Speed Scaling To Manage Temperature Nikhil Bansal IBM T.J. Watson Kirk Pruhs University of Pittsburgh

February 25, 2005STACS Microprocessor Power Increasing Exponentially P6 Pentium ® Year Power (Watts) Source: Borkar, De Intel 

February 25, 2005STACS Why worry about power ? Most Obvious Answer: Batteries have finite energy Expected battery lifetime increase over the next 5 years: 30 to 40% From Rabaey, Rechargable Lithium Year Nickel-Cadmium Ni-Metal Hydride Nominal Capacity (W-hr/lb)

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

February 25, 2005STACS 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 ( code- named 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, 2005STACS Laptops may damage male fertility  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, 2005STACS Pentium 4

February 25, 2005STACS Power (Heat) Dissipation Illustration

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

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

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

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

February 25, 2005STACS Offline YDS Algorithm [YDS 95]  Repeat  Find the interval I with maximum intensity  Intensity of time interval I = Σ w i / |I| Where the sum is over tasks i with [r i, d i ] in I  During I  speed = to the intensity of I  earliest deadline first scheduling policy  Remove I, and the jobs completed in I

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

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

February 25, 2005STACS YDS Example(3)  Final YDS Schedule  Height = processor speed  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=∞.  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, 2005STACS  Algorithm Description: Speed k(t) at time t = e * maximum over all t 2 > t of Σ w i / (t 2 – t 1 )  Sum is over jobs i with t 1 = et – (e-1)t 2 < r i < t and d i < t 2 BKP Algorithm tt2t2 riri didi didi t 1 = et – (e-1)t 2 current time

February 25, 2005STACS BKP Analysis  Theorem [BKP 2004] BKP completes all jobs by their deadlines  Main Theorem: BKP is O(1)-competitive with respect to temperature  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  Hilbert’s Theorem, Hardy and Littlewood inequalities

February 25, 2005STACS Conclusion: Future Work  Try to understand speed scaling better by studying other scheduling problems/objectives  Some results on flow time in [PUW 2004]  Consider the energy-bound and/or temperature- bound variation of your favorite scheduling problem  Energy-bound constraint: Total energy used ≤ E = Energy in battery  Temperature-bound constraint: Maximum temperature ≤ T max = Thermal threshold of the device  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.