Power Management Algorithms

1
Power Management Algorithms

• An effort to minimize Processor Temperature and
  Energy Consumption

2
Motivation

• Microprocessor power consumption is increasing
  exponentially

3
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
• www.cs.pitt.edu/kirk/cool.avi
• Laptops may damage male fertility due to
  increased temperature (Reuters December 9, 2004)

4
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

5
Processor Technologies for Power Management

• Speed Scaling
• Processor can operate on multiple speeds
• Intels SpeedStep 2 speeds
• AMDs PowerNow 9 speeds
• Intels Foxton technology 64 speeds
• Power Down
• Processor can operate on multiple power levels
• Can operate on any power level L0, L1, , Ln.
• Ln is normal state. L0, , Ln-1 are idle states
• It costs to bring back processor to Ln

6
Relationship Between Speed and Energy

• P c V2 s
• Minimum voltage V required to run processor at
  speed s. V is roughly linear to s
• Therefore, P c s3
• Generalize to P sp, for some constant p 1
• Energy ?Time P dt
• Speed goes up(down) ? Energy consumption goes up
  (down)

7
Relationship Between Speed and Temperature

• Key Assumption fixed ambient temperature Ta
• First order approximation of temperature
• dT/dt a P b (T Ta) a P b T
• T Temprature
• t time
• P supplied power
• a,b some constants
• For simplicity rescale so that Ta 0

8
Problem Formulation

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

9
Summary of Results
10
Offline YDS Algorithm (1995)

• Repeat
• Find the time interval I with maximum intensity
• Intensity of time interval I S wi / I
• Where the sum is over tasks i with ri,di in I
• During I
• speed to the intensity of I
• Earliest Deadline First policy
• Remove I and the jobs completed in I

11
YDS Example
Release time
deadline
time
12
YDS Example
First Interval
Intensity
Second Interval Intensity green work blue work
Length of solid green line
13
YDS Example

• Final YDS schedule
• 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 8.
• Bansal, Pruhs Consequence of KKT optimality
• Bansal, Pruhs The YDS is at worst 20-competitive
  with respect to temperature for all cooling
  parameters b

14
Why is YDS optimal?

• Convex program
• They are called KKT optimality conditions

The problem has solution if these conditions hold
15
Why is YDS optimal?

• YDS as convex problem
• Break time into intervals t0,tm at release times
  and deadlines
• J(i) tasks feasibly executed in Ii ti,ti1
• Wi,j for j in J(i) work done on j during
  ti,ti1

KKT optimality conditions hold
It took 10 years to prove YDSs optimality!!!
16
Online AVR Algorithm (1995)

• Each job i has av. rate requirement or density
• avri wi/(di ri)
• while(t lt max dj)
• s(t) Savrj(t)
• 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)
17
Online OA Algorithm (1995)

• After each arrival
• 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

18
BKP Algorithm (2004)

• Algorithm description
• Speed k(t) at time t
• e maximum over all t2 gt t of
• Swi/(t2 - t1)
• Sum is over jobs i with t1 et (e-1)t2 lt ri lt
  t and di lt t2
• Bansal, Pruhs BKP is O(1)-competitive with
  respect to temperature

Can be computed by an online algorithm
t1 et (e-1)t2
t
t2
ri
di
di
current time
19
BKP example

• Suppose e 2.7
• t 4

3
5
4
0 1 2 3
4 5 6
20
BKP example

• Suppose e 2.7
• t 4
• For t 5
• t1 et (e 1)t 2.74 (2.7 1)5 10.8
  8.5 2.3

3
5
4
0 1 2 3
4 5 6
21
BKP example

• Suppose e 2.7
• t 4
• For t 5
• t1 et (e 1)t 2.74 (2.7 1)5 10.8
  8.5 2.3
• w(t,t1,t) w(4,2,5) 4

3
5
4
0 1 2 3
4 5 6
22
BKP example

• Suppose e 2.7
• t 4
• For t 5
• t1 et (e 1)t 2.74 (2.7 1)5 10.8
  8.5 2.3
• w(t,t1,t) w(4,2,5) 4
• w(t,t1,t) /e(t-t) w(4,2,5)/2.7(5-4) 4/2.7
  1.5

3
5
4
0 1 2 3
4 5 6
23
BKP example

• Suppose e 2.7
• t 4
• For t 6
• t1 et (e 1)t 2.74 (2.7 1)6 10.8
  10.2 0
• w(t,t1,t) w(4,0,6) 4 5 3 12
• w(t,t1,t) /e(t-t) w(4,0,6)/2.7(6-4) 12/5.4
  2.22

3
5
4
0 1 2 3
4 5 6
24
BKP example

• Suppose e 2.7
• t 4
• So t2 6
• s(4) e2.22 2.7 2.22 6
• Bansal, Pruhs BKP is O(1)-competitive with
  respect to temperature

3
5
4
0 1 2 3
4 5 6
25
Future Work

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