Dynamic Voltage Scaling for Multitasking Real-Time Systems with Uncertain Execution Time

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1 Dynamic Voltage Scaling for Multitasking Real-Time Systems with Uncertain Execution Time ABSTRACT Changjiu Xian Department of Computer Science urdue University West Lafayette, IN Dynamic voltage scaling (DVS) for real- systems has een extensively studied to save energy. revious studies consider the proailistic distriutions of tasks execution to assist DVS in task scheduling. These studies use proaility information for intra-task frequency scheduling ut do not sufficiently explore the opportunities for intertask scheduling to save more energy. This paper presents a new approach to integrate intra-task and inter-task frequency scheduling for etter energy savings in hard real systems with uncertain task execution. Our approach has two steps: We calculate statistically optimal frequency schedules for multiple periodic tasks using earliest deadline first (EDF) scheduling for processors that can change frequencies continuously. () For processors with a limited range of discrete frequencies, we further present a heuristic algorithm to construct frequency schedules. Our evaluation shows that our approach saves up to 23% more energy than existing solutions. Categories and Suject Descriptors C.4 [erformance of Systems]: design studies General Terms Algorithms, Design, erformance Keywords Dynamic voltage scaling, hard real-, proaility, multitasking, low energy. INTRODUCTION Energy consumption is an important design issue for atteryoperated emedded systems. In these systems, the CU is one major energy consumer. Dynamic voltage scaling (DVS) is an effective technique to reduce CU energy ecause quadratic energy savings can e achieved with only ermission to make digital or hard copies oll or part of this work for personal or classroom use is granted without fee provided that copies are not made or distriuted for profit or commercial advantage and that copies ear this notice and the full citation on the first page. To copy otherwise, to repulish, to post on servers or to redistriute to lists, requires prior specific permission and/or a fee. GLSVLSI 06, April 30 May 2, 2006, hiladelphia, A, USA. Copyright 2006 ACM /06/ $5.00. Yung-Hsiang Lu School of Electrical and Computer Engineering urdue University West Lafayette, IN yunglu@purdue.edu linear decrease of the CU frequency. Various DVS algorithms have een proposed, especially for real- systems. A real- task has a deadline that is the maximum within which the task must completes its execution. A real scheduling algorithm can e either offline or online []. A hard real- system does not allow missing any deadline. Consequently, for a task with uncertain execution, the scheduler must consider the worst-case execution (WCET) to guarantee meeting the deadline. Since this paper considers frequency and voltage scaling, we use execution cycles instead of execution to express the workload o task. The task s execution is the execution cycles divided y the processor s frequency. One solution to meet the deadline is assigning the processor a single frequency equal to the ratio of the worst-case execution cycles (WCEC) to the allowed execution [8]. This can e too conservative if the average workload is much lighter than the worst case. Alternatively, the proaility information of the workload can e used for etter frequency scheduling. I task requires variale numers of cycles in different execution instances, the later cycles have lower proailities to execute than the earlier cycles. For example, suppose three execution instances o task require one million, two million, and three million cycles. Then the first million cycles execute three s, the second million cycles twice, and the third million only once. Some studies [2, 7, 0, ] propose to assign lower frequencies to the earlier cycles and higher frequencies to the later cycles (called accelerating frequency schedule ) such that the deadline can still e met and the earlier cycles save energy. Even though the later cycles consume more energy when executing, they are rarely needed due to their lower proaility. This can save more energy than assigning all cycles a single frequency. The existing studies on accelerating frequency scheduling have two major limitations: They use the proaility information for only intra-task scheduling and do not sufficiently explore the opportunities for inter-task scheduling to save more energy. Inter-task scheduling determines how to allocate for different tasks. () For processors with a limited range of discrete frequencies, the existing studies adjust the frequencies of each task individually to match the availale frequencies without considering the effects of such adjustment on the other tasks. This paper presents a new approach to integrate inter-task and intra-task scheduling in hard real- multitasking systems with uncertain workloads. The key idea is comining inter-task and intra-task scheduling into a single optimization prolem such that the frequency schedule of each task

2 f f is determined y oth its own proaility distriution and the proaility information from all other tasks. In other words, gloal information is used to uild frequency schedules for individual tasks. Our approach has two steps: We calculate statistically optimal frequency schedules for multiple periodic tasks using earliest deadline first (EDF) scheduling for processors that can change frequencies continuously. () For processors with a limited range of discrete frequencies, we further present a heuristic algorithm to construct frequency schedules ased on gloal proaility information and schedulaility constraints. The computation of frequency schedules is offline and has no run overhead. Our evaluation shows that our approach saves up to 23% more energy than existing solutions. 2. BACKGROUND 2. roaility-based DVS Some studies have een conducted for DVS y considering the proaility distriutions of tasks cycle demands. Lorch et al. [7] derive an accelerating frequency schedule for a single task and treat concurrent tasks as a single joint workload. Gruian [2] constructs accelerating intra-task schedules and allocates to multiple tasks ased on their WCEC. Yuan et al. [] comine accelerating schedules with soft real- constraints for muldia applications. Xu et al. [0] study accelerating scheduling in systems with discrete frequencies. These studies use the proaility information for only intratask scheduling. In contrast, we can save more energy y comining the proaility information from all tasks to integrate inter-task and intra-task scheduling. Zhang et al. [2] and Xu et al. [9] have considered multiple tasks in a special case where all periodic tasks share a common period such that all the tasks execute sequentially in each period and finish efore the end of the period. Our approach allows different periods and uses EDF scheduling. 2.2 Accelerating Frequency Schedule Suppose a task demands at most W cycles and the distriution of its cycle demand is expressed y the cumulative distriution function (CDF) shown in Figure. The proaility that the j th cycle is needed is (j) = CDF(j ), as shown in Figure (). Note that is nonincreasing ecause CDF is non-decreasing. Since a task may demand millions of cycles, it is impractical to store the distriution in cycles. Suppose the range [0, W] is partitioned into M ins and each in contains cycles ( = W M ). The CDF is then a function of the ins, as shown in Figure (d). The proaility that the j th in is needed is (j) = CDF(j ), as shown in Figure (e). Bins represent a more general form of granularity than cycles ecause we can choose for and make M = W. We introduce the accelerating frequency schedule ased on ins. The frequency assigned to the j th in is f j and the execution for this in is f j. The processor s power is proportional to v 2 nd v f (v: voltage). The energy for this in is (vjf 2 j) f j fj 2. The expected energy consumption for this in is proportional to the product of the energy and the proaility: fj 2 (j). Suppose the task is released at zero and the deadline is t. The goal is to find a schedule {f, f 2,..., f M } to minimize the total expected energy. This is formulated as follows. CDF CDF cycles ins (d) W cycles W cycles W () ins (e) Figure : CDF, proaility () of cycles or ins, and frequency schedule (f) using the granularity of () cycles and (d)(e)(f) ins. minimize X j M suject to X j M ins (f) f 2 j (j) () f j t (2) Based on earlier studies [7, 0, ], the optimal solutions can e otained y assigning f j: f j = M 3p (j) t 3p (j) This frequency schedule is shown in Figure for = and in Figure (f) for. Both frequency schedules are non-decreasing. The reason is that (j) (j + ) so the denominator decreases and f j f j+ in Equation (3). 3. INTEGRATED TASK SCHEDULING This section presents our scheduling scheme for DVS. We first provide a motivational example to illustrate the asic concept. Then we explain how to integrate inter-task and intra-task scheduling to minimize the expected energy consumption in hard-real systems with uncertain workloads. We extend the solution to processors with only a limited range of discrete frequencies in Section Motivational Example Consider two periodic tasks K and K2, shown in Figure 2, starting from the same with periods T = 3s and T 2 = 6s. Both tasks have WCEC of 3 million cycles, divided into three equal-sized ins. Task K always uses 3 million cycles (3 ins), so the proailities of the three ins are all 00% (see Section 2.2 for the definition of such proaility). Task K 2 has 90% proaility to use million cycles (the first in), 5% proaility to use 2 million cycles (the first two ins), and 5% proaility to use 3 million cycles (all three ins). The proaility that the first in of K 2 is used is 00%, the second in 0%, and the third in 5%. Each task s deadline is the same as the task s period. We use EDF scheduling so K 2 executes after K s first instance. The second instance of K has the same deadline as K 2 so it does not preempt K 2 and executes after K 2. We consider two different methods to assign frequencies so that oth tasks can meet their deadlines. (3)

3 Speed(MHz) 3 2 T T K: T2 K2: 0 6 () Speed(MHz) K K2 K KK K2 K Figure 2: Frequency schedules from different allocation methods. The first method (shown in Figure 2 ()) allocates the same amount of, 2 seconds, to the three instances ecause they have the same WCEC. Within each 2 seconds, the optimal intra-task scheduling explained in Equation (3) is adopted. Figure 2 () shows the frequencies of the schedule. The frequency assigned to K is the same across the 2-second duration ecause the actual cycle demand is the same as the WCEC. The frequency assigned to K 2 gradually increases. This solution is the state-of-the-art proailityased frequency scheduling for multiple tasks with EDF []. Our method (shown in Figure 2 ) allocates more to K than K 2 ecause K requires more cycles than K 2 in most cases. Specifically, we allocate 2.3 seconds for K and.4 seconds for K 2 and save 2% more energy than the first method. Section 3.3 explains how to otain the duration assignments of 2.3 and.4 seconds y integrating inter-task and intra-task scheduling. 3.2 Task and System Model We consider a set of periodic independent tasks denoted as {K, K 2,..., K N }. The tasks are preemptive. Task K i has the following parameters: is its period. Each task has an execution instance per period. () W i is its WCEC, i.e., the maximum numer of cycles needed y one execution instance. The range [0, W i] is divided into M i ins and each contains cycles. D i is its deadline. Every execution instance must finish efore its deadline. We assume D i = in this study. (d) i(j) is the proaility that the j th in of cycles is needed in each period. We first consider processors whose frequencies can e adjusted continuously from 0 to infinity. In Section 3.4, we consider processors with finite and discrete frequencies. The to switch from one frequency to another is in the microseconds range [3, ] and the execution for tasks is normally the milliseconds range. Thus, we ignore the frequency switching. We assume the processors consume oth dynamic and static power during usy periods and only static power during idle periods. The dynamic power is proportional to v 2 nd the static power is a constant. We assume DVS can adjust only dynamic power so we ignore the static power ecause it is constant throughout the whole duration. Overall, the energy per cycle is proportional to v 2 f f f2 since v f. The scheduling is ased on EDF. 3.3 Statistically Optimal Frequency Schedule We integrate inter-task and intra-task scheduling such that the proailities of the ins from all tasks are considered to uild frequency schedules for individual tasks. The 3 2 ins (from all tasks) with higher proailities can use lower frequencies to save energy. For schedulaility, the other ins may have to use higher frequencies. Even though these higher frequencies consume more energy, they are rarely needed due to their lower proailities. Since a task with a shorter period executes more often than a task with a longer period, we divide task K i s proaility i y to normalize the proaility over. The optimization prolem is then to find a frequency for the j th ( j M i) in of task K i ( i N) to minimize the total expected energy oll tasks ased on the ins proailities and under the schedulaility constraint of EDF. EDF can schedule a set of tasks if the total CU utilization is no more than one [6]. The mathematic formulation is as follows. minimize suject to M X i Mi fij 2 i(j) (4) (5) Equation (4) is the total expected energy from all tasks, where fij 2 and i(j) are the energy consumption and the normalized proaility of the j th in of task K i, respectively. Equation (5) guarantees the schedulaility of EDF, where is the worst-case execution of task K i. Note Mi that Equation (5) implies that M i and is the deadline. Based on the convexity of f 2, Equations (4) and (5) can e solved ased on Jensen s inequality [5]. All the optimization equations in this paper are solved with the same rationale. The derivation is omitted in this paper due to the space limit. The minimum expected total energy is Mi 3p i(j) The j th in of task K i is assigned frequency : t i = = Mi N 3 i (j) T p i 3 i(j)! 3 (6) The (t i) allocated to task K i depends on the frequency schedule: p M X i 3 i(j) = Mi (7) N Mi 3 i (j) (8) The statistically optimal frequency schedules and the allocated for each task can e computed from the periods (), the ins (M i, ), and the ins proailities ((i)). Equation (7) indicates that the frequency for each task is accelerating ecause the dominator 3p i(j) decreases as j increases. Equation (8) is used y the second method in the motivational example (Section 3.) to calculate the allocation for the two tasks. 3.4 Finite and Discrete Frequencies Section 3.3 assumes continuous frequencies from 0 to infinity. However, a practical processor has a maximum frequency and a minimum frequency. Meanwhile, the frequencies etween and are discrete. Equation (7) has no guarantee to satisfy this constraint. This

4 t K K K t f max t" B K2 B 2 B3 t 2 () B B 2 B 3 B B 2 B 3 K2 t 2 t"2 f K2 max B 2 B 22 B 23 B 2 B 22 B 23 B 2 B 22 B 23 Figure 3: Steps to confine frequency schedules etween [, ]: Compute schedules for tasks K and K 2 for continuous frequencies. B ij means the j th in of cycles for task K i. () Assign to B 3 and recompute the frequencies for the remaining ins: B, B 2, B 2, B 22, B 23. Assign to B 2 and recompute the frequencies for the remaining ins: B, B 2, B 22, B 23. section presents a heuristic algorithm to adjust the frequencies within and ased on gloal proaility information and schedulaility constraints. We first assume the frequencies are continuous within [, ] and then restrict the frequencies to e discrete. Figure 3 shows the frequency schedules for two tasks K and K 2 using continuous frequencies from Equation (7). The frequencies of the first ins (B and B 2) are lower than. The frequency for B 3 is higher than. Task K is allocated t and task K 2 is allocated t 2 (Equation (8)). We first consider reducing the frequencies higher than, specifically, the frequency of B 3. Since we always assign higher frequencies to the ins with lower proailities, B 3 must have a proaility lower than the other ins. Decreasing the frequency of in B 3 will prolong its execution. Due to schedulaility constraints, at least one of the other ins has to shorten its execution and consequently use a higher frequency. Since all the other ins have higher proailities than B 3, their frequencies should e kept low to minimize the expected energy. Hence, we decrease the frequency of B 3 to only for minimum increasing of other ins frequencies. To decrease B 3 s frequency to, two methods can adjust the frequencies of the other ins to keep schedulaility. The first adjusts only task K within its allocated t. Namely, ins B and B 2 raise their frequencies to use less such that the three ins of K still use t. This method does not consider the opportunity for gloal adjustment. The second method recomputes the frequency schedule for all the remaining ins from oth tasks. Time t and t 2 depend on the new frequency schedules. This is illustrated in Figure 3 (). B 3 is assigned and the frequencies of the remaining ins of oth tasks are adjusted. This concept can e generalized to multiple ins whose frequencies are higher than. Let H i denote the numer of ins that are assigned frequencies higher than for task K i from Equation (7). We assign to these ins and and use the following formulation to calculate the frequencies of the other ins j M i H i from all tasks. minimize suject to M i H X i f 2 j M i H i H i + i(j) (9) (0) The minimum energy is achieved if = N p 3 i(j) Mi H i 3 i (j) N H i fmax «() If this new frequency schedule for the j M i H i ins still contains frequencies higher than, we assign to these ins and recursively perform the aove adjustment for the remaining ins until all ins have frequencies no greater than. We next handle the ins with frequencies lower than. In Figure 3 (), B 2 s frequency is lower than. Because we always assign lower frequencies to ins with higher proailities, B 2 must have a higher proaility than the other ins. The frequency of B 2 should e kept low to minimize the expected energy. Hence, we assign to B 2, as illustrated in Figure 3. Since this shortens the execution of in B 2, the spare can e assigned to the other ins to lower their frequencies. We redistriute such spare to the remaining ins oll tasks y recomputing their frequency schedules. Let L i denote the numer of ins with frequencies lower than for task K i. We determine the frequencies for the remaining (L i + j M i H i) ins using the following formulation. suject to minimize M i H X i j=l i + L i + H i + The minimum energy is achieved if = N p 3 i(j) f 2 j M i H i i(j) (2) j=l i + (3) Mi j=l i + 3 i (j) N L i + H i fmax «(4) After the previous procedure, all ins have frequencies etween and. We next consider availale discrete frequencies. A simple method replaces each frequency with its two adjacent discrete frequencies [2, 4]. However, this method can waste energy when multiple ins have their frequencies etween the same two adjacent frequencies. Figure 4 shows an example when three ins B, B 2, and B 3

5 f B B 2 B B 3 f B 2 B 3 B a B 2a B3a () f B a B Figure 4: Three ins frequencies f, f 2, f 3 are etween two adjacent discrete frequencies and f. () Each in reaks into two su-ins to use and f. The three ins are comined into two ins to use and f. have frequencies etween two adjacent discrete frequencies and f. Figure 4 () shows that each in reaks into two su-ins to use and f within the same execution of the original in. For example, B reaks into B a and B. The frequency rises and falls etween and f and the schedule is no longer accelerating. The early ins (e.g., B ) with higher proailities may use f while the later ins (e.g., B 3a) with lower proailities may use ; consequently, the expected total energy is not minimized. Alternatively, we propose to comine the three ins into two ins and use and f within the duration of the three original ins, as illustrated in Figure 4. This guarantees only one frequency switching etween any two adjacent frequencies and the early in uses the lower frequency. There can e more than three ins with frequencies etween and f. Let n e the numer of ins with frequencies (f k, k n) etween two adjacent discrete frequencies and f. The total numer of cycles of the n ins is n and total duration is n k= f k. We need to comine the n ins into two ins. Let x e the numer of cycles assigned to the first in. Then n x is the numer of cycles assigned to the second in. The two ins should use the same duration as the original n ins. The value of x thus can e found y solving equation n k= f k = x + n x f. 3.5 Implementation Issues Our method can e implemented in practical systems. First, the proaility distriutions of the cycle demands of tasks can e otained y profiling their offline traces. Second, our method computes the frequency schedules for a task set offline, so there is no computation overhead at run. The computed schedules can e stored in the system memory for run usage. Third, at run, we only need to count how many cycles have een consumed y the current task and determine whether to switch to a higher frequency y looking up the accelerating frequency schedule. revious study [] has implemented cycle counting together with an EDF scheduler and shows the feasiility of using accelerating frequency schedule at run. 4. EXERIMENTAL RESULTS This section descries the simulation setup and presents the simulation results comparing the energy savings from our method and the existing solutions for oth finite and continuous frequencies. When the numer of finite frequencies increases, we are approaching the ideal case where continuous frequencies are availale. Using continuous frequencies allows evaluating the maximum potential of our method. 4. Method of Simulation Tale : XScale s frequency/voltage and power. Frequency(MHz) Voltage(V) ower(mw) ower (mw) XScale s discrete power Least square fitting with ax Frequency (MHz) Figure 5: Discrete and continuous power. We use the frequency/voltage settings and power consumption of Intel XScale [0] (as shown in Tale ) for finite frequencies. For continuous frequencies, we otain their power numers y applying least-square curve fitting using af 3 + (a = , = 60) to the XScale s frequency and power, as shown in Figure 5. Both muldia applications and synthetic workloads are used in our evaluation. A task set of four muldia programs is used: mpegplay, madplay, tmndec, and toast, as shown in Tale 2. All these programs except toast have the same period, ut their WCEC vary consideraly. The distriutions of their cycle demands are otained y profiling offline traces with ins each containing 00,000 cycles. We also perform evaluation over a synthetic task set with 30 tasks and with a wider range of periods, WCEC, and distriutions. We construct the task set in three steps. The first step assigns each task a period randomly chosen uniformly etween 0 milliseconds and second. The second step chooses for each task its WCEC (W) randomly uniformly etween 00,000 cycles and 00,000,000 cycles under the constraint that the worst-case CU utilization at 000MHz is no more than one. The third step determines a distriution function of cycle demand for each task. We consider three types of distriutions (Gaussian, exponential, and uniform) for cycle demands as suggested in [0, 2]. We use the same type of distriutions for all tasks and test the three distriution types separately. Gaussian distriution has two parameters: mean (µ) and standard deviation (σ). For task K i, µ i is randomly chosen within the range (0, W i] and σ i = W i/6. Exponential distriution has one parameter λ (µ =, λ σ2 = ). We choose for each task its µ λ 2 i (or λ i ) also randomly within the range (0, W i]. Uniform distriution is a constant equal to W i for task K i. Four methods that handle multiple tasks are compared: separated continuous (SC): Separating inter-task and intra-task scheduling. SC allocates to tasks ased on their WCEC [] and then constructs a frequency schedule (Equation (3)) for each task. SC considers continuous frequencies. separated discrete (SD): SD considers finite and discrete frequencies [0] and allocates to multiple tasks ased on their WCEC. integrated continuous (IC): IC integrates inter-task and intra-task scheduling for continuous frequencies as explained in Section 3.3. integrated discrete (ID): ID integrates inter-task and intra-task scheduling for finite and discrete frequencies as explained in Section 3.4.

6 Energy Savings (%) separated continous (SC) integrated continous (IC) separated discrete (SD) integrated discrete (ID) Muldia Gaussian Exponential Uniform Figure 6: Energy savings from the set of muldia enchmarks and the three synthetic task sets with Gaussian, exponential, and uniform distriutions, respectively. Tale 2: Muldia applications. thousand cycles Application Description T(ms) W mpegplay MEG video decoder 30 0,500 madplay M3 audio decoder tmndec H263 video decoder 30 2,700 toast GSM speech encoder The first two methods are existing solutions. The third and fourth methods are proposed in this paper. The aseline method used to compare the four methods uses a single frequency N (Wi/Ti) for all tasks. 4.2 Energy Savings Figure 6 shows the simulation results. The four groups ors show the energy savings from four task sets: the set of muldia enchmarks and the three synthetic task sets with Gaussian, exponential, and uniform distriutions, respectively. For each task set, the four ars represent the energy savings y the four methods. Figure 6 shows that IC always saves more energy (ranging from 8% to 23%) than SC. Method ID always saves more energy (ranging from 7% to 9%) than SD. The reason is that the integrated scheduling methods consider the periods, the worst-case cycle demands, and the proailistic distriutions oll tasks to perform gloal adjustment for etter energy savings. The figure also shows that, for finite and discrete frequencies, oth separated and integrated scheduling methods save less energy than for continuous frequencies. Specifically, SD saves 3% to 26% less energy than SC and ID saves 7% to 30% less energy than IC. This is ecause finite and discrete frequencies restrict the options in the frequency schedules and produce inferior energy savings. The muldia task set has only four tasks while each of the other task sets has 30 tasks. Intuitively, the overall cycle demands or CU utilization y the muldia set may e significantly lower and there is much more room for energy reduction. However, Figure 6 shows that the energy savings otained y all methods from the muldia task set are not significantly higher than the savings from the other data sets. The reason is that the four muldia programs all have very small periods and their cycle demands are often close to their worst cases. This makes their overall CU utilization actually close to the the other task sets. 5. CONCLUSIONS This paper presents a DVS scheme for multitasking real systems with uncertain execution. We use the proaility distriutions of the cycle demands of tasks and integrate inter-task and intra-task frequency scheduling for etter energy savings. We consider oth continuous frequencies and finite discrete frequencies. Our evaluation shows that our method outperforms the existing solutions for muldia applications and synthetic workloads. Overall, our approach saves 7% to 23% more energy than the existing solutions. For future work, we plan to perform sensitivity analysis of the in size (granularity) of tasks. 6. ACKNOWLEDGMENTS This work is supported in part y National Science Foundation Career CNS Any opinions, findings, and conclusions or recommendations are those of the authors and do not necessarily reflect the views of the sponsors. 7. REFERENCES [] G.C. Buttazzo. Hard Real-Time Computing Systems. Kluwer, 997. [2] F. Gruian. Hard Real-Time Scheduling for Low-Energy Using Stochastic Data and DVS rocessors. In the ISLED, pages 46 5, August 200. [3] Intel XScale Microarchitecture. ( [4] T. Ishihara and H. Yasuura. Voltage Scheduling rolem for Dynamically Variale Voltage rocessors. In ISLED, pages , 998. [5] J. L. W. V. Jensen. Sur Les Fonctions Convexes Et Les Inegalites Entre Les Valeurs Moyennes. Acta Math, 30:75 93, 906. [6] C. L. Liu and J. W. Layland. Scheduling Algorithms for Multiprogramming in a Hard-Real-Time Environment. Journal of the Association for Computing Machinary, 20():46 6, January 973. [7] J. R. Lorch and A. J. Smith. Improving Dynamic Voltage Scaling Algorithms with ACE. In the 200 ACM SIGMETRICS, pages 50 6, June 200. [8]. illai and K. G. Shin. Real- Dynamic Voltage Scaling for Low-power Emedded Operating Systems. In ACM SOS, pages 89 02, 200. [9] R. Xu, D. Moss, and R. Melhem. Minimize Expected Energy in Real-Time Emedded Systems. In the 5th ACM EmSoft, pages , Septemer [0] R. Xu, C. Xi, R. Melhem, and D. Moss. ractical ACE for Emedded Systems. In the 4th ACM EmSoft, pages 54 63, Septemer [] W. Yuan and K. Nahrstedt. Energy-Efficient Soft Real-Time CU Scheduling for Moile Muldia Systems. In ACM SOS, pages 49 63, [2] Y. Zhang, Z. Lu, J. Lach, K. Skadron, and M. R. Stan. Optimal rocrastinating Voltage Scheduling for Hard Real-Time Systems. In the 42nd DAC, pages , June 2005.