Low-Complexity Policies for Energy-Performance Tradeoff in Chip-Multi-Processors

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1 Low-Comlexity Policies for Energy-Performance Tradeoff in Chi-Multi-Processors Avshalom Elyada, Ran Ginosar, Uri Weiser Det. Of Electrical Engineering Technion, Israel Institute of Technology Haifa 3000, Israel Abstract Chi-Multi-Processors (CMP) utilize multile energy-efficient Processing Elements (PEs) to deliver high erformance while maintaining an efficient ratio of erformance to energy-consumtion. In order to utilize CMP resources, the software alication is slit into multile tasks that are executed in arallel on the PEs. Dynamic frequency-voltage Scaling (DVS) balances erformance and energy consumtion by dynamically varying a PE's frequency-voltage workoint in order to save energy while meeting erformance requirements. This work addresses DVS olicies for CMP. We consider multi-task CMP alications with unknown workloads. We dynamically set frequency-voltage workoints for each PE in the CMP, attemting to minimize a defined energy-erformance criterion.. Other DVS methods tyically use high comlexity otimization techniques, which limits the ossibility of real-time imlementation in erformance-driven, energy-aware systems. In contrast, we investigate simle heuristic DVS olicies for simlified serial/arallel task-grahs. We comare the results of our olices to a theoretical best-case solution and show that these lightweight heuristics achieve good results with low comlexity. In most cases the very simle Constant olicy is the most cost-effective.. Introduction Chi-Multi-Processors (CMP) achieve high erformance while maintaining an accetable ratio of erformance to energy consumtion, in comarison to traditional single-core architectures. Performamce imrovement techniques of single-core architectures, mainly include () taking advantage of shrinking gatedelays in order to increase the oerating frequency, and () using the increased transistor density to add erformance-enhancing microarchitecture features [],. Increasing the oeration frequency beyond a certain oint is energy inefficient, since energy consumtion is roughly quadratic in frequency, Features such as large caches, dee execution ies and comlex branch redictors yield a decreasing erformance/energy return []. High energy consumtion shortens the battery life of mobile devices, and may cause ower delivery and heat disersion roblems, which consequently limit feasible frequencies and erformance. CMP architectures, on the other hand, integrate multile relatively small and simle PEs, otentially enabling linear scaling of erformance [3]. Dynamic frequency-voltage Scaling (DVS) is a widely racticed [4] and researched [5-7] technique for energy-erformance tradeoff. When using DVS in CMP, a PE s frequency is altered dynamically to meet current erformance requirementsm while consuming no more energy than is necessary. PE suly voltage is also adjusted in conjunction with frequency; usually ket at the lowest feasible value that still enables circuit oeration and timing at the current frequency. Scaling the frequency-voltage workoint (f,v) can result in near quadratic energy savings [6]. In a CMP running multile deendent tasks, DVS may save energy without degrading erformance. Tyically, at any given time, one of the tasks constitutes the erformance bottleneck. Other PEs can be slowed down, saving energy without affecting total erformance. We refer to the tactic of slowing down non-critical tasks as slack-utilization.

2 When all task workloads are known in advance, the DVS olicy sets PE frequencies to utilize recisely all available time-slacks and thus save the maximum ossible energy without affecting erformance. Tyically however, task workloads are unknown in advance and efficient slack utilization is non-trivial. A DVS olicy which assumes worst-case workloads achieves limited energy savings, since aggressive workoints need to be set to maintain required erformance in the worst case. Conversely, overestimating slack can lead to erformance degradation. When workloads are unknown they must be estimated, and the method of estimation is a rimary factor of the efficiency of DVS olicies. Selecting a criterion for DVS olicy efficiency is not a clear-cut issue, since different DVS-caable CMPs for different alications have different energy- and owererformance requirements. For examle, mobile batteryoerated devices aim at long battery life as well as erformance, while deskto systems and servers tyically otimize ower rather than energy consumtion. Consider a system whose only objective is to maximize erformance. The best olicy would be to run at maximum frequency at all times (in this work we refer to this olicy as the f-max olicy). Conversely, a system which is concerned only with energy minimization should always run at the minimum frequency (f-min olicy). A more interesting and more ractical case is that of a system which strives to balance between the two: tuning u frequencies only to the oint where the increased energy consumtion is deemed justifiable, and likewise saving energy by running slower, but only to the oint considered as tolerable erformance. This choice is reflected in the criterion selected to assess alternative DVS olicies, as discussed in Section.3. DVS olicies vary in comutational comlexuty. If the required calculations for olicy imlementation are to be integrated into the system itself and done in realtime rather than offline and externally, then it is essential that the energy-consumtion and delay of the calculation itself be minimal. Sending a substantial amount of execution time and energy just to calculate each workoint may considerably offset the savings aimed at, making it imractical for use in a erformance- and energy-aware system. Thus, a ractical slack-utilization comutation must weigh savings accomlished versus its own comlexity. Previously ublished DVS olicies for CMP formulate recise otimization roblems [8-0] seeking otimal solutions, tyically at the cost of high comutational comlexity. In contrast, we introduce lightweight heuristic DVS-olices, and show that they achieve good results comared to theoretical bounds. We show further that in most cases the simlest olicy is the most cost-effective. This aer is organized as follows. Section defines the minimization criterion and formulates the DVS minimization roblem. The various lightweight huristic DVS olicies are resented in Section 3, and analyzed in Section 4. Summary and conclusions are drawn in Section 5.. Definitions and Problem Formulation Consider a CMP latform running a multi-task alication. At any given time, each PE accommodates one software task. Each PE is indeendently DVScaable, i.e., the workoint of each PE is controlled indeendently of the other PEs. A DVS olicy dynamically assigns (f,v) workoints to each PE, in order to minimize a secified erformance and energyconsumtion criterion. For simlicity, all PEs are identical, although these results can be generalized to heterogeneous-pe systems [3,, ] with few modifications... DVS Hardware Model DVS-Caable PE Model We assume that each PE in the system is caable of oerating at a clock frequency within the range cycles f [ f, f ] min max sec, and may also be in a standby mode. At any frequency, the PE oerates at the minimum feasible suly voltage, defining a frequencyvoltage (f,v) oeration curve (see [3] for details). ote that for simlicity we emloy a continuous frequency model, while tyical rocessors oerate at only a finite set of discrete frequencies. The rate of changing the frequency is also limited in ractice due to transition cost in both energy and erformance, and also comlexity of frequent recalculation of the workoint. It is unrealistic to change the frequency too often, and we assume a small number of frequency changes,, therefore we can neglect the transition overhead. PE Power, Energy, and Execution Time We denote the total ower consumtion of a PE by P f, where f is the PE's current frequency. As ( ) joules sec

3 mentioned above, the oerating frequency imlicitly defines a corresonding suly voltage. We consider P ( 0) to be the standby ower, consumed by the PE when it is not doing any work. We further define the energy consumed er cycle, denoted by e( f ) joules cycle. By definition: P( f ) ( ) e f =. () For a task of unknown workload, the cumulative density function cdfw ( w ) of the workload W is defined as the robability that the task will be comleted within w cycles or less: cdf ( W w ) = Pr ( W w). Hence the robability that the task will take w cycles or more to execute is cdfw ( w ). Some examle distributions are dislayed in Figure below, which shows robability density functions, conventionally defined as dfw( w) = cdf W( w). These distributions are used in our simulations, as described in Section 4. (i) (ii) (iii) (iv) (v) Figure : Workload distributions: Examles on a 6-PE CMP. If the cumulative density and energy-er-cycle functions are known, we can formulate the exected energy required to execute a task of workload W on a PE by the following exression: f E = e f cdf w, () ( )[ ( ) ] w W w= where f w is the frequency at cycle w. Eq. () sums the energy-er-cycle times the robability that the task will still be running at that cycle. Convergence is assured since the task comletes within a finite number of cycles. Suose that the task starts at time t = 0 and comletes by time t = T. If it comletes before that, the PE goes to a standby state until t = T. We can reformulate Eq. () to include the energy consumed when the PE is in standby, by dividing the ower into standby ower P ( 0) and active ower Pact ( f ) = P( f ) P( 0). If we define the corresonding act ( ) active energy-er-cycle ( ) P f eact f = f then total energy can be rewritten as: E = eact ( fw)[ cdfw( w ) ] + T P( 0). (3) w= The energy-er-cycle functions e(f) and e act (f) are comuted from the PE ower function P(f). We used 3 P( f ) = af + b as the PE ower function for our simulations in Section 4. This ower function is justified as an aroximation, since the scaling of the oerating voltage is roortional to the frequency scaling, while dynamic ower consumtion is roortional to frequency times the voltage squared, hence the ower is aroximately roortional to the third ower of frequency. In [3] we show that with correct choice of 3 fitting arameters a and b, P( f ) = af + b is emirically quite close to real ower measurements, although the fitting arameters do not reresent meaningful dynamic or static ower coefficients. Alternatively, any other ower function can be used with the formulations herein, whether in closed or numeric form. To obtain the task execution exected time, T, we sum the delay-er-cycle times the robability that the task will still be executing at that cycle, in a manner similar to Eq. (): cdfw ( w ) T =. (4) f w=.. DVS Alication Model In this study we model the alication as an execution timeline comrsising alternating serial and arallel hases [3], as shown in Figure (a), and focus on DVS olicies for the arallel hases. This simle execution model is aroriate for numerous alications [4, 5] and rogramming models [6]. w 3

4 . Figure : An execution timeline with (a) equal workloads, and (b) unequal workloads. Slack in (b) is utilized in (c) to save energy with no erformance degradation. Reduced frequencies are indicated by thinner lines. Tasks of equal workloads running on identical PEs at the same frequency-voltage work-oint will achieve identical run-times, as shown in Figure (a). A tyical examle is multile task instances erforming the same work in arallel on different, equally sized data-sets (i.e., Data Decomosition [7]). In such symmetrical cases, there is no slack to utilize although the work-oint can be controlled, any energy saved will necessarily come at the exense of erformance. A timeline of unequal workloads, as shown in Figure (b), exhibits slack and thus resents oortunity for energy saving without erformance degradation. Unequal workloads can occur when tasks execute in arallel on different data sizes or when different tyes of tasks execute in arallel (i.e., Functional Decomosition [7]). A timeline such as Figure (b) can also occur in a heterogeneous-pe system [3, ] where the PEs have unequal comutation throughut. Combinations of the factors mentioned above are also ossible. Usually, the actual workload of a task is known only once the task comletes. We must therefore estimate the workload of each task, and emloy this estimation to assign (f,v) workoints. The estimation is based on ast erformance of tasks of the same tye. (a) (b) (c).3. CMP Problem Formulation The total system energy E of a arallel hase is the sum of the energies of all PEs =.., while the combined execution time T of the arallel eriod is the maximum of task execution times: E = E, T = max T. (5) = =.. T is determined by the comletion time of the last task, i.e. the critical task running on the critical PE. ote that it is wasteful for any PE to comlete before the combined execution time T, since it otentially could have run slower and consumed less energy, finishing at time T and causing no erformance degradation. We would like to utilize the slack of non-critical tasks in order to save energy. This notion is illustrated in Figure (c), where the reduced frequencies are indicated by thinner lines. Alternative DVS olicies are judged according to the balance that they manage to achieve between energy and erformance. Different criteria may assign different weight to energy and erformance. In this study we α emloy the criterion of minimal ET, i.e., minimize the roduct of energy E, and execution-time T to some ower α. The exonent ρ is used to control the relative weight of execution time and energy. We refer α=, since ET has the useful characteristic of frequency invariance [8]: ET measures olicy quality regardless of the actual frequencies used. A task's energy consumtion is aroximately roortional to the square of the voltage, which is in turn aroximately roortional to the frequency, hence E f. On the ρ α other hand, T so ET f. In suort of this we f find in our simulations that using α < gives inherent advantage to the f-min olicy, while α > favors the f- max olicy. Further suorting this choice is the fact that any olicy which always uses a single constant frequency (for examle f-min and f-max, see Figure 9) achieves a constant ET measure, regardless of frequency. Otimization A straightforward aroach to finding an otimal olicy is to solve the following minimization roblem: min ET st.. f ( w) 0 [ fmin, fmax ], (6) =.., w=,.. amely, minimize ET while frequencies are constrained to an oerating range, or 0 if the PE is idle. 4

5 Substituting E and T from Eqs. (5) into Eq. (6), and further substituting E and T (the individual PE energy and delay) from Eq. (3) and (4), we note that the ensuing otimization roblem seems very hard, desite the simlifications already assumed in the model. Rigorous analysis of this minimization roblem is beyond the scoe of the resent work. Moreover, we suggest that although this roblem may be solvable, yielding some otimal frequency assignment er set of task distributions, the comutation required is likely to be too great to justify imlementation in any ractical system. In a erformance-driven energy-aware system, and assuming frequency assignment is integrated as art of the system s comutational requirements in real-time, we must weigh the imrovement that a comutation offers versus the cost of the comutation itself, both in energy and erformance. This observation rationalizes the aroach of searching for simle lightweight heuristic DVS olicies. 3. DVS Policies In this section we describe a grou of relatively simle frequency assignment olicies for a CMP. Since all olicies share the same outline and differ only in certain details, we first describe the common outline and then secify the differences for each olicy. 3.. Common Outline for All Policies At the t=0 fork-oint of the arallel hase in the timeline, all olicies erform the same stes: () Find the PE (task) estimated to have the most remaining work, which is referred to as the estimated-critical PE (ECP). () Set a joint-target-time (JTT) for all PEs to comlete their tasks, and assign PE frequencies (and voltages). (3) Run until the ECP finishes (or time interval elased). (4) Udate workload estimations. (5) Loo back to Ste () above. In Ste (): We comute the exected value of the (remaining) estimated work for all PEs (currently running tasks): Wˆ = E[ W W ] (7), rem, com ote that at time t=0, the comleted work W com, = 0. From this estimation we find the ECP, the PE for which the estimated work is maximal: Wˆ ˆ ECP, rem = max{ W, rem} =... (8) ECP = arg max{ Wˆ } =.., rem In Ste (): The ECP is assigned f max. Intuitive reasoning for this is that ET is frequency invariant if we ignore P (0), which means that results deend solely on the amount of utilized slack, while actual PE frequencies are insignificant. However, P (0) cannot be ignored. Therefore, assigning f max to the ECP, i.e., running as fast as ossible, will clearly minimize standby energy consumtion. Otimality of assigning f max to the critical PE is roven in Section 3. for a case of known workloads. To utilize the available time-slack, we want all the other (non-critical) PEs to comlete together with the critical PE. We therefore set a joint-target-time for all PEs, which is the exected comletion time of the ECP: W ECP, rem JTT =, (9) f where W ECP, rem is the estimated (remaining) work of the ECP. Frequency assignment of the other (noncritical) PEs is described er each olicy below. In Ste (3): All PEs run at their assigned frequencies until the ECP comletes. (In case of the Interval olicy described in 3.4, re-estimation and re-assignment are erformed also at intermediate fixed time intervals.) The ECP is exected to comlete last by definition, and this will tyically be the case. However since workloads are statistical it is ossible that the ECP will comlete before other PEs. In Ste (4): If the ECP comletes (or a time-interval elases) while other PEs still have remaining work, reestimation is erformed, taking into account the work done by each PE so far. The cycle is reeated until all PEs have comleted their work. Figure 3 shows a flowchart of the common outline. The secific variations of the olicies are described next. max 5

6 ET d dt W = T E = T W e + P( 0) T T act = = W = T W e + P T + = T ( ET ) act ( 0) W W + T We act P + ( 0) = = T T = T W e W W e W + 3 P(0) T act act = T = T ow if we assume the ower model described in Section., then eact ( x) = ax, e act ( x) = ax and substituting in (0) we find that the two sums cancel out: Figure 3: Flowchart of common outline for all olices 3.. The "Oracle" Policy The Oracle olicy is a non-causal, hyothetical olicy which assumes future knowledge of the workloads. When simulating the Oracle olicy we still generate workloads statistically, however we assume that they are known in advance, before run time. We use the Oracle olicy results as a lower bound for comaring to causal, imlementable olicies in which the exact workloads are not known in advance. Given that the workloads are known, we calculate the otimal frequencies f to assign to each PE =... In articular we show that f max is otimal for the critical PE. We denote the known task workloads on each PE W, and define T to be the joint execution time for all PEs. Because the workload of each PE is known and the execution time is set, energy can be minimized by running at a constant frequency that is just fast enough to finish the task by its deadline. This is a well established result which is due to the convexity of e(f), the energyer-cycle [9]. Therefore we assign constant frequencies W f = to each PE. We initially assume that T frequencies are not restricted to any range, and later incororate frequency bounds. All PEs comlete exactly at T, so that full slack utilization is attained. Assuming that there exists a T as above and it minimizes ET, that T can be found by differentiating ET w.r.t. T and equating the derivative to 0: W W = T Wa W ia + 3 P(0) T = = d ( ET ) dt T T = 3 P(0) T. () ote that if P(0)=0 then this means the criterion is truly frequency invariant, since the measure does not deend on the oerating frequencies. Otherwise, T=0 is required in order to minimize the criterion, but since that is not ossible, the shortest execution time we can set is T = max ( W ) fmax, so all frequencies are corresondingly set as follows: W f = ma = T W. () ma f max, =.. max( W ) o re-estimation is necessary since all task workloads are known. The ma () function mas the frequency in () to a feasible frequency. In the simle case of a continuous frequency range, this merely means trimming out-of-range values to f min and f max corresondingly. (Also note that in the articular case of Eq. () the inner result is already bounded by f max.) All PEs comlete recisely at T, with the excetion of noncritical PEs with a small workload which run at f according to Eq. (), finishing before T. min 3.3. The Constant Policy The Constant olicy is a simle olicy in which a constant frequency is assigned to each PE. After calculating the JTT and assigning the ECP to run at the maximum frequency f max, we set non-critical PE 6

7 frequencies with an aim to comlete at the joint-targettime: W rem, + λσ rem, f = ma( ) = JTT W rem, + λσ rem, ma( fmax ), =.. W ECP, rem (3) where W, rem is the remaining work estimation of PE, σ rem, is the standard deviation of W, rem, and λ 0 is the bias arameter. At time t=0 no work has been done so the remaining work is the total work. The ma () function mas the result of Eq. (3) to a feasible PE frequency as described above. For λ = 0, f is set so that PE comletes W, rem work during the time it takes the ECP to comlete W ECP, rem work. However since delay outweighs energy for the ET criterion, we can achieve better results by setting the bias arameter λ > 0, as described below in Section 4.. ote that the critical PE frequency can also be formulated using Eq. (3) if we assume λ 0 and substitute W, rem with W ECP, rem. Early Comletion of the Estimatedcritical PE If the ECP comletes while other PEs still have remaining work (ste (3) in the outline), then the assumtions by which frequencies were assigned in ste () no longer hold. In this case we udate the estimations W, rem σ, rem to reflect the work done so far, W,com, and reeat stes () and (): set a new joint-target-time and a new ECP, and assign new PE frequencies. We continue stes () to (4) reeatedly until all PEs have comleted. Figure 4 shows an examle of alying the Constant olicy in a 3-PE system. Figure 4(a) shows the workload distributions, and then two scenarios are illustrated. In Figure 4(b), the workloads are such that the ECP (red, dotted) comletes last. This is the exected, common scenario. In Figure 4(c), the ECP unexectedly finishes first, causing re-estimation; a new ECP is selected (green, dashed) and work-oints are reassigned. If we regard the comlexity of one (f,v) work oint assignment (including the receding remaining workload estimation) as O(), then the comlexity of the Constant olicy can be aroximately regarded as O(), being the number of PEs in the system. This aroximation ignores the occasional work-oint re-estimation that is required when the ECP comletes before other PEs. However, this is justified since the robability of this scenario is (a) generally small, (b) distribution deendent, (c) very hard to calculate and incororate into comlexity estimations, and (d) common among all described olicies, so it generally shouldn t affect comaring them. (a) (b) (c) Figure 4: Constant olicy examle with three PEs: (a) workload distributions, (b) the ECP (red, dotted) finishes last as exected, (c) the ECP finishes first, re-estimation is erformed The Interval Policy The Interval olicy is an enhancement of the Constant olicy described above. The Interval olicy assigns constant frequencies in the same way as the Constant olicy. But in the Interval olicy, the critical PE is re-chosen and frequencies are reassigned at intermediate fixed time intervals. For each time interval, estimated remaining workloads are used to choose the critical PE and frequencies for the next interval. Reestimating remaining workloads and re-assigning workoints following the re-estimation may offer a significant advantage: the difference between the estimated remaining workloads at time t=0, comared to the estimations at a later time when all PEs have done a certain amount of work, may be substantial. Interval can be viewed as a refinement of the Constant olicy: the shorter the time interval, the more accuracy can be achieved. ote that it is ossible to calculate frequency assignments not just for the current interval but for future intervals as well, since all the needed information is available at time t=0. The only information that is not available at time t=0 is actual task workloads. So at any 7

8 given time we can calculate frequencies to be assigned for future intervals, and these calculations will be valid until the time the ECP comletes. Therefore, although it is easier to understand Interval as a rocess of recalculating frequencies at each interval, in ractice Interval calculates frequencies at time t=0, and recalculates only when an ECP comletes. A bias arameter λ exists also for the Interval olicy and allows tuning in the same way as for the Constant olicy. Figure 5 shows an examle of the Interval olicy with PEs. of the ECP, thus the other (blue, solid) PE is designated the new ECP. (a) (a) (b) (c) Figure 5 : Interval olicy examle with two PEs: (a) workload distributions, (b) the ECP (green, dashed) finishes last as exected, (c) the ECP finishes first. In tyical examles, the frequency of non-critical PEs increases with time, as shown in Figure 5. This behavior is similar to the behavior of PACE [0], as exlained in the following section. However, decreasing frequencies can occur, since PE frequencies are a function of the estimated remaining workload relative to the estimated remaining workload of the ECP at each interval. otably, when there is a substantial difference in both the mean and variance of workload distributions, both increasing and decreasing frequencies can occur, as shown in Figure 6 below. ote that in Figure 6, there is an ECP switch at a certain time during the run. The ECP (green, dashed) is initially estimated to have more remaining work, and thus it is designated as the ECP and assigned f max. However as time rogresses the ECP does more work relative to the other (blue, solid) PE, until a time where the other PE s estimated remaining work surasses that (b) Figure 6: Another interval olicy examle, with PEs having substantial difference in both the mean and variance of their workload distributions, (a). Both increasing and decreasing frequencies are observed, (b). The PE marked in dashed green is initially the ECP, but after some time the PE marked in blue becomes the new ECP. Since the comlexity of the Constant olicy is O(), the comlexity of the Interval olicy is O(k), where k is the number of time intervals at which re-estimation occurs The Multi-PACE Policy Energy-erformance tradeoff in a single standalone PE has been studied extensively [9, -4]. The Multi- PACE olicy resented in this section is an attemt to generalize from the well-known single-rocessor aroach PACE [0] onto CMP systems. PACE Scheduling for a Single Processor Consider a task of given workload robability distribution dfw ( w ), running on a PE with a continuous frequency range f [ fmin, fmax ], with some soft deadline D, i.e. a deadline that is required to be attained only in a certain fraction of cases, not necessarily all the time. We reresent this by the robability PMD (Probability of Meeting the Deadline). Given dfw ( w) of a task, a deadline D, and PMD, we can find the maximum workload w PMD for which a task will meet its deadline. The PACE scheme is an analytically otimal method for minimizing the exected energy subject to robabilistically meeting the deadline [0]. Given the above inuts, PACE comutes the otimal frequency schedule er cycle: 8

9 f( w) = PACE([ fmin, fmax ], dfw ( w), D, PMD),. (4) w [0, w ] PMD Conversion from f(w), which exresses the frequency as a function of work cycles done, to a more intuitive f(t) time function, is straightforward. As mentioned above in Section 3., running at a constant frequency is otimal in case workloads are known, but not so when workloads are unknown. PACE stands for Processor Acceleration to Conserve Energy, reflecting the fact that the otimal frequency schedule is an increasing function when workloads are unknown. An intuitive exlanation for this is that a task workload may be small or large, so it is worthwhile to run slowly at first. If the workload is small then the task can easily comlete by the deadline, saving energy by running at a low frequency. As the deadline aroaches, if the task has not yet comleted the frequency is gradually increased in order to assure meeting the deadline with robability PMD. Multi-PACE (f,v) Work-oint Assignment We introduce Multi-PACE, which utilizes PACE to form a DVS olicy for a CMP, as follows. After setting the ECP to run at fmax and setting the JTT as in Eq. (9), non-critical PE frequencies are set according to PACE with the JTT as a deadline: f ( w) = PACE([ fmin, fmax ], dfw ( w), JTT, PMD),. (5) rem, w [0, w ] PMD To clarify, note that JTT is not an alication deadline. Rather, it is comuted following Eq. (9). We use the PACE deadline mechanism to synchronize comletion times between PEs. PACE does not secifically define which frequency to use for the ost-deadline art (i.e., cycles greater than w PMD ). Multi-PACE runs at f max during the cycles subsequent to w PMD in an attemt to minimize delay ast the JTT. Figure 7 shows an examle of alying Multi- PACE, using the same workload distributions as in Figure 4(a). (a) (b) Figure 7: Multi-PACE examle with two PEs, workload distributions as in Workload distributions are as in Figure 4. (a) the ECP (green, dashed) finishes last as exected; (b) the ECP finishes first. The value of PMD has a significant effect on the overall results. If PMD is too high, multi-pace sets overly aggressive frequency schedules, resulting in excessive energy consumtion. On the other hand, choosing PMD too low increases the robability of missing the JTT, causing increased overall execution time. In Section 4. we exeriment with different PMD values for Multi-PACE. The use of PMD is similar to the use of the λ bias arameter for the Constant and Interval olicies. Multi-PACE requires PEs to have a continuous frequency range and to be able to change frequency every cycle, but this is not ractical for reasons reviously discussed. Practical methods of imlementing PACE, which can aly to Multi-PACE as well, are described in [0, 3]. As exlained above, the comlexities of the Constant and Interval olicies are O() and O(k) resectively, where is the number of PEs and k is the number of intervals. In rincile, frequencies are recalculated in Multi-PACE for each cycle. In a ractical system, however, the workload distributions would be built by collecting data into a histogram, and their granularity would therefore be according to the number of histogram bins, denoted by B. Thus the number of frequency changes in multi-pace is in ractice roortional to B, and thus the comlexity of the multi-pace olicy is O(B) [0]. 4. Simulations and Results In this section we resent simulation results of the roosed DVS olicies. We used a few sets of synthetic robability distributions to reresent the task workloads, as shown above in Figure. We simulated a system with six identical PEs, each with a continuous frequency range of 0.3GHz to.5ghz. Energy was calculated using the ower model 3 of P( f) = af + b with a = 0.8 Watt 3, b = 0.[ Watt] GHz as fitting arameters, as described in section.. 9

10 4.. Choosing Bias Parameters Prior to comaring the olicies, we consider the issue of selecting bias arameters: λ for Constant and Interval olicies, and PMD for Multi-PACE. As reviously mentioned, setting PE work-oints so tasks comlete their estimated workloads at the JTT is subotimal. Delay outweighs energy for the ET criterion, therefore missing the JTT by a certain time margin incurs a greater enalty than finishing before the JTT by that same time margin. Better results can be achieved by setting the bias arameter λ > 0 in Eq. (3), thereby running faster. A similar effect can be achieved for the Multi-PACE olicy by tweaking the PMD arameter in Eq. (5). Figure 8 shows olicy results, in terms of ET values, normalized to the Oracle olicy, for distributions (i), (ii), and (v) of Figure, using different bias values λ and PMD. As can be seen in Figure 8, each case shows a certain otimal choice of the bias arameters λ and PMD. However, the results are not very sensitive to small variations, and thus it is reasonable to use the same bias arameter ( λ or PMD) for all distributions. Emirically, a good choice lies in the range of The effect of λ on ET is less accentuated in Interval than in Constant, since Interval () erforms eriodic re-estimations of W i () and σ i. rem rem ormalized ET (c) Multi-PACE dist. (i) dist. (ii) dist. (v) PMD Figure 8: Comarison of bias values for olicies (a) Constant, (b) Interval (50ms), and (c) Multi-PACE olicies. Distributions (i), (ii), and (v) (see Figure ) are shown. 4.. DVS Policy Comarison We simulated each of the workload distributions shown in Figure using Oracle, Constant, Multi-PACE, and Interval olicies. For Interval, we used intervals lengths of 50, 00, 50 and 00 milliseconds which corresond to roughly 6, 8, 4, and intervals throughout the simulated execution time. We chose bias arameters λ = 0.8 for Constant, λ = 0.5 for Interval and PMD=80% for Multi-PACE, following the conclusions of Section 4.. ET (ormalized).8 Policy Comarison.7 ormalized ET ormalized ET (a) Constant dist. (i) dist. (ii) dist. (v) Bias (b) Interval dist. (i) dist. (ii) dist. (v) Bias (i) (ii) (iii) (iv) (v) distribution Figure 9: ET comared across all olicies Multi-PACE Constant Interval 50ms Interval 00ms Interval 50ms Interval 00ms Oracle F-min F-max For each distribution, Figure 9 shows ET comared across all olicies, normalized as above to the results of Oracle, which we regard as a lower bound. Additionally, the results show that f-min and f-max olicies reach the same results in ET for all distributions. This follows from the aroximate frequency invariance of ET, and likewise holds for any other olicy that always uses a single constant frequency. Thus we exect our olicies to achieve results that are in between Oracle and f-max/f-min, i.e., better than running at an arbitrary constant frequency (which we regard as 0% imrovement), but worse than the otimal olicy in which exact workloads are known in advance (00%). As can be seen, this is indeed the case. 0

11 The Interval olicy usually achieves the best results while Constant, the simlest olicy, usually achieves the worst results. However, the difference between the olicies is generally quite small, with no more than 4-3% difference between the best and worst olicies (excet for examle iv, which is discussed next.) Desite the above similarities, distribution (iv) of Figure is an examle where the Interval olicy stands out. As can be seen in Figure (iv), two of the six task workloads of examle (iv) have a bimodal distribution 5 5 ( 0 i or 0i0 cycles) while the other four are known 5 ( 7.5i0 cycles). At time t=0, the estimated critical 5 workload is 7.5i0 cycles. However, once the bimodaldistributed tasks comlete 0 cycles work, the 5 estimation may change to 0 6 cycles, enabling the known-workload tasks to run slower, saving considerable energy. Interval erforms much better than all other olicies since it is the only olicy that reestimates the critical workload. With further regard to the Interval olicy, we note that increasing the interval resolution (increasing the number of re-estimations) rovides only minor, insignificant imrovement. A few re-estimations over a relatively large eriod of time can drastically change the outcome, as demonstrated by examle (iv), while additional re-estimations have only a marginal effect. Multi-PACE on the whole achieves better results than Constant and worse than Interval in the simulated examles. Multi-PACE is more deendent on correct estimation of the critical task than the other olicies, and therefore roduces slightly better results in cases where there is little uncertainty regarding the critical task (v), comared to cases where the uncertainty is greater (iii). Comutational Comlexity In Section 3, we concluded that the comlexities of Constant, Interval, and Multi-PACE are O(), O(k), and O(B), resectively; where is the number of PEs, k the number of intervals, and B the number of workload histogram bins. Since increasing the number of intervals for the Interval olicy beyond a small number rovides only marginal imrovement (note the marginal imrovement when using shorter intervals in Figure 9), it is reasonable to assume small values for k. On the other hand, Multi-PACE erforms no re-estimation, so it needs a considerable large number of bins B [0]. B does not necessarily need to be of cycle granularity, but will be several orders larger than k, i.e., k<<b. Following this reasoning, we conclude that the relative comlexity of the olicies is O() < O(k) << O(B). 5. Summary and Conclusions In this work, we started by formulating an energyerformance tradeoff otimization roblem of an alication running on a CMP. We noted the comlexity of the roblem, which makes it virtually imractical for imlementation. As an alternative to direct otimization, we described several simle heuristic DVS olicies for energyerformance tradeoff. These olicies try to utilize available time-slack in order to save energy in a erformance-aware manner. The frequency-invariant ET criterion was emloyed for comaring the olicies. The olicies described were: Constant, a olicy that tries to estimate the best constant frequency to assign to each PE; Interval, which works in a manner similar to Constant but reassigns new frequencies at fixed time intervals, and Multi-PACE, alying PACE, an otimal scheme for a single-core system with a deadline requirement, for use in a CMP. We comared these olicies using various distributions, and resented several examles. We showed that, excet for some isolated cases, all olicies reach comarable results. Increasing the number of reestimations (using Interval) imroves results comared to estimating merely once at the beginning (using Constant). However, the marginal return sharly diminishes with the number of re-estimations. Multi- PACE roduces results that are anywhere between Interval and Constant, occasionally aearing at the to or bottom of the results list, deending on the distribution. We analyzed the olicy comlexities, and showed that Constant is the least comlex, followed by Interval, while Multi-PACE has the highest comlexity, significantly higher than Constant and Interval. Since the results are usually quite close for all olicies, we conclude that the least comlex olicy, Constant, is usually referred. In individual cases, such as distribution (iv) shown in Figure, there is justification for using Interval. Based on these findings, a scheme could be contemlated whereby the number of intervals is chosen dynamically based on certain characteristics of the distribution, or alternatively, start with a default number of intervals, and assess the result over time to determine if the number of intervals can be decreased. Multi-PACE generally does not achieve better results

12 than any of the other two, and has a very high comlexity, so it is not referred. Frequency-voltage transitions, which are not considered in this work, may degrade the results since each transition is accomanied by erformance and energy enalties [6]. When the cost of transitions is considered, simle olicies such as Constant become even more attractive because they use fewer transitions. The following issues are left for future research:. Study of more comlex task-grahs.. Discret (f-v) workoint sets. 3. With regard to ), the interval olicy may be enhanced to consider re-estimation at flexible times. Such an interval olicy would determine when to jum to an adjacent discrete workoint, rather than directly calculating a new workoint at an arbitrary time. 4. Test cases based on real alication traces. 5. Alications may data assisting in estimation of their own remaining work, which can imrove the accuracy of remaining workload estimations. 6. Real-time alications, which need to achieve a eriodic deadline, can be modeled by relacing the execution time T in the criterion with a relative T D measure which results in enalty only to the extent by which the alication missed its deadline. References [] F. Pollack, "ew Microarchitecture Challenges in the Coming Generations of CMOS Process Technologies," in Micro 3, 999. htt:// [] E. Grochowski, R. Ronen, J. Shen, and H. Wang, "Best of Both Latency and Throughut," in Proceedings of the IEEE International Conference on Comuter Design (ICCD'04) - Volume 00: IEEE Comuter Society, 004. [3] T. Y. Morad, U. C. Weiser, A. Kolodny, M. Valero, and E. Ayguade, "Performance, Power Efficiency and Scalability of Asymmetric Cluster Chi Multirocessors," IEEE Comuter Architecture Letters, vol. 5, 006. [4] "Intel Enhanced SeedSte(R) Technology" htt:// htt:// [5] T. Pering, T. Burd, and R. Brodersen, "The simulation and evaluation of dynamic voltage scaling algorithms," in Proceedings of the 998 international symosium on low ower electronics and design. Monterey, California, United States, 998, [6] S. M. Martin, K. Flautner, T. Mudge, and D. Blaauw, "Combined dynamic voltage scaling and adative body biasing for lower ower microrocessors under dynamic workloads," in Proceedings of the 00 IEEE/ACM international conference on Comuter-aided design. San Jose, California, 00. [7] K. Flautner, S. Reinhardt, and T. Mudge, "Automatic erformance setting for dynamic voltage scaling," Wireless etworks, vol. 8, , 00. [8] S. Yaldiz, A. Demir, S. Tasiran, P. Ienne, and Y. Leblebici, "Characterizing and exloiting task load variability and correlation for energy management in multi core systems," in 3rd Worksho on Embedded Systems for Real- Time Multimedia, 005, 005, [9] Y. Zhang, X. S. Hu, and D. Z. Chen, "Task scheduling and voltage selection for energy minimization," in Proceedings of the 39th conference on Design automation. ew Orleans, Louisiana, USA, 00, [0] A. Andrei, M. Schmitz, P. Eles, Z. Peng, and B. M. Al-Hashimi, "Overhead-conscious voltage selection for dynamic and leakage energy reduction of time-constrained systems," Comuters and Digital Techniques, IEE Proceedings-, vol. 5,. 8-38, 005. [] R. Kumar, D. M. Tullsen, P. Ranganathan,. P. Joui, and K. I. Farkas, "Single-ISA Heterogeneous Multi-Core Architectures for Multithreaded Workload Performance," in Proceedings of the 3st annual international symosium on Comuter architecture. Munchen, Germany: IEEE Comuter Society, 004. [] S. Ghiasi, T. Keller, and F. Rawson, "Scheduling for heterogeneous rocessors in server systems," in Proceedings of the nd conference on Comuting frontiers. Ischia, Italy, 005, [3] A. Elyada, "Low Comlexity Policies for Energy-Performance Tradeoff in Chi-Multi-Processors," in Electrical Engineering. Haifa: Techion, Israel Institute of Technology, 007. [4] M. L. Crow and M. Ilic, "The arallel imlementation of the waveform relaxation method for transient stability simulations," IEEE Trans. on Power Systems, vol. 5,. 9-93, Aug 990. [5] R. A. Saleh, K. A. Gallivan, M.-C. Chang, I.. Hajj, D. Smart, and T.. Trick, "Parallel circuit simulation on suercomuters," Proceedings of the IEEE, vol. 77, , Dec 989.

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