# Speed Scaling To Manage Temperature Nikhil Bansal IBM

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Speed Scaling To Manage Temperature Nikhil Bansal IBM T. J. Watson Kirk Pruhs University of Pittsburgh February 25, 2005 STACS 2005

Microprocessor Power Increasing Exponentially Power (Watts) 100 P 6 Pentium ® 10 8086 286 8085 1 8080 8008 4004 486 386 0. 1 1974 1978 1985 Year 1992 2000 Source: Borkar, De Intel February 25, 2005 STACS 2005 2

Why worry about power ? Most Obvious Answer: Batteries have finite energy 50 Nominal Capacity (W-hr/lb) Rechargable Lithium 40 Ni-Metal Hydride 30 20 Nickel-Cadmium 10 0 65 70 75 80 Year 85 90 95 From Rabaey, 1995 Expected battery lifetime increase over the next 5 years: 30 to 40% February 25, 2005 STACS 2005 3

Why worry about power ? Less Obvious Answer 2: Chips get hot February 25, 2005 STACS 2005 4

Intel Hits “Thermal Wall” Reuters Friday May 7, SAN FRANCISCO, May 7 (Reuters) - Intel Corp. said on Friday it has scrapped the development of two new computer chips ( codenamed Tejas and Jayhawk) for desktop/server systems in order to rush to the marketplace a more efficient chip technology more than a year ahead of schedule. Analysts said the move showed how eager the world's largest chip maker was to cut back on the heat its chips generate. Intel's method of cranking up chip speed was beginning to require expensive and noisy cooling systems for computers. February 25, 2005 STACS 2005 5

Laptops may damage male fertility q q Reuters: December 9, 2004 Men should keep their laptops off their laps because they could damage fertility, an expert said on Thursday. Laptops, which reach high internal operating temperatures, can heat up the scrotum which could affect the quality and quantity of men’s sperm. “The increase in scrotal temperature is significant enough to cause changes in sperm parameters, ” said Dr Yefim Sheynkin, an associate professor of urology at the State University of New York at Stony Brook. STACS PC February 25, 2005 STACS 2005 6

Pentium 4 February 25, 2005 STACS 2005 7

Power (Heat) Dissipation Illustration February 25, 2005 STACS 2005 8

Problem Statement: Speed Scaling with Deadlines q Input: A collection of tasks, where task i has m m m q The processor must perform wi units of work on each task i between time ri and time di m q q Release time ri when it arrives in the system Deadline di when it must finish by Work requirement wi Preemption is allowed Objective: minimize the maximum temperature For each time, the scheduler must specify both m m Job Selection: which job to run Ø wlog, may assume Earliest Deadline First policy Speed Setting: at what speed the processor should run at February 25, 2005 STACS 2005 9

The Relationship Between Speed and Power q P = c V 2 s m There is a minimum voltage V required to run the processor at speed s, and V is roughly linear in s. m Therefore P = c s 3 m Generalize q to P = sp for some constant p ≥ 1 Energy E = ∫Time P dt February 25, 2005 STACS 2005 10

Our Basic Temperature Equation q q Key Assumption: fixed ambient temperature Ta Basic temperature equation d. T/dt = a P – b (T – Ta) = a P – b T Fourier Law of Heat Conduction = rate of cooling is proportional to the temperature difference m T = Temperature t = time P = supplied power a, b are constants m For simplicity rescale so that Ta = 0 m m m February 25, 2005 STACS 2005 11

Summary of Results (New, Main Result) Recall d. T/dt = a. P–b. T Energy b=0 Equals Maxt ∫tt+x P dt x=∞ Temperature 0<b<∞ x= Θ(1/b) Maximum Power b=∞ x=infinites imal February 25, 2005 Offline Online Optimal YDS algorithm O(1)-competitive algorithms YDS 1995 Cute correctness proof Ellipsoid Exact BKP 2004 YDS is O(1)-approximation Optimal YDS algorithm YDS 1995 STACS 2005 OA AVR : YDS 1995 BKP : BKP 2004 BKP is O(1)-competitive BKP is strongly O(1)-competitive BKP 2004 12

Offline YDS Algorithm [YDS 95] q Repeat m Find the interval I with maximum intensity Ø Intensity n of time interval I = Σ wi / |I| Where the sum is over tasks i with [ri, di] in I m During I Ø speed = to the intensity of I Ø earliest deadline first scheduling policy m Remove February 25, 2005 I, and the jobs completed in I STACS 2005 13

YDS Example(1) q Input release time February 25, 2005 Area = work of job STACS 2005 deadline 14

YDS Example(2) First Interval Intensity Second Interval Intensity = green work + blue work Length of solid green line February 25, 2005 STACS 2005 15

YDS Example(3) q Final YDS Schedule m q YDS Theorem: The YDS schedule is optimal for energy, or equivalently temperature when b=0. And YDS is optimal for maximum power, or equivalently when b=∞. m q Height = processor speed Our Proof: A cute consequence of KKT optimality Our Theorem: The YDS schedule is at worst 20 -competitive with respect to temperature for all cooling parameters b February 25, 2005 STACS 2005 16

BKP Algorithm q Algorithm Description: Speed k(t) at time t = e * maximum over all t 2 > t of Σ wi / (t 2 – t 1) m Sum is over jobs i with t 1 = et – (e-1)t 2 < ri < t and di < t 2 t 1= et – (e-1)t 2 ri di t 2 current time February 25, 2005 STACS 2005 17

BKP Analysis q q Theorem [BKP 2004] BKP completes all jobs by their deadlines Main Theorem: BKP is O(1)-competitive with respect to temperature m m m Proof: If YDS does y(t) work at time t, then we modify the instance so that y(t) work arrives at time t with deadline t+1 This transformation doesn’t effect YDS and won’t decrease speed/temperature for BKP Show that ∫t t+1/b k(t) dt (an upper bound for the energy used by BKP during a interval of length 1/b) is O(1) times the energy that YDS uses during that interval Ø Hilbert’s Theorem, Hardy and Littlewood inequalities February 25, 2005 STACS 2005 18

Conclusion: Future Work q Try to understand speed scaling better by studying other scheduling problems/objectives m q Consider the energy-bound and/or temperature-bound variation of your favorite scheduling problem m m q Some results on flow time in [PUW 2004] Energy-bound constraint: Total energy used ≤ E = Energy in battery Temperature-bound constraint: Maximum temperature ≤ Tmax = Thermal threshold of the device Ø A cooling oblivious algorithm, that is one that works for all cooling parameters b, will also give an energy bound result Speed scaling can make many scheduling problems more difficult and interesting. Lots of nice problems here. February 25, 2005 STACS 2005 19