl s law to make assetions about the oganization of multicoe chips. In paticula, should a multicoe chip consist of many small and simple coes, a few lage, high-pefomance coes, o some mixtue of both? To do so, an aea-pefomance model is needed to estimate the numbe of coes a chip can contain, and the pefomance of a coe as a function of its size (in numbe of tansistos). Like Hill and Maty, we assume that a multicoe chip of a given size implemented in a given technology node can contain n Base Coe Equivalents (BCEs) and each BCE has a pefomance of 1. Futhemoe, a coe with an aea of BCEs has a pefomance of pef(), whee pef() is between 1 and. pef() can be an abitay function, but like Hill and Maty, in all speedup gaphs we assume pef() =, which coesponds to Pollack s / Boka s ule [15, ]. 3.1 Symmetic Multicoes In a symmetic multicoe all coes have the same size and pefomance. So a symmetic multicoe chip of n BCEs can contain n/ coes of BCEs each, and the pefomance of each coe is pef(). Unde Hill and Maty s colloay of Amdahl s law, the speedup of a symmetic multicoe ove a single-bce coe is a function of the faction that is paallelizable (f), the chip aea in BCEs (n), and the size of each coe in BCEs (). The seial pat is executed sequentially by one coe at pefomance pef(), and the paallel pat f is executed in paallel by all n/ coes, each with a pefomance of pef(). Using this easoning Hill and Maty obtained the following equation fo symmetic multicoes: S Amdahl symmetic(f, n, ) = = 1 + f pef() pef() n/ 1 + f pef() pef() n. ()

5 We efe to this equation as Amdahl s law fo symmetic multicoes. The same easoning can be applied to Gustafson s law. We use one coe to execute the seial pat 1 f and all n/ coes to execute the paallel pat. The paallel pat, howeve, is now lage. The unnomalized paallel faction on a single coe is not f but f n, since the speedup is elative to a 1-BCE coe. Consequently, we obtain: S Gustafson symmetic (f,n,) = +f n = + f pef() pef() ( +f n) pef(). (5) +f This equation will be efeed to as Gustafson s law fo symmetic multicoes. Similaly, fo the GSSE we obtain S Geneal symmetic(f,n,) = = +f scale(n) + f scale(n) pef() n/ pef() ( +f scale(n)) pef().() + f scale(n) n Again we obseve that if scale(n) = 1, this equation is identical to Amdahl s law fo symmetic multicoes (Eq. ()), and when scale(n) = n, it is identical to Gustafson s law fo symmetic multicoes (Eq. (5)). Figue (a) depicts the esults obtained using Amdahl s law fo symmetic multicoes fo n = 1 and as a function of the size of each coe. The esults obtained using Gustafson s law and the GSSE fo symmetic multicoes ae shown in Figue (b) and Figue (c), espectively. In all figues we assume that pef() = and that scale(n) = n. The esults indicate that when application scaling is consideed leads to fundamentally diffeent esults than when Amdahl s law is applied. Fo example, Amdahl s law fo symmetic multicoes indicates that using 1 coes of 1 BCE each is not necessaily the optimal solution, depending on the paallel faction f. When f =.5, the optimal solution unde Amdahl s law is to use a single lage coe that occupies all chip esouces. In fact, it can be shown analytically that when f.5, a single lage coe will always achieve the highest pefomance unde Amdahl s law fo symmetic multicoes, independent of n, since n. Even when f =.9, the optimal solution is not a single lage coe, but coes of BCEs each. On the othe hand, Gustafson s law fo symmetic multicoes indicates that 1 1-BCE coes achieve the highest pefomance fo each value of f.5. Thus, unde the assumption that the paallel faction scales linealy with the chip aea n, many simple coes is the best appoach. The impession changes slightly when non-pefect application scalability is assumed(figue (c)). Fo f.9, many simple coes is still the best appoach, but fo f =.5, coes of BCEs each pefom slighlty bette. Howeve, the pefomance diffeence between the optimal oganization and 1 coes of 1 BCE each is much smalle when the GSSE is assumed (with scale(n) = n) than suggested by Amdahl s law fo symmetic multicoes (1.5x vesus.15x). Based on Amdahl s law fo symmetic multicoes, Hill and Maty concluded that eseaches should seek methods of inceasing coe pefomance even at high cost. While we agee with this conclusion, this conclusion cannot be dawn solely on the basis of Amdahl s law. Rathe, it is likely that Symmetic speedup Symmetic speedup Symmetic speedup (a) Amdahl, n = (b) Gustafson, n = (c) GSSE, n = 1. f =.999 f =.99 f =.9 f =.5 f =.999 f =.99 f =.9 f =.5 f =.999 f =.99 f =.9 f =.5 Figue : Speedup of symmetic multicoes of n = 1 BCEs ove a single-bce coe assuming Amdahl s law, Gustafson s law, and the GSSE. thee will be abundant single-theaded legacy codes fo a few decades to come. An impotant question is how these esults change when Mooe s law allows many moe BCEs pe chip. To answe this question, Figue 7 depicts the esults fo n = 5 BCEs. While the pecise esults ae slightly diffeent, in geneal the same conclusions can be dawn. While Amdahl s law fo symmetic multicoes indicates that 5 single-bce coes neve yield the highest pefomance (unless when f =.999), Gustafson s law suggests the opposite. Futhemoe, assuming less that pefect scaling of the paallel pat, the GSSE shows that now non-single BCE coes can povide a pefomance advantage, not only fo f =.5 but also fo f =.9, but fo f =.5 the pefomance advantage is much smalle than pedicted by Amdahl s law(.13x vesus.3x),

6 Symmetic speedup Symmetic speedup Symmetic speedup (a) Amdahl, n = 5. (b) Gustafson, n = 5. (c) GSSE, n = 5. f =.999 f =.99 f =.9 f =.5 f =.999 f =.99 f =.9 f =.5 f =.999 f =.99 f =.9 f =.5 Figue 7: Speedup of symmetic multicoes of n = 5 BCEs ove a single-bce coe assuming Amdahl s law, Gustafson s law, and the GSSE. and fo f =.9 the pefomance advantage is eally mino (less than 1.x). It is questionable, howeve, if lage symmetic multicoes should be oganized in such a way as to obtain optimal pefomance fo applications with a paallel faction f of Asymmetic Multicoes In an asymmetic multicoe, one o moe coes ae lage and moe poweful than the othes. Asymmetic multicoes ae also called pefomance heteogeneous multicoes, since all coes implement the same instuction set achitectue (ISA) but at diffeent pefomance levels. Besides pefomance heteogeneous, thee ae functionally heteogeneous multicoes, whee diffeent coes suppot diffeent ISAs. An example of an asymmetic multicoe is the single-isa heteogeneous multicoe poposed by Kuma et al. [13]. Examples of functionally heteogeneous multicoes in industy and academia ae the Cell pocesso [9] and the SARC achitectue [1]. Amdahl s law makes a case fo asymmetic multicoes with one lage coe to acceleate the seial pat of the execution time of an application, while many small coes ae used to execute the paallel pat. In this section we investigate if the same holds unde the assumptions of Gustafson s law and the GSSE. Unde Amdahl s law fo asymmetic multicoes, the seial pat is executed by one lage coe of size BCEs and pefomance pef(). The paallel pat, on the othe hand, is executed by both the n single-bce coes and the lage coe. Oveall, this yields S Amdahl asymmetic(f, n, ) = pef() + 1 f pef()+n. (7) Note that this equation assumes that pefect dynamic scheduling is pefomed, i.e, the lage coe pefoms a lage pat of the paallel faction than the single-bce coes. Similaly, we obtain and S Gustafson asymmetic (f,n,) = +f n + f n pef() pef()+n Sasymmetic(f,n,) Geneal +f scale(n) = + f scale(n) pef() pef()+n These equations will be efeed to as Gustafson s law and the GSSE fo asymmetic multicoes, espectively. Figue depicts the speedup attained by asymmetic multicoes ove a single-bce coe fo n = 1 BCEs, while Figue 9 depicts the speedup cuves fo n = 5. Oveall, the esults indicate that asymmetic multicoes indeed povide a pefomance advantage compaed to symmetic multicoes, but when application scaling is consideed, the pefomance advantage is smalle, occus fo smalle values of f, and the optimal size of the lage coe is smalle than unde Amdahl s law fo assymetic multicoes. Fo example, when n = 1 and assuming pefect application scaling (Gustafson s law), Figue (b) shows that asymmetic multicoes only povide a pefomance advantage fo f =.5 and in that case the pefomance benefit (compaed to when = 1) is limited to 1.x. When non-pefect application scaling is assumed, Figue (c) shows that asymmetic multicoes also impove pefomance fo f =.9, but the pefomance benefit is small (1.x). When f =.5, the pefomance benefit pedicted by the GSSE fo asymmetic multicoes is lage, but smalle than pedicted by Amdahl s law fo asymmetic multicoes (1.73x vesus.3x). When n = 5, the speedup benefits of asymmetic multicoes incease, but still the conclusions ae diffeent when application scaling is consideed. As depicted in Figue 9(b), Gustafson s law indicates that asymmetic designs still only povide a substantial pefomance impovement fo f =.5, while Amdahl s law suggests that they povide substantial impovements fo any value of f. The GSSE, on the othe hand, indicates that asymmetic multicoes of n = 5 BCEs also yield substantially highe pefomance fo f =.9, but unlike Amdahl s law fo asymmetic multicoes, not fo f =.99. The equations that assume application scaling also suggest that the optimal size of the lage coe is smalle () (9)

7 1 5 Asymmetic speedup 1 1 f =.999 f =.99 f =.9 f =.5 Asymmetic speedup f =.999 f =.99 f =.9 f = (a) Amdahl, n = 1. (a) Amdahl, n = Asymmetic speedup 1 1 f =.999 f =.99 f =.9 f =.5 Asymmetic speedup f =.999 f =.99 f =.9 f = (b) Gustafson, n = 1. (b) Gustafson, n = Asymmetic speedup 1 1 f =.999 f =.99 f =.9 f =.5 Asymmetic speedup f =.999 f =.99 f =.9 f = (c) GSSE, n = 1. (c) GSSE, n = 5. Figue : Speedup of asymmetic multicoes assuming Amdahl s law, Gustafson s law, and the GSSE fo n = 1 BCEs. Figue 9: Speedup of asymmetic multicoes assuming Amdahl s law, Gustafson s law, and the GSSE fo n = 5 BCEs. than the size pedicted by Amdahl s law. Fo example, when f =.5, Amdahl s law indicates that half the chip esouces (1 BCEs) should be devoted to the lage coe and still the speedup is limited to.9. Gustafson s law (Figue 9(b)), on the othe hand, indicates that when the lage coe is 1 BCEs (.3% of the chip esouces), nea optimal pefomance and a speedup of 19 ae achieved. This is a much moe optimistic esult, since it is questionable if a coe that is 1 times lage than a base coe can and should be built with a pefomance that is 11.3 times as high. It is even moe questionable if the chip aea should be statically divided such that optimal pefomance is achieved fo pooly scalable applications with a seial faction as lage as.5, especially consideing that such a static division will hut the pefomance of applications with lage paallel factions. Unde the GSSE when f =.9, an asymmetic multicoe with a lage coe of 1 BCEs pefoms slightly wose (1.5x) than the optimal design (lage coe of 3 BCEs). 3.3 Dynamic Multicoes In a dynamic multicoe, it is assumed that up to coes can be tempoaily aggegated to acceleate the sequential components of an application. Duing the paallel phases, the esouces ae divided into n 1-BCE coes again to attain maximum speedup duing the paallel phases. As indicated by Hill and Maty, helpe theads conceptually boost the pefomance of a single coe, since the helpe theads may e.g. pefetch data needed by the sequential main thead. Futhemoe, two dynamic multicoe designs wee pesented at ISCA 1: WiDGET [1] and Fowadflow [7].

8 Amdahl s law suggests that we should use a lage dynamic coe of BCEs duing the seial phases and n single-bce coes duing the paallel phases. The speedup achieved by such a dynamic multicoe pocesso ove a single-bce coe is given by: S Amdahl dynamic(f,n,) = 1 +. (1) f pef() n Gustafson s law and the GSSE fo dynamic multicoes ae obtained similaly: and S Gustafson dynamic +f n (f,n,) = = S Geneal dynamic(f,n,) = + f n pef() n +f n pef() +f (11) +f scale(n). (1) + f scale(n) pef() n Figue 1 (fo n = 1 BCEs) and Figue 11 (fo n = 5 BCEs) depict the speedups calculated using these equations as a function of the size (in BCEs) of the lage dynamic coe. While these figues display the esults fo up to a lage dynamic coe of n BCEs, pactical consideations might keep much smalle than n. Obviously, in all cases the speedup inceases with the size of the lage dynamic coe. But simila to asymmetic designs, the advantage of a dynamic multicoe is moe ponounced unde Amdahl s law than unde Gustafson s law and the GSSE. Fo example, when n = 1, Gustafson s law and the GSSE indicate that dynamic multicoes only povide a significant (moe than %) pefomance impovement when f =.5. A lage analytical pefomance impovement is impotant, since a dynamic multicoe natually incus a highe ovehead than asymmetic designs, as additional data paths ae needed to be able to aggegate seveal coes. Even when n = 5, if pefect application scaling is assumed (Figue 11(b)), dynamic multicoes only povide a significant pefomance impovement fo f =.5. When less than pefect application scaling is assumed (Figue 11(c)), dynamic multicoes also povide a significant advantage fo f =.9, but if the dynamic lage coe consists of 1 BCEs with a pefomance that is times highe than that of a single BCE, a speedup of 17.5 is attained (vesus 3 fo = 5), which is much bette than the speedup of 35.1 fo = 1 (vesus 1 fo = 5) achieved unde Amdahl s assumptions. This also has a positive implication, since it is questionable if a dynamic lage coe of 5 BCEs with a pefomance that is 1 times highe than that of a single BCE can be constucted. If we optimistically assume that it takes one cycle to oute a signal though one BCE and that the BCEs have to be layed out in D space, it takes at least 1 cycles to oute an opeation fom the middle of the chip to a functional unit of a coe at the cone. It is doubtful if such lage opeation latencies can be completely hidden. Figue 1 depicts the speedup of the optimal dynamic design (with a dynamic lage coe of = n BCEs) ove the optimal asymmetic design. It is easy to see that this can be at most, since we can devote half of the asymmetic multicoe chip to the lage coe and the othe half to n/ small (single-bce) coes. Not supisingly, unde all thee scaling equations, dynamic multicoes povide a pefomance advantage ove asymmetic designs (by at most 1.3x). It emains to be seen, howeve, if this is achievable in pactice Dynamic speedup Dynamic speedup Dynamic speedup (a) Amdahl, n = (b) Gustafson, n = (c) GSSE, n = 1. f =.999 f =.99 f =.9 f =.5 f =.999 f =.99 f =.9 f =.5 f =.999 f =.99 f =.9 f =.5 Figue 1: Speedup of dynamic multicoes assuming Amdahl s law, Gustafson s law, and the GSSE fo n = 1 BCEs. since dynamic multicoes natually incu a highe ovehead than asymmetic designs. Unde Gustafson s law, howeve, dynamic multicoes only povide a significant pefomance advantage if f =.5. Somewhat supisingly, the lagest impovement is obtained unde the GSSE (fo f =.5 and n = 5) Howeve, it needs to be kept in mind that this is fo a dynamic lage coe of = 5 BCEs, while pactical consideations might keep much smalle than its maximum of n. If we limit the size of the lage coe to 1 BCEs, the speedup of the optimal dynamic design ove the optimal asymmetic design is less than 1.1 when the GSSE is assumed. Futhemoe, the pefomance advantage of dynamic multicoes ove asymmetic designs diminishes when f inceases unde Gustafson s law and the GSSE, while unde Amdahl s law this is not necessaily the case.

9 Dynamic speedup (a) Amdahl, n = 5. f =.999 f =.99 f =.9 f =.5 Speedup dynamic ove assymmetic Amdahl Gustafson GSSE f=.5 f=.9 f=.99 (a) n = 1 BCEs. 5 Dynamic speedup (b) Gustafson, n = 5. f =.999 f =.99 f =.9 f =.5 Speedup dynamic ove assymmetic Amdahl Gustafson GSSE f=.5 f=.9 f=.99 5 (b) n = 5 BCEs. Dynamic speedup f =.999 f =.99 f =.9 f =.5 Figue 1: Speedup of optimal dynamic design ove optimal asymmetic design assuming Amdahl s law, Gustafson s law, and the GSSE fo n = 1 and n = 5 BCEs. (c) GSSE, n = 5. Figue 11: Speedup of dynamic multicoes assuming Amdahl s law, Gustafson s law, and the GSSE fo n = 5 BCEs.. CONCLUSIONS The main contibution of this pape is two-fold. Fist, we have pesented a genealized scaled speedup equation (GSSE) that encompasses both Almdahl s and Gustafson s law by substituting the appopiate application scaling function. Second, we have applied Amdahl s and Gustafson s law and the genealize scaled speedup equation to the aeapefomance model developed by Hill and Maty, and showed that substantially diffeent esults ae obtained. While Amdahl s law makes a stong case fo asymmetic and dynamic multicoes, Gustafson s law and the GSSE show that asymmetic and dynamic multicoes can still povide a pefomance advantage ove symmetic multicoes, but much less so than unde Amdahl s assumptions. The point of this pape is not to question the contibution of Hill and Maty. On the contay, we thank them fo stating a stimulating discussion (see Acknowledgment). Futhemoe, unde the assumption that futue multicoe will be used to acceleate fixed-size applications, thei conclusions still hold. The main point of this pape is, howeve, that one has to conside the scaling popeties of the tageted applications. One cannot simply take Amdahl s law and use it to detemine the oganization of next-geneation multicoes. Moeove, it seems unlikely that multicoes should be oganized such that optimal pefomance is achieved fo paallel applications with a seial faction as lage as.5. It also seems unlikely that geneal-pupose multicoe pocessoswillbetime-shaedandthusatanytimeexecuteasingle application. It is moe likely that they will be space-shaed and time-shaed between seveal applications. Such a scenaio indicates that a design with a few (but not one) lage (but not huge) coes and seveal (but not too many) small coes will povide optimal thoughput. The few lage coes will be used to execute single-theaded applications and applications with lage seial factions as well as to acceleate the seial phases of applications with modeate seial factions. The seveal small coes will be used to execute the paallel phases of seveal applications in a time-shaed manne.

10 Many possibilities fo futue wok exist. Like Hill and Maty, we have assumed that the pefomance is limited by aea. One could also conside, fo example, powe constaints [], pin count constaints, as well as Themal Design Powe, aea/pefomance, powe/pefomance, ITRS scaling factos, memoy bandwidth, wokload behavio, etc.[5]. One could conside a multi-application scenaio as descibed above and analyze how this affects the esults. Anothe possibility would be to detemine how the paallel faction scales with the poblem size fo typical applications. We note that this would be somewhat eminiscent to iso-efficiency analysis []. Iso-efficiency analysis, howeve, can only be applied to elatively simple and well-undestood paallel algoithms and achitectues. The simplicity of Amdahl s law, Gustafson s law, and the GSSE is both thei stength as well as thei weakness. 5. ACKNOWLEDGMENTS Thanks to Mak Hill and Michael Maty fo challenging us to develop a bette model. Thanks to Pete Hofstee fo suggesting the title. Thanks to Chi Ching Chi fo making some of the figues.. REFERENCES [1] G. Amdahl. Validity of the Single Pocesso Appoach to Achieving Lage Scale Computing Capabilities. AFIPS Confeence Poceedings, 3():3 5, 197. [] S. Boka. Getting Gigascale Chips: Challenges and Oppotunities in Continuing Mooe s Law. ACM Queue, 1: 33, Octobe 3. [3] C. C. Chi and B. Juulink. A QHD-Capable Paallel H. Decode. In Poc. Int. Conf. on Supecomputing, ICS 11, 11. [] S. Cho and R. Melhem. Coollaies to Amdahl s Law fo Enegy. IEEE Compute Achitectue Lettes, 1, 7. [5] H. Esmaeilzadeh, E. Blem, R. St. Amant, K. Sankaalingam, and D. Buge. Dak silicon and the end of multicoe scaling. SIGARCH Comput. Achit. News, 39(3):35 37, June 11. [] S. Eyeman and L. Eeckhout. Modeling Citical Sections in Amdahl s Law and its Implications fo Multicoe Design. SIGARCH Comput. Achit. News, 3:3 37, June 1. [7] D. Gibson and D. A. Wood. Fowadflow: a scalable coe fo powe-constained CMPs. In Poc. 37th annual Int. Symp. on Compute Achitectue, ISCA 1, 1. [] A. Y. Gama, A. Gupta, and V. Kuma. Isoefficiency: Measuing the Scalability of Paallel Algoithms and Achitectues. IEEE Concuency, pages 1 1, August [9] M. Gschwind, H. P. Hofstee, B. K. Flachs, M. Hopkins, Y. Watanabe, and T. Yamazaki. Synegistic Pocessing in Cell s Multicoe Achitectue. IEEE Mico, ():1,. [1] J. Gustafson. Reevaluating Amdahl s Law. Communications of the ACM, 31(5):53 533, 19. [11] M. D. Hill and M. R. Maty. Amdahl s Law in the Multicoe Ea. IEEE Compute, 1(7):33 3,. [1] T. Kauanne. Intoducing Paallel Computes into Opeational Weathe Foecasting. PhD thesis, Lappeenanta Univesity of Technology,. [13] R. Kuma, D. M. Tullsen, P. Ranganathan, N. P. Jouppi, and K. I. Fakas. Single-ISA Heteogeneous Multi-Coe Achitectues fo Multitheaded Wokload Pefomance. In Poc. 31st Int. Symp. on Compute Achitectue, ISCA, Washington, DC, USA,. IEEE Compute Society. [] C. Meendeinck and B. Juulink. (When) Will CMPs hit the Powe Wall? In Poceedings of the Euo-Pa Wokshops (HPPC), August. [15] F. J. Pollack. New micoachitectue challenges in the coming geneations of CMOS pocess technologies (keynote addess)(abstact only). In Poc. 3nd annual ACM/IEEE Int. Symp. on Micoachitectue, MICRO 3, [1] A. Ramiez, F. Cabacas, B. Juulink, M. A. Mesa, F. Sanchez, A. Azevedo, C. Meendeinck, C. Ciobanu, S. Isaza, and G. Gaydadjiev. The SARC Achitectue. IEEE Mico, pages 1 9, Sept./Oct. 1. [17] X. Sun and L. Ni. Anothe View on Paallel Speedup. Poc. of Supecomputing 9, pages 3 333, 199. [1] Y. Watanabe, J. D. Davis, and D. A. Wood. WiDGET: Wisconsin Decoupled Gid Execution Tiles. In Poc. 37th annual Int. Symp. on Compute Achitectue, ISCA 1, pages 13, 1.