Multiprocessor Energy-Efficient Scheduling with Task Migration Considerations

Transcription

1 ultiprocessor Energy-Efficient cheduling with Tas igration Considerations Jian-Jia Chen, Heng-Ruey Hsu, Kai-Hsiang Chuang, Chia-Lin Yang, Ai-Chun ang, and Tei-Wei Kuo epartment of Computer cience and Information Engineering National Taiwan University, Taipei, Taiwan 106, ROC. E-ails: Abstract This paper targets energy-efficient scheduling of tass over multiple processors, where tass share a common deadline. istinct from many research results on heuristics-based energy-efficient scheduling, we propose approximation algorithms with different approximation bounds for processors with/without constraints on the maximum processor speed, where no tas migration is allowed. When there is no constraint on processor speeds, we propose an approximation algorithm for two-processor scheduling to provide trade-offs among the specified error, the running time, the approximation ratio, and the memory space complexity. An approximation algorithm with a 1.13-approximation ratio for -processor systems is also derived ( > ). When there is an upper bound on processor speeds, an artificial-bound approach is taen to minimize the energy consumption with a 1.13-approximation ratio. An optimal scheduling algorithm is then proposed in the minimization of the energy consumption when tas migration is allowed. Keywords: Energy-Efficient cheduling, Real- Time Tas cheduling, ower anagement, Real-Time ystems, ultiprocessor cheduling. 1. Introduction While an energy-efficient design has become a focus on various systems, voltage-scaling CU s and poweraware subsystems are now adopted in many modern computer systems. The design of CU circuitry is usually done such that a higher supply voltage results in upport in parts by research grants from ROC National cience Council (NC-9-13-E , NC-9-13-E-00-09, and NC-9-0-E ). a higher execution speed (or higher frequency). An example energy consumption function [1, 14], as follows, shows the energy consumption of a processor as a function of the processor speed: (s) =C ef Vdd s, (1) where s = (V dd V t) V dd, and, s, C ef,v t,v dd, and denote the energy consumption, the processor speed, the effective switch capacitance, the threshold voltage, the supply voltage, and a hardware-design-specific constant, respectively (V dd V t 0, >0, and C ef > 0). The energy consumption function of a processor is usually a convex function of the processor speed, and each specific function is highly dependent on the design of the corresponding processor. 1 Energy-efficient scheduling has been an active research topic in the past decade. In particular, Yao, et al. [15] proposed an off-line scheduling algorithm and an on-line competitive algorithm to minimize the energy consumption of tas executions in a uniprocessor environment, where the processor under considerations has an infinite number of continuous processor speeds. In [10], Ishihara and Yasuura showed that an optimal schedule in the minimization of energy consumption with only two processor speeds when the processor has only a finite number of discrete processor speeds, and all tass are ready at time 0 and have a common deadline. Note that the results could only be applied to processors with an energy consumption function equal to Formula (1). While an energy consumption function could be any convex function, Chen, et al. [3] showed that the result in [10] remains. Although many excellent results were proposed for uniprocessor energy-efficient scheduling, little wor has been done for multiprocessor envi- 1 f(x) is a convex function if f(αx +(1 α)y) αf(x) +(1 α)f(y) for any α (0, 1) and any x, y [5].

2 ronments. In recent years, energy-efficient design has been outlined as a critical issue by the industry in business operations, e.g., [], where various configurations of server farms are adopted. Unfortunately, multiprocessor energy-efficient scheduling is often N-hard under various application constraints. Gruian [7] proposed a simulated annealing (A) approach in multiprocessor energy-efficient scheduling with the considerations of precedence constraints and a predictable execution time for each tas. In [8], a power-aware scheduling algorithm based on a list heuristics with a dynamic priority assignment was proposed to determine the amount of time allocated to each tas. Zhang, et al. [16] proposed a heuristic algorithm in which each tas was first assigned to a proper processor, and the processor speed in executing each tas was then chosen without violating the precedence and timing constraints. ishra, et al. [1] explored scheduling issues on the communication delay of tass. Zhu, et al. [17] explored on-line scheduling for a set of independent/dependent frame-based tass, where all tass in a frame-based tas set are ready at time 0 and share a common deadline. Given an off-line schedule with worst-case tas execution times, on-line strategies were proposed to reclaim the slacs resulted from the early completion times of tass observed in the run time. Although some wor has been done on multiprocessor energy-efficient scheduling, many previous results are mainly on heuristics-based energy-efficient scheduling. istinct from the past wor, the objective of this paper is to propose approximation algorithms with different approximation bounds for processors with/without constraints on the maximum processor speed, where tas migration is considered. We first show that there does not exist any polynomial-time approximation algorithm with an approximation bound (1 + ɛ) inthe minimization of energy consumption for multiprocessor scheduling over processors with an upper bound on the processor speed, where ɛ could be any positive real. When there is no constraint on processor speeds, we propose an approximation algorithm for two-processor scheduling to provide trade-offs among the specified error, the running time, the approximation ratio, and the memory space complexity. An approximation algorithm with a 1.13-approximation ratio for -processor systems is also derived ( >). When there is an upper bound on processor speeds, an artificial-bound approach is taen to minimize the energy consumption with a 1.13-approximation ratio. An optimal schedul- uch a constraint violation study was first introduced in [11]. ing algorithm is then proposed in the minimization of the energy consumption when tas migration is allowed. The rest of this paper is organized as follows: ection formally defines the multiprocessor energy-efficient scheduling problems and show their hardness. ection 3 presents approximation algorithms for multiprocessor energy-efficient scheduling for the two-processor case and general cases, where no tas migration is allowed. In ection 4, an optimal polynomial-time scheduling algorithm is proposed in the minimization of the energy consumption when tas migration is allowed. ection 5 is the conclusion.. roblem efinitions and N- Hardness.1. roblem efinitions This paper is interested in multiprocessor energyefficient scheduling with/without constraints on the maximum processor speed and with tas migration considerations. We assume a homogeneous multiprocessor environment, where each of the identical processors has the same energy consumption function (s) of a given processor speed s. Inthispaper, (s) is assumed being a convex and increasing function. An example energy consumption function in [1, 14] is q 4Vt s (s) =C ef s(( s +V t) Vt+ s + Vt ), which is a reformulation of Formula (1). Let U max denote the maximum available processor speed for processors under considerations such that tass could be executed at any processor speed in [0,U max ]. When V t = 0, (s) =αs 3,whereα = C ef. It is reasonable to consider only cases for a given energy consumption func- tion (s) where (s 1 ) >(s )fors 1 >s.lettheenergy consumed for a processor in the execution of tass at the processor speed s for t time units be (s)t. We assume that the number of CU cycles executed in a time interval is linearly proportional to the processor speed. We denote the amount of required CU cycles for a tas running at a speed s for t time units is the multiplication of s and t. Tas migration might or might not be allowed in the exploring of energy-efficient scheduling in this paper. When tas migration is allowed, migration cost is assumed being negligible. No tas could execute simultaneously on more than one processors. For the rest of ection, we first formally define the multiprocessor scheduling problems with the minimization of en-

3 ergy consumption with/without tas migration in this paper. We then show the N-hardness of the problems. efinition 1 ultiprocessor cheduling with the inimization of Energy Consumption with Tas igration (E): Consider a set T of independent tass over identical processors with an energy consumption function (s), where all tass in T are ready at time 0 and share a common deadline. Each tas τ i T is associated with a computation requirement equal to c i CU-cycles. The problem is to minimize the energy consumption in the scheduling of tass in T without missing the common deadline, where tas migration is allowed. A variation of the E problem without tas migration could be defined similarly as follows: efinition ultiprocessor cheduling with the inimization of Energy Consumption without Tas igration (E): The input and output of the E problem are as the same as their counterparts of the E problem, where no tas migration is allowed. A schedule of a tas set is a mapping of the executions of the tass in the set to processors in the system with an assignment of processor speeds for the corresponding time intervals of the tass. A schedule is feasible if all processor speeds assigned for its time intervals are valid, no tas misses the deadline, andthegiven tas migration constraint is satisfied. The energy consumptionof a schedule C is denoted as Φ(C)(lease see the first paragraph of this section for the definition of the energy consumption). A schedule is optimal if it is feasible, and its energy consumption is equal to the minimum energy consumption of all feasible schedules. If there does not exist any feasible schedule for an input instance, then the minimum energy consumption is denoted as... Hardness of the E roblem We shall show the N-hardness of the E problem in this section and then propose an optimal algorithm for the E problem in a later section: Lemma 1 (Chen, Kuo, and Yang [3]) There exists an optimal schedule for any tas set T executing on τ a single processor at the single processor speed i T ci, where the processor under considerations has an infinite number of continuous processor speeds, and all tass in T are ready at time 0 and have a common deadline. Although multiprocessor scheduling is N-hard [4] when no tas migration is allowed, this does not imply the N-hardness of the E problem directly. For example, when () is a linear function( (s) s), any feasible schedule is an optimal solution. Theorem 1 The E problem is N-hard when. roof: The N-hardness is proved by a reduction from the 3-ARTITION problem [4], where () is a strict convex and increasing function, follows from Lemma 1. 3 A polynomial-time (1 + ɛ)-approximation algorithm must have a polynomial-time complexity of the input size and derive a solution with a bound (1 + ɛ) onthe given objective function [13]. That is, when E(OT ) represents the value of an optimal solution for the objective function, any solution derived from a (1 + ɛ)- approximation algorithm should have a value of the objective function no more than (1 + ɛ)e(ot )(for minimization problems). Theorem There does not exist a polynomial-time (1 + ɛ) approximation algorithm for the E problem (ɛ >0) when U max,unless = N. roof: This theorem can be proved by contradiction: uppose that there exists a polynomial-time (1 + ɛ) approximation algorithm for the E problem, called ALG. Given an instance of the ARTI- TION problem [4] (which is N-complete), the problem is to find a subset A in a given set A such that a w(a i A i) = a w(a i A A i), where each element a i in A is associated with a size w(a i ) Z +.Let U max be an arbitrarily positive real and U max. The instance of the ARTITION problem could be reduced to an instance of the E problem such that a unique tas τ i is created for each element a i A, and the required CU cycles for τ i is w(a i ) U max. All tass are ready at time 0, and the common deadline is set as = i A w(ai) a.letthenumber of processors be. By applying the approximation algorithm ALG to the resulting instance of the E problem, the energy consumption of the derived schedule would be bounded by the multiplication of (1 + ɛ) and the energy consumption of an optimal schedule if there exists would be any feasible schedule. However, any feasible schedule could not execute tass at a speed over U max. In other words, if there exists a feasible schedule, then ALG must already identify a subset A of A such that a i A w(a i )= a i A A w(a i ). If there does not exist a feasible schedule, then ALG would report the failure by returning. incealg is 3 f() is strict convex if f(αx +(1 α)y) <αf(x)+(1 α)f(y) for any α (0, 1) and any x, y. For example, (s) s 3 when s 0.

4 a polynomial-time algorithm, such a conclusion contradicts with the N-Completeness of the ARTITION problem (unless =N). Theorem implies that there does not exist any polynomial-time approximation algorithm for the E problem when U max unless = N, since ɛ could be any positive real. 3. ultiprocessor cheduling without Tas igration In this section, we present approximation algorithms for the E problem. We first consider the case in which U max = for two and an arbitrary number of processors, respectively. We then show the proposed algorithms can be proved to bound the maximum processor speed by constant factors, when U max. Note that, since all tass are ready at time 0 and share a common deadline, the tass assigned on a processor can be executed in any order. That is, the execution order for the tass assigned on a processor does not affect the feasibility and the energy consumption for any feasible schedule. The following formula resulted from the convexity of the energy consumption function is used in this section: ( c 1 ) + (c ) (c 3 ) + (c 4 ), () when c 1 + c = c 3 + c 4 and 0 c 1 <c 3 <c 4 <c. Based on Formula () and Lemma 1, it is clear that the executing of a tas τ i on the processor i from time 0to at the speed ci results in an optimal schedule, when T. In the following of this section, only non-trivial cases, T >, are considered ultiprocessor cheduling over Two Identical rocessors When U max = We shall show how to obtain a fully polynomial time approximation scheme (FTA) for two identical processors by a reduction to the aximum ubset um problem [4]. 4 Given a set A of positive numbers a 1,a,,a A and an arbitrary number W,theaximum ubset um problem [4] (which is N-hard) is to find a subset A of A such that a a i A i W and a i A a i is maximized. Lemma (Ibarra and Kim [9]) The aximum ubset um problem admits a fully polynomial-time 4 An algorithm A for a minimization problem is an FTA if A is executed in polynomial time in the size of the input and 1 ɛ, and the approximation ratio of algorithm A is 1 + ɛ [13], where 0 <ɛisauserinput parameter. Note that the approximation 1 ratio is for a maximization problem, where 0 <ɛ<1. 1 ɛ 1 (1 δ)-approximation algorithm subset () for any 0 <δ<1, where the time complexity is O( A ( 3 δ ) ) and the space complexity is O( A +( 3 δ )3 ). ue to Lemma 1 and non-migration of tass, there must exist an optimal schedule for the E problem with two processors which assigns two subsets T 1 and T (T 1 T = T ) of tass on the processors 1 and at τ i T 1 ci τ i T ci the speeds and, respectively. Without loss of generality, let τ c i T 1 i τ c i T i.because of the convexity of the energy consumption func- tions in Formula (), achieving the optimal schedule for the E problem is to generate a subset T 1 of τ i T c i and τ i T 1 c i is T such that τ c i T 1 i 1 maximized. We develop Algorithm basic for the E problem with two processors by applying the subset routine in Lemma with a proper δ. The input parameter ɛ in Algorithm basic is a specified amount of error tolerant to users, which is a necessary requirement for an FTA. It is obvious that the correctness of Algorithm basic is guaranteed. In the following theorems, we show that setting δ = ɛ leads Algorithm basic to be a fully polynomial time (1 + ɛ)-approximation algorithm for the E problem when the energy consumption function satisfies Formula (1). Algorithm 1 : basic Input: (T,,ɛ); Output: A feasible schedule C with minimal energy consumption; 1: let W = τ i T c i ; : C = subset `c 1,c,,c T,W,δ with δ = p ɛ/; let T 1 be the corresponding tas set of C. 3: output the schedule C which executes all tass in T 1 at τ i T 1 c i the speed on the processor 1 and all tass in T T 1 τ at the speed i T T 1 c i on the processor ; Lemma 3 f(x, γ) = (γx)3 +(1 γx) 3 x 3 +(1 x) γ 4γ +3for 3 any fixed γ,where1 γ>0and 1 x>0. roof: It is solved when f(x,γ) x =0. Theorem 3 Algorithm basic is a (1+ɛ)-approximation algorithm for the E problem for any 0 <ɛ< when (s) s 3, i.e., V t =0in Formula (1). roof: Let OT denote a subset of T, where τ c i OT i W and τ c i OT i is maximized. For the simplicity of representation, we use C(X) tode- note τ c i X i for any subset X of tass. Let C opt be the schedule which executes the tass in OT at the speed C(OT ) on the processor 1 and the tass

5 C(T OT ) in T OT at the speed on the processor. C opt is an optimal solution for the E problem and Φ(C opt )=( ( C(OT ) ) C(OT ) )+(C(T )). Without loss of generality, let C(OT ) = C(T ) x and C(T OT )=C(T) (1 x). ince C(OT ) C(T OT ), we have 0 <x 1.WenowC(T1 )= γ C(OT ), where 1 γ 1 δ due to the approximation ratio of Algorithm subset. The ratio of the energy consumption of C to that of C opt is defined as a function f(): f(x, γ) = Φ(C) Φ(C opt ) = (γx)3 +(1 γx) 3 x 3 +(1 x) 3 γ 4γ+3, where the inequality comes from Lemma 3. Note that both γ and x are unnown during the calculation. f(x, 1 ɛ ) 1+ɛ by solving 1 + ɛ =γ 4γ +3. ince γ 4γ + 3 is a decreasing function of γ for any 0 <γ 1, f(x, γ) 1+ɛ if δ = ɛ. Therefore, by setting δ = ɛ, we conclude that Algorithm basic is a (1 + ɛ)-approximation algorithm for the E problem. The time complexity of Algorithm basic is O( T 18 ɛ ), and the space complexity is O( T +( 18 ɛ )1.5 ). We can also prove that Algorithm basic is an F- TAevenwhenV t 0 in Formula (1) in the following theorem. Theorem 4 Algorithm basic is a (1+ɛ)-approximation algorithm for the E problem for any 0 < ɛ < when (s) =C ef ( s3 + Vts + svt V ts 4 + V V t s ts 4 ), i.e., V t 0in Formula (1). 3.. ultiprocessor cheduling over an Arbitrary Number of rocessors When U max = In this section, we present a scheduling algorithm with a 1.13-approximation ratio for the E problem when the maximum available processor speed is infinite. Our proposed algorithm shown in Algorithm (Algorithm ltf) adopts the Largest-Tas-First strategy. That is, tass are considered in a non-increasing order of their computation requirements. Let p m denote the load on the processor m. Theload of a processor is defined as the total amount of the computation requirements of the tass assigned to that processor. Let T m denote the set of the tass assigned to the processor m. Note that the tas set T is a sorted set in a non-increasing order of the computation requirement of each tas, i.e., c i c j if i<j. Algorithm ltf τ i Tm ci. assigns a tas to the processor with the smallest load by the tas order in T. To achieve the minimal energy consumption, based on Lemma 1, each tas on the processor m should be executed at the speed The time complexity of Algorithm ltf is O( T log T ), which is dominated by the sorting of the tass. ince each tas is assigned to one processor without missing the common deadline, the correctness of Algorithm ltf is guaranteed. For the simplicity of representation, the schedule derived from Algorithm ltf is denoted as C T,LTF. Algorithm : ltf Input: (T,,); Output: A feasible schedule C T,LTF with minimal energy consumption; 1: sort all tass in a non-increasing order of the computation requirement of each tas; : set p 1,p,,p to 0, and T 1,T,,T to φ; 3: for i =1to T do 4: find the smallest p m; (brea ties arbitrarily) 5: T m T m {τ i} and p m p m + c i; 6: return the schedule C T,LTF which executes all of the tass in T m (1 m ) at the speed pm on the processor m; Lemma 4 Algorithm ltf is an optimal algorithm if T and c i+ 1 c i+1 for all 1 i T. roof: It can be proved by transforming any feasible solution into C T,LTF without increasing the energy consumption. The next step is to derive the lower bound of the E problem by relaxing the problem constraint. Let be the largest index satisfying and c i+ 1 c i+1 for all 1 i. T represents the set of the first tass of T.Notethat,if T < and T T φ, wenowc T +1 < 1 c T.Werelax the constraint of the E problem so that any tas in T T couldbeexecutedonmorethanoneprocessor simultaneously. Below, we describe the scheduling method for the relaxed E problem. We assign the tass in T according to Algorithm ltf.letp m denote the load of the processor m after performing the tas assignment. There exists a positive value min that satisfies the following equation: ( min p m )δ m = m=1 τ i T T c i, (3) where δ m is 1 if min >p m and 0 otherwise. ince tas migration and simultaneous execution of a tas on multiple processors are allowed for the tass in T T,we can distribute the computation of these tass among

6 min T T τ 1 τ 11 τ 10 τ 9 for each processor m, wehavec c T +1. Wenow min o min p since Algorithm ltf adds the tass in T T to the processor with the minimal load. ue to the definition of T, c T +1 1 min o. 5 Combining all inequality relations mentioned above, we have τ 1 τ τ 3 τ 4 τ 5 τ 6 τ 7 τ 8 p 1 p p 3 p 4 p 5 p 6 p 7 p 8 Figure 1. The tas assignment of C T,LTF for =8and T =1. The computation requirements of the tass in T T are distributedx over the processors 3, 4,and7 (the patterned regions). the processors. If min >p m,( min p m ) CU cycles of T T are distributed on the processor m. Eachprocessor m then performs computation at the speed pm if p m > min and min otherwise. Let C T,LTF denote the resulting schedule. Figure 1 illustrates the loads of processors in C T,LTF for the relaxed E problem. ue to the optimality provided in Lemma 4, it is clear that C T,LTF consumes no more energy than C T,opt,whereC T,opt is an optimal schedule for the E problem. Lemma 5 Φ(CT,LTF ) R = max{ Φ(C T,opt) Φ(CT,LTF) Φ(C T,LTF ) l i L ( l i ) ( R,where } for any positive integer,positive reals and,andanysetlof positive reals that ) satisfy l l i L i = and max li Ll i 3 min l i Ll i. roof: ince Φ(C T,LTF ) Φ(C T,opt), the first inequality is proved. Let o 1,o,,o denote the load on each processor for the tas assignment generated by Algorithm ltf for T and p 1,p,,p for T. max p, min p,andmin o are the values with max{p i }, min{p i }, and min{o i }, respectively. It is clear that max p min, min p min, and Φ(C T,LTF ) τ i T ci ( ). If max p = min, then min p = min and Algorithm ltf generates an optimal solution. Therefore, we only consider the condition where max p > min min p and T T φ. We now prove the second inequality. We first consider the case where o m min for each processor m. Letm be the processor with the largest load in C T,LTF, i.e., p m = max p, and τ be the last tas added into T m in C T,LTF. Once a processor m satisfies p m min, Algorithm ltf will not assign any more tas to the processor m. Therefore, we have τ i T m {τ } c i < min.itis clear that min p τ i T m {τ } c i; otherwise, Algorithm ltf will not assign tas τ to the processor m. Therefore, max p min p c. Because of o m min max p min p c c T +1 1 min o 1 min p. Therefore, max p 3 min p which proves the second inequality. We now consider the case where some o m > min. Let these processors be J, where J 1. For each processor m in J, C T,LTF and C T,LTF assign the same tass on the processor m, e.g., the processors 1,, 5, 6, and 8 in Figure 1. By assuming Φ(CT,LTF) Φ(C T,LTF ) >R,we conclude that Φ(CT,LTF) j J (oj/) Φ(C T,LTF ) j J (oj/) > R.This contradicts the case for = J and T = T j J T j. Therefore, the approximation ratio for Algorithm ltf is R. We now derive the value of R when the energy consumption function satisfies Formula (1). Theorem 5 Algorithm ltf is a 1.13-approximation algorithm for the E problem when (s) s 3, i.e., V t =0in Formula (1). roof: We prove this theorem by showing that R By the definition of R in Lemma 5, there must exist at least one real number x for a set L, where x l i 3x for all l i L. It is clear that x 3x. For each element l i L, wenow ( li ) 3x li x ( x li x )+ x ( 3x ).6 Therefore, we have l ( li i L ) ( 3x )+( ) ( x )forareal satisfying = x x (rephrasing of 3x +x ( ) = ). R is obtained by finding the proper x which maximizes the function f(x) =( 3x )+( ) ( x ). Without loss of generality, let (s) =αs 3,whereαis a constant. By solving f (x) =0andshowingf (x) < 0 for all x satisfying x 3, f(x) is maximized when x = and the maximum value of f(x) is α / α 3 3. Therefore, R Corollary 1 Algorithm ltf is a 1.13-approximation algorithm for the E problem when (s) =C ef ( s3 V ts + svt V ts 4 + V t s V ts 4 ). + 5 There are two cases: 1. If min o = c T,thenc T +1 < 1 c T = 1 mino;.ifmino = c i + c j for some i, j > T, then relations c T +1 c i and c T +1 c j result in c T +1 1 mino. 6 The inequality comes from that (αx +(1 α)y) α (x)+ (1 α) (y) for any α (0, 1) and any x, y. The coefficients of ( x )and( 3x ) are obtained by solving a, b in the following equations: l i = a x + b 3x and 1 = a + b.

7 3.3. ultiprocessor cheduling When U max By adopting the constraint-violation approach [11], we propose an artificial-bound approach by first setting an artificial upper bound on the processor speed and then derive feasible schedules in the minimization of energy consumption. We show that Algorithms basic and ltf bound the maximum processor speed by the factors of ɛ and ( ), respectively. For the simplicity of representation, we assume that tass in T are sorted in a non-increasing order of their computation requirements. We first prove that Algorithm ltf could derive a schedule with a 1.13-approximation ratio without violating the maximum processor speed for certain input instances (lease see Theorem 6): Theorem 6 Algorithm ltf is a 1.13-approximation algorithm if the given input instance satisfies c +1 U max and c 1 U max. τ i T ci + roof: This theorem is proved by contradiction. We assume that there exists a processor m in C T,LTF, where τ i T m c i > U max.letτ be the last tas added into T m in Algorithm ltf. Two cases are considered. If, we now c 1 c > U max. This contradicts the assumption. If >,wehave τ i T m {τ } c i >U max c U max c +1.In τ Algorithm ltf,oncep m i T ci, no more tass can τ be assigned on the processor m. Therefore, i T ci > τ i T m {τ } c i. Based on the above inequalities, we τ have i T ci + c +1 >U max. This contradicts our assumption. We now show that Algorithms basic and ltf bound the maximum processor speed by the factors of ɛ and ( ), respectively. Theorem 7 Given an input instance with a feasible schedule for the E problem, no schedule derived from Algorithm ltf uses any processor speed larger than ( )U max. roof: Let O be a feasible schedule for the input instance. Without loss of generality, O partitions T into disjoint subsets of tass. We assume that m is the processor with the largest load in O. inceo is a feasible schedule, the load on the processor m, says p m, must be no more than U max. Let n be the processor with the largest load in C T,LTF.By rephrasing the processing time into computation requirement in the aespan problem, we now that p n ( )p m ( )U max since the Longest- rocessing-time-first algorithm was proved to be a ( )-approximation algorithm for the aespan problem in [6]. 7 We complete the proof. Theorem 8 Given an input instance with a feasible schedule for the E problem over two processors, no schedule derived from Algorithm basic uses any processor speed larger than (1 + ɛ )U max. roof: It comes from the setting of δ in Algorithm basic. 4. Tas igration: An Optimal Algorithm Algorithm 3 : ltf-m Input: (T,,); Output: An optimal schedule with minimum energy consumption; 1: sort T in a non-increasing order of the computation requirement of each tas; let C τ i T ci; C : if >Umax or τi T such that c i >Umax then 3: return non-existence of any feasible schedule; 4: let i 1; 5: while i T do 6: if c i > C then 7: schedule τ i to be executed at the speed c i on the processor from time 0 to ; 8: C C c i, i i +1,and 1; 9: else 10: brea; 11: let C and t 0; 1: while i T do 13: if t + c i >then 14: schedule τ i to be executed at the speed on the processor 1fromtime0tot + c i andonthe the processor from time t to ; 1; 15: else 16: schedule τ i to be executed on the processor at the speed from time t to t + c i ; 17: i i +1andt (t + c i ) mod ; 18: return the schedule of all tass; In this section, an efficient optimal algorithm is proposed for the E problem, where tas migration is allowed. If T, based on Formula () shown in ection 3, it is clear that the executing of each tas τ i on the processor i from time 0 to at the speed ci results in an optimal schedule. Our proposed Algorithm ltf-m (Algorithm 3) adopts the Largest-Tas-First strategy again, and the time complexity O( T log T ) comes from the sorting of T in line 1. We can prove the following two lemmas. 7 The aespan problem is as follows: Given processing time for n tass, findanassignment of thetass to identical processors so that the completion time for these tass is minimized.

8 Lemma 6 If c 1 >and T, then there exists an optimal schedule which executes onlyτ 1 on a processor at the speed c1 from 0 to,where = τ i T ci. Lemma 7 If c 1 and T, then there exists an optimal schedule which executes each tas in T on at most two processors at the speed,where = τ i T ci. Theorem 9 Any schedule derived from Algorithm ltf-m is an optimal schedule. τ i T ci roof: If c 1 >where =, then Algorithm ltf-m executes τ 1 on at the speed c1,andthe remaining tass T {τ 1 } and 1 processors form a subproblem of the E problem; otherwise, Algorithm ltf-m executes each tas in T over at most two processors at the speed. Based on Lemmas 6 and 7, we conclude this proof by repeating the above procedure in solving the E subproblems. 5. Conclusion This paper targets energy-efficient scheduling of tass over multiple processors, where tass share a common deadline. istinct from the past wor, this paper proposes approximation algorithms with different approximation bounds for processors with/without constraints on the maximum processor speed. We show the non-existence of polynomial-time approximation algorithms in the minimization of energy consumption for multiprocessor scheduling over processors with an upper bound on the processor speed, unless = N. When there is no constraint on processor speeds, we propose an approximation algorithm for two-processor scheduling to provide trade-offs among the specified error, the running time, the approximation ratio, and the memory space complexity. An approximation algorithm with a 1.13-approximation ratio for -processor systems is also derived ( >). When there is an upper bound on processor speeds, an artificial-bound approach is taen to minimize the energy consumption with a 1.13-approximation ratio. Furthermore, an optimal polynomial-time scheduling algorithm is proposed for the minimization of the energy consumption when tas migration is allowed. For future research, we shall explore multiprocessor energy-efficient scheduling for tas sets with arbitrary deadlines and arrival times. References [1] A. Chandraasan,. heng, and R. Broderson. Lower- ower CO digital design. IEEE Journal of of olid- tate Circuit, 7(4): , 199. [] J.. Chase,. C. Anderson,. N. Thaar, A. Vahdat, and R.. oyle. anaging energy and server resources in hosting centres. In ymposium on Operating ystems rinciples, pages AC ress, 001. [3] J.-J. Chen, T.-W. Kuo, and C.-L. Yang. rofit-driven uniprocessor scheduling with timing and energy constraints. In AC ymposium on Applied Computing, pages AC ress, 004. [4].R.Gareyand..Johnson. Computers and intractability: A guide to the theory of N-completeness. W.H. Freeman and Co, [5] G. Golub and J. Ortega. cientific Computing and ifferential Equations. Academic ress, 199. [6] R. Graham. Bounds on multiprocessing timing anomalies. IA Journal on Applied athematics, 17:63 69, [7] F. Gruian. ystem-level design methods for low-energy architectures containing variable voltage processors. In ower-aware Computing ystems, pages 1 1, 000. [8] F. Gruian and K. Kuchcinsi. Lenes: Tas scheduling for low energy systems using variable supply voltage processors. In roceedings of Asia outh acific esign Automation Conference, pages , 001. [9] O. H. Ibarra and C. E. Kim. Fast approximation algorithms for the napsac and sum of subsets problems. Journal of the AC, (4): , [10] T. Ishihara and H. Yasuura. Voltage scheduling problems for dynamically variable voltage processors. In roceedings of the International ymposium on Low ower Electroncs and esign, pages 197 0, [11] J.-H. Lin and J.. Vitter. ɛ-approximations with minimum pacing constraint violation. In ymposium on Theory of Computing, pages AC ress, 199. [1] R. ishra, N. Rastogi,. Zhu,. osse, and R. elhem. Energy aware scheduling for distributed real-time systems. In International arallel and istributed rocessing ymposium, page 1, 003. [13] V. V. Vazirani. Approximation Algorithms. pringer, 001. [14]. Weiser, B. Welch, A. emers, and. hener. cheduling for reduced CU energy. In roceedings of ymposium on Operating ystems esign and Implementation, pages 13 3, [15] F. Yao, A. emers, and. hanar. A scheduling model for reduced CU energy. In roceedings of the 36th Annual ymposium on Foundations of Computer cience, pages IEEE, [16] Y. Zhang, X. Hu, and. Z. Chen. Tas scheduling and voltage selection for energy minimization. In Annual AC IEEE esign Automation Conference, pages , 00. [17]. Zhu, R. elhem, and B. Childers. cheduling with dynamic voltage/speed adjustment using slac reclamation in multi-processor real-time systems. In roceedings of IEEE th Real-Time ystem ymposium, pages 84 94, 001.