Post on 11-Feb-2016
description
Optimality, Fairness and Robustness in Speed Scaling Designs
Lachlan Andrew
Joint work with
Minghong Lin, Caltech
Adam Wierman, Caltech
Supported by ARC, NSF
speedpo
wergreener
faster
Systems must balance delay and energy use
Speed scaling
state-dependent controllable service rates
Goal: min [ ] [ ] E Delay E Energy
( )P s power to run at speed s
A simple model
( )P s s (often α ≈ 2)CPUs, disks:
A simple model
The speed of the server
What order to serve jobs
??
Fundamental questions about speed scaling
Can on-line speed scaling be optimal?What structure do optimal algorithms have?
How does speed scaling interact with scheduling?
Are sophisticated speed scalers better?
Are there drawbacks of speed scaling?
A simple model
The speed of the server
What order to serve jobs
?
What can we say about the optimal design?
?
A simple model
The speed of the server
What order to serve jobs
?SRPTShortest Remaining
Processing Time
A simple model
The speed of the server
What order to serve jobs
?SRPT…but in practice often…
PSProcessor Sharing
An example – batch arrivals under SRPTn jobs of size xi arrive at time 0 (no other arrivals)
Cost
1 11 1
1 1 1( ) ( )n n n nn n n n
n nx P s x P ss s s s
SRPT
1( )1n n
nP s s s P
“Energy-proportional”
1 11
1 1
( ) ( )n n n n nn n
n n n n n
x x x x xP s P ss s s s s
Observation: Speeds only depend on number of jobs in system (not on sizes)
2 design choices
The speed of the server
What order to serve jobs
SRPTorPS
sn =?
SRPT
PS
Stochastic analysis(M/GI/1)
Worst-case analysis(no assumptions)
Two styles of analysis
Worst case analysis
Undertest
optimum cost
cost
At most factor k worse= “k-competitive”
How can we find kwithout knowing OPT?
Amortized Local competitiveness
• (0)=0• () 0• has no upward jumps
𝑑𝑐𝑜𝑠𝑡 (𝐴)𝑑𝑡 ≤𝑘 𝑑𝑐𝑜𝑠𝑡 (𝑂𝑃𝑇 )
𝑑𝑡 − 𝑑𝜑𝑑𝑡
Potential function
SRPT
PS
Stochastic analysis(M/GI/1)
Worst-case analysis(no assumptions)
(SRPT, P-1(n))Exactly 2-competitive
(PS, P-1(n))(4-2)-competitive?
PS
M/GI/1
PS
M/M/1
FCFS
M/M/1
0 1 2 3λ λ λ λ
s1 s2 s3 s4
“Textbook” stochastic control problem:
Can “solve” numerically with dynamic programming[Bertsekas] [Ross] [George & Harrison 01] …
The analysis
1/32 min ,2nn s n
n
~ns n
Theorem:In an M/GI/1 queue with α=2, the optimal speeds satisfy
For large n,
arriving load
Our Results
Wierman, A., Tang: Allerton 2008, INFOCOM’09
SRPT
PS
Stochastic analysis(M/GI/1)
Worst-case analysis(no assumptions)
(SRPT, P-1(n))Exactly 2-competitive
(PS, P-1(n))(4-2)-competitive
ns n
ns n ( 2)
( 2)
SRPTlower bound
100
10
11 1000.01
Cos
t per
job
low load and/orlow energy cost
high load and/orhigh energy cost
PS
Switching to SRPT can’t improve
performance much!
How well does it work?
Fundamental questions about speed scaling
Can on-line speed scaling be optimal?What structure do optimal algorithms have?
^and answers
2-competitive by using SRPT and 1( )ns P n
Fundamental questions about speed scaling
Can on-line speed scaling be optimal?What structure do optimal algorithms have?
How does speed scaling interact with scheduling?
Are sophisticated speed scalers better?
Are there drawbacks to speed scaling?
^and answers
They can be decoupled with littleperformance loss
Energy proportionalityworks well under SRPT & PS
Can on-line speed scaling be optimal?What structure do optimal algorithms have?
2 design choices
The speed of the server
What order to serve jobs
SRPTorPS Time
Gated static
Time
Dynamic
spee
d
SRPT
PS
Stochastic analysis(M/GI/1)
Worst-case analysis(no assumptions)
[ ] ( )(1 / ) log(1/ (1 / ))
E N P ss s
1s
(heavy traffic)
Gated static speed scaling
( 2)
Gated Static vs. Dynamic (under SRPT)
dynamicgated static
100
10
11 1000.01
Cos
t per
job
low load and/orlow energy cost
high load and/orhigh energy cost
Why is dynamicspeed scaling used?
…same holds under PS
Robustness
Why is dynamicspeed scaling used?
real ρ
design ρ
Cos
t per
job
gated static
Dynamic(stochastic control)
Dynamic (worst-case control)
Robustness/Optimality Tradeoff
Theorem:For P(s)=s2, speeds optimal for PS with ρ are
2(2ρ2 +2ρ + 1)-competitive under SRPT 6(2ρ2 +2ρ + 1)-competitive under PS
Tradeoff betweenoptimality and robustnessthrough tuning estimated load
worst-case result for optimal stochastic control
Can we improverobustness further?
real ρ
design ρ
Cos
t per
job
gated static
dynamic
linear: ns n
For α=22 2( ) 2 ( )linear dynamiccost cost
Robust against arrival rateif arrivals are Poisson
Fundamental questions about speed scaling
Can on-line speed scaling be optimal?What structure do optimal algorithms have?
How does speed scaling interact with scheduling?
Are sophisticated speed scalers better?
^and answers
Benefit of dynamic is robustness,not mean performance
Fundamental questions about speed scaling^
and answers
Can on-line speed scaling be optimal?What structure do optimal algorithms have?
How does speed scaling interact with scheduling?
Are sophisticated speed scalers better?
Are there drawbacks to speed scaling?Speed scaling can “magnify/create” unfairness
Job types that are often served when the queue is large get faster service
Which jobs are run fast?
Magnifies the bias towards small jobs
Creates a bias towards big jobs
Remains fair
SRPT
FCFS
PS
…the ones run when the queue is big
Theorem:With dynamic speed scaling:
[ ( )] [ ( )]lim limSRPT PS
x x
E T x E T xx x
[ ( )] [ ( )]lim limSRPT PS
x x
E T x E T xx x
Bias against large jobs is magnified
SRPT is “fair” to large job sizes
With constant speeds:
[Sigman, Harchol-Balter, Wierman 02]
Fundamental questions about speed scaling
Can on-line speed scaling be optimal?What structure do optimal algorithms have?
How does speed scaling interact with scheduling?
Are sophisticated speed scalers better?
Are there drawbacks of speed scaling?
^and answers
Speed scaling can “magnify/create” unfairness
Optimality, Fairness and Robustnessin Speed Scaling design
Optimality, Fairness and Robustness
(SRPT, P-1(n))
Optimality, Fairness and Robustness
(SRPT, P-1(n)) (PS, P-1(n))
Optimality, Fairness and Robustness
(SRPT, P-1(n)) (PS, P-1(n))
(SRPT, gated-static)