Power Management Algorithms An effort to minimize Processor Temperature and Energy Consumption

Motivation  Microprocessor power consumption is increasing exponentially

Motivation  Battery capacity is increasing linearly  Expected battery life increase in the next 5 years: 30 to 40%  Chip manufacturers are close to “thermal wall”  Increase in speed  increase in heat generation  Expensive and noisy cooling systems required  Intel: Tejas and Jayhawk   Laptops may damage male fertility due to increased temperature (Reuters: December 9, 2004)

Motivation  Information Technology (IT) consumes about 8% of energy in US  Exponential growth  50% of energy consumption  Analysis from Intel: 25,000-square-foot server farm with approximately 8,000 servers consumes 2 megawatts -- 25% of the cost of such a facility

Processor Technologies for Power Management  Speed Scaling  Processor can operate on multiple speeds o Intel’s SpeedStep — 2 speeds o AMD’s PowerNow — 9 speeds o Intel’s Foxton technology — 64 speeds  Power Down  Processor can operate on multiple power levels o Can operate on any power level L 0, L 1, …, L n. o L n is normal state. L 0, …, L n-1 are idle states o It costs to bring back processor to L n

Relationship Between Speed and Energy  P = c V 2 s o Minimum voltage V required to run processor at speed s. V is roughly linear to s o Therefore, P = c s 3 o Generalize to P = s p, for some constant p ≥ 1  Energy = ∫ Time P dt  Speed goes up(down)  Energy consumption goes up (down)

Relationship Between Speed and Temperature  Key Assumption: fixed ambient temperature T a  First order approximation of temperature dT/dt = a P – b (T – T a ) = a P – b T  T = Temprature  t = time  P = supplied power  a,b some constants  For simplicity rescale so that T a = 0

Problem Formulation  Input: A collection of tasks, where task I has: o Release time r i when it arrives in the system o Deadline d i when it must finish by o Work requirement w i (number of cycles)  The processor must perform w i units of work between time r i and time d i o Preemption is allowed  Objectives o Minimize energy consumption o Minimize maximum temperature  For each time, the scheduler must specify both o Job Selection: which job to run  may assume Earliest Deadline First policy o Speed Setting: at what speed the processor should run at

Summary of Results

Offline YDS Algorithm (1995)  Repeat o Find the time 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 o During I  speed = to the intensity of I  Earliest Deadline First policy o Remove I and the jobs completed in I

YDS Example Release timedeadline time

YDS Example First Interval Intensity Second Interval Intensity = green work + blue work Length of solid green line

YDS Example  Final YDS schedule o Height = processor speed  YDS theorem: The YDS schedule is optimal for energy, or equivalently for temperature when b = 0. And YDS is optimal for maximum power, or equivalently when b = ∞. o Bansal, Pruhs: Consequence of KKT optimality  Bansal, Pruhs: The YDS is at worst 20-competitive with respect to temperature for all cooling parameters b

Why is YDS optimal?  Convex program o They are called KKT optimality conditions The problem has solution if these conditions hold:

Why is YDS optimal?  YDS as convex problem o Break time into intervals t 0,…t m at release times and deadlines o J(i): tasks feasibly executed in I i = [t i,t i+1 ] o W i,j for j in J(i): work done on j during [t i,t i+1 ] KKT optimality conditions hold It took 10 years to prove YDS’s optimality!!!

Online AVR Algorithm (1995)  Each job i has av. rate requirement or density avr i =w i /(d i – r i )  while(t < max d j ) o s(t) = Σavr j (t) o Apply Earliest deadline First policy  Yao, Demers, Schenker: 4 ≤ AVR ratio ≤ 8 with respect to energy  Bansal, Pruhs: AVR is not O(1)-competitive with respect to temperature AVR(t)

Online OA Algorithm (1995)  After each arrival o Recompute an optimal schedule (YDS alg.) consisting of  Newly arrived job j  Remaining portions of other jobs  Bansal, Pruhs: OA is not O(1)-competitive with respect to temperature

BKP Algorithm (2004)  Algorithm description Speed k(t) at time t = e * maximum over all t 2 > t of Σw i /(t 2 - t 1 ) o Sum is over jobs i with t 1 = et – (e-1)t 2 < r i < t and d i < t 2  Bansal, Pruhs: BKP is O(1)-competitive with respect to temperature tt2t2 riri didi didi t 1 = et – (e-1)t 2 current time Can be computed by an online algorithm

BKP example  Suppose e = 2.7  t =

BKP example  Suppose e = 2.7  t = 4 For t’ = 5 t 1 = et – (e – 1)t’ = 2.7*4 – (2.7 – 1)*5 = 10.8 – 8.5 =

BKP example  Suppose e = 2.7  t = 4 For t’ = 5 t 1 = et – (e – 1)t’ = 2.7*4 – (2.7 – 1)*5 = 10.8 – 8.5 = 2.3 w(t,t 1,t’) = w(4,2,5) =

BKP example  Suppose e = 2.7  t = 4 For t’ = 5 t 1 = et – (e – 1)t’ = 2.7*4 – (2.7 – 1)*5 = 10.8 – 8.5 = 2.3 w(t,t 1,t’) = w(4,2,5) = 4 w(t,t 1,t’) /e(t’-t) = w(4,2,5)/2.7(5-4) = 4/2.7 =

BKP example  Suppose e = 2.7  t = 4 For t’ = 6 t 1 = et – (e – 1)t’ = 2.7*4 – (2.7 – 1)*6 = 10.8 – 10.2 = 0 w(t,t 1,t’) = w(4,0,6) = = 12 w(t,t 1,t’) /e(t’-t) = w(4,0,6)/2.7*(6-4) = 12/5.4 =

BKP example  Suppose e = 2.7  t = 4 So t 2 = 6 s(4) = e*2.22 = 2.7 * 2.22 = 6  Bansal, Pruhs: BKP is O(1)-competitive with respect to temperature

Future Work  Combination of Speed Scaling and Power Down  What about multicore processors?  What about systems with rejuvinative sources (i.e. solar cells)?