Generalized Network Flow Techniques for Dynamic Voltage Scaling in Hard Real-Time Systems

Transcription

1 Generalized Network Flow Techniques for Dynamic Voltage Scaling in Hard Real-Time Systems Vishnu Swaminathan and Krishnendu Chakrabarty Department of Electrical & Computer Engineering Duke University Durham, NC 27708, USA ABSTRACT Energy consumption is an important performance parameter for portable and wireless embedded systems However, energy consumption must be carefully balanced with real-time responsiveness in hard real-time systems We present an optimal offline dynamic voltage scaling (DVS) scheme for dynamic power management in such systems A generalized network flow model for the uniprocessor DVS problem is developed and solved optimally using an efficient network flow algorithm The proposed method outperforms existing DVS schemes for several popular embedded processors where the number of processor speeds is limited to a few values The GNF model provides theoretical lower bounds on energy consumption using DVS in hard real-time systems 1 Introduction Energy consumption is an important performance parameter for battery-operated embedded systems An effective approach to power reduction in embedded systems is based on dynamic voltage scaling (DVS), a run-time technique that exploits the quadratic dependence of power consumption of a CMOS processor on its operating voltage However, voltage reduction results in a drop in the CPU operating frequency and an increase in task execution times Therefore, to ensure that no task deadlines are missed, DVS must be performed judiciously in hard real-time systems In this paper, we present an optimal offline DVS scheme for hard real-time systems that minimizes the energy consumed by a given set of jobs The DVS problem is modeled as a generalized network flow (GNF) graph and solved using the generalized network simplex algorithm [1] We consider a scenario where a set of jobs execute on a processor that is capable of running at a limited number of speeds The solution of the GNF model results in an offline job schedule and an assignment of speeds to jobs such that (i) the total energy consumed by the set of jobs is minimized, and (ii) no job deadlines are missed We also prove that the proposed method solves the DVS problem optimally The models scale well to real-life task sets for a small number of processor speeds Experimental results indicate that the GNF model for several embedded processors with a limited number of speeds leads to better solutions than one of the most efficient offline DVS schemes presented in the 1 This research was supported in part by DARPA under grant no N , and in part by a graduate fellowship from the North Carolina Networking Initiative, It was also sponsored in part by DARPA, and administered by the Army Research Office under Emergent Surveillance Plexus MURI Award No DAAD Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the sponsoring agencies Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee ICCAD 03, November 11-13, 2003, San Jose, California, USA Copyright 2003 ACM /03/0011 $ literature [9] Our results also show that the number of available processor speeds is an important consideration for the design and evaluation of DVS algorithms 2 Related Prior Work Voltage scheduling with voltage switching during task execution has been studied in [7] Offline, preemptive task scheduling for minimum-energy has been studied in [11] An online DVS technique based on the rate-monotonic algorithm is presented in [10] In [9], a near-optimal offline fixed-priority scheduling algorithm is described These methods are some of the most efficient DVS schemes published thus far in the literature However, a drawback of the DVS methods listed above is that they either assume a processor model with a large range of operating frequencies and voltages, or they are based on a specific scheduling strategy For example, [9, 10] perform fixed-priority scheduling for a processor with a frequency range of 100MHz to 8MHz, adjustable in increments of 1MHz, while [11] describes dynamic-priority scheduling with a continuously variable voltage spectrum Many embedded processors do not provide such a fine-grained or continuous range of frequencies The Transmeta Crusoe processor, for example, has a frequency range of MHz, with the frequency adjustable in increments of 100MHz [3] The AMD K-6 also uses a small range of frequencies from MHz, adjustable in 50MHz increments [2] Therefore, in practice, there may only be a limited set of available speeds at which tasks can be executed Here, we show that with this limitation on processor speeds, the DVS problem can be modeled as a network flow graph and such models can be solved optimally in reasonable periods of time Our experimental results also show that the effectiveness of the techniques described in prior work degrades with a reduction in the number of processor speeds On the other hand, the proposed method, which is based on network flow theory, is very effective when the processor model assumes a limited set of speeds 3 Problem Statement 31 Problem Description P cpu: We are provided with a set J = {j 1, j 2,, j n} of n jobs and a processor that is capable of operating at one of k speeds S 1 > S 2 > > S k These speeds correspond to k unique CPU operating voltages V 1 > V 2 > > V k Each job j i J is characterized by (i) a release time a i, (ii) a deadline d i, and (iii) worst-case execution times c i1 < c i2 < < c ik at the k different operating speeds

2 Job nodes Speed nodes Interval nodes The goal is to determine a voltage v i {V 1, V 2,, V k } and a start time t i for each job j i such that: (i) the total energy, measured in terms of n v2 i c i, c i {c i1, c i2,, c ik }, is minimized, and (ii) t i a i and t i + c i d i We present here a GNF model for P cpu with two speeds S h and S l GNF models for more than two speeds can be easily developed in a manner similar to the 2-speed case The details are omitted here due to lack of space 32 Network Flow Models The objective of a network flow model is to move some entity (electricity, water, time, etc) from one point to another through an underlying network as efficiently as possible Let G = (N, A) represent a directed network defined by a set N of n nodes and a set A of m directed arcs Each arc (i, j) has an associated cost C ij that denotes the cost per unit flow on that arc The flow cost varies linearly with the amount of flow Also associated with each arc (i, j) is an upper bound u ij on the arc capacity that denotes the maximum flow on the arc and a lower bound l ij that denotes the minimum flow on the arc A positive multiplier µ ij exists for every arc (i, j) of the network and if 1 unit of flow is sent from node i to node j along arc (i, j), then µ ij units of flow arrive at node j Each node i N is characterized by an integer b(i) representing its supply/demand The decision variables in the generalized network flow problem are the arc flows on each arc (i, j) and are represented by x ij The generalized network flow problem is an optimization problem that can be formulated as follows: Minimize subject to 1 j:(i,j) A x ij 2 l ij x ij u ij (i,j) A j:(j,i) A C ijx ij µ jix ji = b(i) i N, and Constraint (1) is called the mass balance constraint It states that the difference between the flow into a node j and the flow out of node j must equal its supply/demand of flow However, in a circulation version of a network flow problem, such as the one used here, b(i) = 0 i N Therefore, no flow can accumulate at any node i N Constraint (2) is called the flow constraint It states that the flow in any arc (i, j) must lie between its lower and upper capacity bounds In any feasible solution of the GNF model, every arc flow x ij satisfies both these constraints All flows, capacities and bounds are assumed to be integers GNF models are solved using the generalized network simplex algorithm The generalized network simplex algorithm maintains a feasible basis structure at every iteration and by performing pivot operations, it transforms a solution into a better one until the solution satisfies specific optimality criteria The worst-case complexity of the generalized network simplex algorithm cannot be bounded by a polynomial function of the number of nodes n and number of arcs m However, empirical studies have revealed that the running time of the algorithm is generally a low-order polynomial of n and m A feasible solution for the GNF model is a set of flows x ij that satisfies all flow constraints and capacity constraints In the GNF s j 1 j 2 j n Speed selection arcs [c ih, c ih, P h, 1] [0, 1, P l c il, c il - c ih ] s 1l s 1h s 2l s 2h s nl s nh 2n-1 [0,, 0, 1] Task scheduling arcs [0,, 0, 1] Interval arcs [0, k, 0, 1] Figure 1 The GNF model for P cpu model for P cpu, the decision variables x ij represent the execution time for each job The identification of the execution times trivially leads to an assignment of speeds to the jobs 4 GNF Model for P cpu In this section, we detail the construction of a GNF model for P cpu with two discrete speeds S h and S l, with supply voltages V h and V l, respectively Depending on the execution speed, a job j i has execution times c ih or c il First, the arrival times and deadlines of the n jobs are sorted in ascending order, resulting in 2n elements in the set P = {p 0, p 1, p 2,, p 2n 1} that divide the hyperperiod H into a sequence of at most 2n 1 distinct sub-intervals Let i represent the time interval [p i, p i 1], ie, i = p i p i 1, 1 i 2n 1 The GNF model for P cpu with two speeds consists of a special source node s, a special sink node t, a set of nodes that represent the jobs, a set of nodes that represent execution speeds for the jobs, and a set of nodes that represent time intervals in which jobs can be scheduled Each job node j i has two associated speed nodes s ih and s il (see Figure 1) Each interval node k represents a distinct time interval in P ( k = [p k, p k 1 ]) The arcs in the GNF model are described below Each arc (i, j) is described by a 4-tuple [l ij, u ij, C ij, µ ij] representing the lower bound, upper bound, cost and multiplier respectively 1 Speed identification These arcs are constructed from source s to each job vertex j i They are described by the 4-tuple [c ih, c ih + 1, 0, 1] The lower bound represents the execution time of j i at the high speed, and the upper bound is simply c ih + 1 The integer restriction on arc flows forces the flow value to one of two values, thereby ensuring that only one of the two allowable speeds is chosen for job execution t 22

3 2 Speed selection These arcs are constructed from job node j i to its associated speed nodes s ih and s il The sum of the flows along this pair of arcs for each job node represents its assigned execution time We describe each type of speed selection arc individually: Job execution at S h The flows along the arcs from j i to s ih represent the execution of job j i at speed S h In any feasible schedule, the flows on these arcs is equal to c ih The associated costs are indicative of the power consumed by job j i at the high speed, ie, C ji,s ih = P h, where P h = V 2 h Job execution at S l A flow of 1 unit along the arcs from j i to s il indicates that j i is executed at the low speed S l 3 Interval assignment These arcs are constructed from speed nodes s ih and s il to each interval node k if a i p k and d i p k+1 They are described by the 4-tuple [0,, 0, 1], These arcs ensure that jobs can be scheduled only within certain intervals, ie, between their arrival times and deadlines 4 Time intervals These arcs are constructed from each interval node k to sink node t with parameters [0, k, 0, 1] The capacities of these arcs represent the amount of time available each interval The solution of the GNF model results in a set of flows x ij along the arcs The flow along each arc (s, j i) represents the execution time of job j i From this, the assigned execution speed for each job is determined The job schedule is determined by inspecting the flows along the (s ih, k ) and (s il, k ) arcs These flows represent the amount of CPU time that is allotted to job j i in interval k In some real-time systems, priority-based execution of tasks is an important consideration In the GNF model, job priorities are not used for the purposes of scheduling Tasks of lower priority can therefore potentially preempt tasks of higher priority However, it can be proved that any feasible solution of the GNF model can always be transformed into a prioritized task schedule The details are omitted here due to lack of space 41 Proof of Energy-Optimality In order to show that the GNF model for P cpu results in energyoptimal solutions, it is sufficient to show that the costs associated with the arcs accurately represent the energy consumptions of the jobs at the different speeds, and to show that the objective function and constraints in the GNF model are equivalent to an optimization problem formulation of P cpu We first show that the cost functions of the arcs accurately model the energy consumption of the jobs at different speeds This is formalized through the following theorem: Theorem 1 The costs associated with the speed selection arcs in the GNF model for P cpu are representative of the energy consumption of the jobs at the different operating speeds Proof: Recall from Section 32 that the flow into a job node j i can be either c ih or c ih +1 Depending on the value of the flow entering j i, job j i consumes different amounts of energy A flow of c ih into j i corresponds to its execution at the high speed, while a flow of c ih + 1 represents its execution at the low speed Since the mass balance constraint states that no flow can accumulate at j i, any flow entering j i must leave through the speed selection arcs Consider the following two cases: Case 1: c ih units of flow entering j i In this case, j i has been assigned the higher speed for execution Since the lower bound for arc (j i, s ih ) is equal to c ih, all flow entering j i exits through this arc The cost function associated with arc (j i, s ih ) is P h (= V 2 h ) Recall that this cost represents the cost of transporting a single unit of flow along an arc Therefore, the cost of transporting c ih units of flow is P h c ih, which is the energy cost of j i s execution at the high speed Case 2: c ih + 1 units entering j i This represents the assignment of the lower operating speed for job j i Recall that when c ih + 1 units of flow enter j i, c ih units exit through (j i, s ih ) and 1 unit leaves through (j i, s il ) The cost of the (j i, s il ) arc is selected to be P l c il P h c ih Summing the total cost of all the flow leaving node j i, we have: P h c ih + (P l c il P h c ih ) 1 = P l c il, (1) which represents the energy cost of executing j i at the lower speed S l This completes the proof of the theorem It is now easy to see that the separable terms in the objective function of the GNF model are representative of the energy costs of the jobs Since the costs of all the arcs other than the speed selection arcs is zero, no other term in the objective function contributes to the overall flow cost in the objective function We next show that the objective function of the GNF model maps to an optimization problem statement for P cpu This is formalized through the following theorem Theorem 2 The objective function and constraints that are modeled in the GNF model for P cpu are equivalent to the optimization problem statement for P cpu Proof Since we wish to minimize the total energy consumed by the set of jobs, and because E i vi 2 c i, the objective function can be written as: Minimize vi 2 c i (2) The execution time c i and execution voltage v i of job j i can be represented in a sum-of-products form shown below: c i = a i1c ih + a i2c il, and vi 2 = a i1vh 2 + a i2vl 2, where a i1 and a i2 are 0 1 binary variables, ie, a i1 + a i2 = 1 Substituting for c i and v i in Equation (2) and noting that after multiplication, the terms containing the product of a i1 and a i2 are equal to zero, we obtain the following simplified objective function: Minimize (a i1c ih Vh 2 + a i2c il Vl 2 ) (3) We next show that the objective function of the GNF model simplifies to Equation (3) The objective function in the GNF model is: Minimize C ijx ij (4) i:(i,j) A Since the only arcs with non-zero costs are (j i, s ih ) and (j i, s il ), Equation (4) can be re-written as: Minimize (C ji,s ih x ji,s ih + C ji,s il x ji,s il ) (5) 23

4 Any feasible solution of the model always has a flow which is exactly equal to c ih along arc (j i, s ih ) Moreover, C ji,s ih = P h = Vh 2 and C ji,s il = Vl 2 c il Vh 2 c ih Equation (5) can therefore be written as: Minimize [V 2 h c ih + (V 2 l c il V 2 h c ih )x ji,s il ] (6) Simplification of this equation results in the following objective function: Minimize Vh 2 c ih (1 x ji,s il ) + Vl 2 c il x ji,s il (7) It is now easy to observe the similarity between Equations (7) and (3) Specifically, the variables x ji,s il are equivalent to the a i2 variables in the optimization problem formulation Since a i1 = 1 a i2, the correspondence between the two objective functions is proved The constraints in the GNF model can be seen to map exactly to the constraints in the optimization problem statement by a simple inspection These constraints are listed below: 1 An entire job is executed at a single speed 2 For every execution speed, there is a corresponding execution time for a job 3 Each job must start no earlier than its arrival time and must complete execution at its assigned speed no later than its deadline This completes the proof of the theorem 42 Graph Complexity Let n be the number of jobs in the job set, and k be the number of available processor speeds The number of nodes in the network flow graph can be bounded from above through the following analysis: the network flow graph has n job nodes, and each job node has k corresponding speed nodes This results in n + nk vertices The hyperperiod H is divided into 2n 1 distinct sub-intervals, thereby contributing at most 2n 1 interval nodes k Therefore, an upper bound on the number of nodes in the network flow graph is (k + 3)n 1 In a similar manner, an upper bound on the number of arcs can be derived through the following analysis: n arcs exist from the source node s to the job nodes j i, and nk arcs from the job nodes to their corresponding speed nodes In a worst-case scenario, arcs could exist from each of the nk speed nodes to each of the 2n 1 interval nodes, resulting in 2n 2 k nk speed-interval arcs Finally, 2n 1 arcs are constructed from the interval nodes to the sink t An upper bound on the number of arcs in the network flow graph is therefore 2n 2 k + 3n 1 5 Experimental Results We next evaluate the GNF model with several different job sets with varying utilization Our results are compared to the offline DVS algorithm from [9] (VSLP) and the online DVS algorithm from [10] (LPFPS) These methods assume a processor with a large range of operating frequencies from 8MHz to 100MHz that is adjustable in increments of 1 MHz To best of our knowledge, the Transmeta Crusoe AMD K6-IIIE Voltage Frequency Voltage Frequency 13V 800 MHz 18V 500 MHz 12V 667 MHz 17V 450 MHz 11V 533 MHz 16V 400 MHz 10V 400 MHz 15V 350 MHz 09V 300 MHz 14V 300 MHz Table 1 Speed and voltage settings for the processor models used in our experiments results presented in [9] are the best results published thus far in the literature To demonstrate that the GNF method performs better than these algorithms for processors with only a few speed settings, we also compare our method with modified versions of VSLP and LPFPS, which we call 5-VSLP and 5-LPFPS, that use only five speeds We use two different processor models for our experiments The first model is that of a Crusoe processor and our second model corresponds to an AMD K-6 processor The voltage and frequency settings for these processors are listed in Table 1 Our job sets are constructed from pure periodic task sets The periods and execution times of these tasks are generated randomly We assume that the deadline of a task is equal to its period Our models are solved using a Microsoft Excel-based GNF solver [5] running on a Pentium IV PC at 500MHz with 256MB of RAM The comparison of the GNF method with VSLP and 5-VSLP are presented in Table 2 For task sets with low processor utilization, the results for 5-GNF and 5-VSLP are identical This is because in these cases, the highest voltage used for scheduling is less than the lowest voltage available in the processor Therefore, the execution voltages of jobs that execute at voltage values less than 09V (for the Crusoe processor) are raised to 09V However, at higher utilization values, 5-VSLP performs worse than 5-GNF VSLP operates by distributing slack uniformly to all tasks within a critical interval In order to do this, it computes the utilization of a critical interval and then scales down the frequency of execution in this critical interval such that each task in the critical interval sees an equal increase in execution time (yet no tasks miss their deadlines) Thus, within each critical interval, all tasks run at a constant voltage For example, consider the case where the CPU is allowed to assume only two frequencies, say, V max, and V max/2 If the constant minimum voltage required by VSLP to schedule a set of tasks in a critical interval is 057V max, the set of tasks cannot be scheduled at 05V max because this exceeds the utilization of the interval! Therefore, the next highest speed, which is V max, is chosen to schedule all the jobs within the critical interval This causes an increase in the energy consumption using VSLP with a processor with a few speeds In the GNF model, this equal allocation of slack does not take place Jobs are allocated slack in a lump-sum manner, which is more effective than an equal distribution of slack to all tasks when processor utilization is high A comparison of 5-GNF with 5-LPFPS is shown in Table 3 For all values of utilization, the energy consumptions of the GNF schedules are lower than the energy consumptions of the schedules generated using 5-LPFPS For low processor utilization, LPFPS outperforms 5-GNF (recall that VSLP and LPFPS assume a large range of processor voltages) LPFPS and 5-LPFPS are online DVS techniques in which the execution voltage is scaled down only when a single task is present in the ready queue This does not occur very 24

5 Processor Processor Energy consumption ( vi 2c i) E = Execution time E model utilization VSLP [9] 5-VSLP 5-GNF 5GNF E 5V SLP % VSLP 5-VSLP GNF E 5V SLP % 001s 001s 6s % 001s 001s 1s Crusoe % 001s 001s 9s % 001s 001s 48s % 002s 002s 1m,3s % 001s 001s 1m, 5s CNC (048) % 05s 05s 1h, 23m % 001s 001s 6s % 001s 001s 1s AMD % 001s 001s 9s % 001s 001s 25s % 002s 002s 1m, 3s % 001s 001s 1m Table 2 Comparison of GNF and VSLP [9] Processor Energy consumption ( vi 2c i) E = E Model Util LPFPS [10] 5-LPFPS 5-GNF 5G E 5L E 5L % % Crusoe % % % % % % AMD % % % % Table 3 Comparison of GNF and LPFPS [10] Processor Voltages Energy ( v 2 i c i) utilization used (V) VSLP GNF E 55,33 (Intel) % ,33,22 (Motorola) % 55,33 (Intel) % ,33,22 (Motorola) % 55,33 (Intel) % ,33,22 (Motorola) % 55,33 (Intel) % ,33,22 (Motorola) % Table 4 Impact of number of speeds on energy frequently and therefore, most task instances are executed at the highest voltage to ensure schedulability In order to demonstrate the scalability of the GNF method, we also solved a network flow graph for a task set used in a real-life CNC machine controller system [6] The 5-GNF method performed at par with 5-VSLP due to the relatively low processor utilization of the CNC task set Finally, we consider the DVS problem for processor models with less than five speeds This comparison highlights the importance of the number of available speeds as an important evaluation parameter for DVS algorithms Our results for these processors are shown in Table 4 Examples of such embedded processors are the embedded SL enhanced Intel486 DX2 processor [4], which operates at 55V and 33V, and the Motorola 6805 [8], which operates at 55V, 33V, and 22V GNF outperforms VSLP for all task sets with high processor utilization 6 Conclusions We have developed a generalized network flow model for the DVS problem in hard real-time systems where the CPU can operate at a small number of admissible speeds The GNF method for discrete-speed processors performs nearly as well as other DVS schemes that use a processor model with a large frequency spectrum For task sets with low processor utilization, the GNF method performs as well as competing methods, and for task sets with high processor utilization, the GNF method outperforms these techniques The solutions generated using the GNF method are provably optimal and represent the theoretical upper bounds on energy savings that can be achieved using DVS and therefore serves as a baseline against which other DVS schemes can be compared References [1] R K Ahuja, T L Magnanti and J B Orlin Network Flows: Theory, Algorithms and Applications Prentice-Hall, Englewood Cliffs, NJ, 1993 [2] AMD PowerNow! Technology Platform Design Guide for Embedded Processors AMD Document number 24267a, December 2000 [3] Transmeta Crusoe TM5500 Data Sheet docshtml [4] Intel embedded SL Enhanced 486DX2 processor [5] P A Jensen and J F Bard Operations Research Models and Methods Available online at wwwormmnet [6] N Kim, M Ryu, S Hong, M Saksena, C Choi and H Shin Visual assessment of a real-time system design: case study on a CNC controller Proc Real-Time Systems Symp, pp , 1996 [7] W-C Kwon and T Kim Optimal voltage allocation techniques for dynamically variable voltage processors Proc DAC, pp , 2003 [8] Motorola 6805 processor [9] G Quan and X Hu Energy efficient fixed-priority scheduling for real-time systems on variable voltage processors Proc DAC, pp , 2001 [10] Y Shin and K Choi Power conscious fixed priority scheduling for hard real-time systems Proc DAC, pp , 1999 [11] F Yao, A Demers and S Shenker A scheduling model for reduced CPU energy Proc IEEE Annual Foundations of Computer Science, pp ,