Algorithmic problems in Scheduling jobs on Variablespeed processors

1
Algorithmic problems in Scheduling jobs on
Variable-speed processors

• Frances Yao
• City University of Hong Kong

2
Background

• Energy and heat
• Workstation and server draw energy and produce
  heat
• Portable electronic devices rely on battery

Analysis from Intel 25,000-square-foot server
farm with approximately 8,000 servers consumes 2
megawatts 25 of the total cost for such a
facility
3
Background

• Financial Times (2000)
• Information Technology (IT) consumes about
• 8 of energy in US
• exponential growth?50 of the energy consumption
• Some techniques
• Add several inactive states
• Processor can be set at one of the states if idle
• Extra energy is required to bring the processor
  back to the normal state

4
DVS Technique(dynamic voltage scaling)

• Variable Voltage Processor
• Processor with multiple speeds
• Voltage is proportional to speed
• Power function sp (pgt1)
• Current and future DVS technology
• Intels SpeedStep 2 speeds
• AMDs PowerNow 9 speeds
• Intels Foxton technology 64 speeds
• Operating systems can save energy by scheduling
  jobs wisely
• -- executing as slowly as possible

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DVS Scheduling Model (Yao, Demers and Shenker
(1995))
arearequired cycles

• A set of n jobs
• ak arrival time
• bk deadline
• Rk required CPU cycles
• Preemptive execution
• Schedule S specifies
• 0s(t)lt
• which job is executed at time t
• Cost
• Whats the optimal (Min-Energy)
• schedule
• Good characterization
• ?efficient computation
• Benchmark for heuristics

s
t
jobk
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The Basics

• Each job will be executed at one uniform speed in
  optimal schedule
• Convexity
• Optimal schedule needs at most n different speeds
• the flatter the better
• Strategy
• Determine peak speed s ,
• apply iterative procedure
• to find 2nd peak speed etc.

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Optimal Schedule

• Whats the peak speed in the optimal schedule?
• defines the speed lower bound over
  any I
• s defines peak speed and critical
  interval I
• s over critical interval is feasible
• Extract critical interval, update jobs and repeat
  

speed
I
time
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Developments

• 1995 Model and characterization of optimal
  schedule
• O(n3)
• 2005 Optimal schedule in the discrete model
• O(n3) ? O(nlogn)
• 2006 New scheduling algorithm for the continuous
  model
• O(n3) ? O(n2logn)
• Yao F, Demers A and Shenker S. A Scheduling Model
  for Reduced CPU Energy. FOCS 1995, 374-382.
• Minming Li, Frances F. Yao. An Efficient
  Algorithm for Computing Optimal Discrete Voltage
  Schedules. SIAM J. Comput. 2005, 35(3) 658-671.
• Minming Li, Andrew C. Yao, Frances F. Yao.
  Discrete and Continuous Min-Energy Schedules for
  Variable Voltage Processors. Proceedings of the
  National Academy of Sciences of the USA,
  2006(103) 3983-3987.

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Scheduling Model (Discrete)

• Discrete speed levels
• Discrete Optimal
• Kwon and Kim (2002)
• Compute optimal schedule for the continuous model
• Adjust each jobs optimal speed to adjacent
  levels

si
s
si1
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Can we obtain discrete optimal without
computing continuous optimal?

• For example, what if only two speed levels are
  available?
• Strategy
• Partition
• sufficient to do repeated Bi-partition
• Two-Level scheduling

Two-Level Scheduling
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Bi-partition (relative to some speed s)

• Let sgt0 be given for job set J
• Can we divide jobs into Jhigh and Jlow correctly?
• Identify segments of Thigh and Tlow

speed
Thigh
Thigh
Sopt(t)
s
Tlow
Tlow
Tlow
time
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Bi-partition

• Main tool s-schedule
• an EDF schedule with constant speed s
• Gaps
• Tight deadlines
• Tight arrival times
• J(Tlow)Jlow
• J(Thigh)Jhigh

Thigh
Thigh
Gap
s
1
5
2
3
4
5
8
6
7
9
10
Tlow
Tlow
Tlow
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Bi-partition (Algorithm Outline)

• Gaps always exist (and only exist) in Tlow
• Expand a gap suitably to identify a connected
• component of Tlow
• Delete all jobs intersecting with this component
• New gaps must exist

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Example Bi-partition Algorithm
gap
1
5
4
2
3
4
6
11
7
8
9
10
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Optimal Discrete Schedule

• Strategy
• Partition J into J1,J2,Jd with Bi-partition
• Find Two-Level schedule for Ji with sisi1

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Optimal Discrete Schedule

• Strategy
• Partition J into J1,J2,Jd with Bi-partition
• Find Two-Level schedule for Ji with si si1

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Two-Level Scheduling
s1
s2

• Given job set J and s1gts2 satisfying
• Optimal speeds of jobs in J are between s1 s2
• Compute an optimal (s1, s2)- schedule
• Observation
• Any feasible (s1,s2)-schedule is optimal
• How to obtain a feasible (s1,s2)-schedule?

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Two-Level Scheduling (Algorithm Outline)

• Compute s1-schedule and s2-schedule
• Process jobs reversely by deadlines jn , jn-1 ,
  j1
• Use up all s2-execution time of each job
• Take extra time (if needed) from its s1-execution
  time (all available)

s1
1

i
i1

n
s2
i1

n
1

i
disjoint
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Correctness and Complexity

• Every iteration preserves the existence of
  feasible schedule for the remaining jobs
• No idle time is left in the end
• Total intervals of the final schedule
• Intersection of sorted lists of s1-schedule and
  s2-schedule blocks (can be pre-computed)
• At most one extra interval is introduced
  when scheduling every job
• O(n log n)

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Optimal Discrete Schedule

• Strategy
• Partition J into J1,J2,Jd with Bi-partition
• O(d nlogn)
• Find Two-Level schedule for Ji with si and si1
• O(nlogn)
• Total time O(d nlogn)

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Lower Bound

• Any deterministic algorithm for computing a
  min-energy Discrete Voltage Schedule with dgt1
  voltage levels will require
• A linear reduction from Integer Element
  Uniqueness (IEU) to this problem

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Interesting By-product
Continuous Model
Discrete Model

• Original Method for Continuous Optimal
• Compute iteratively peak speed via convex program
• New Method
• Calculate successive approximations to the entire
  optimal speed curve
• Complexity

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Optimal Continuous Schedule
speed
savr(J)
time
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Online Heuristics
AVR(t)

• Competitive ratio
• AVR (Average Rate)
• Lower bound 4 and upper bound 8
• Yao, Demers and Shenker
  (1995)
• Tight bound 4 for some special job sets
• Li, Liu and Yao (2005)
• Can be adapted to the discrete model with
  competitive ratio 2(k1)2/k, where
  k max ratio of two adjacent speeds
• OPA (Optimal Available)
• Tight bound 4
• Bansal, Kimbrel and Pruhs
  (2004)

t
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AVR for Discrete Model

• Discrete Speed Levels
• kmax si/si1
• Adjustment change speed s to its
• adjacent speed levels
• AVRDoff off-line adjustment of AVR
• AVRDon on-line adjustment of AVR
• (running at higher speed first)

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AVR for Discrete Model
s1
s1
s
s
s2
s2
t
t

• AVRDon
  AVRDoff (knows the future)

• It can be proved that
• E(AVRDon)E(AVRDoff)
• E(AVRDoff) 2(k1)2/k ? E(AVR)
• True for a class of online heuristics

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Analysis of OPA Bansal, Kimbrel and Pruhs
(2004)

• Defining a potential function f
• Let ?f(t) denote the change in the potential due
  to a job arrival at time t. Then ?f(t)0
• At any time t between arrivals,
  saopa(t) - aa saopt(t) df(t)/dt 0
• f(t0)f(t ) 0

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Temperature Model

• The rate of cooling follows Fouriers law
• The rate of cooling is proportional to the
  difference in temperature between the object and
  the ambient environmental temperature
• First order approximation
• T(t)aP(t)-bT(t)
• P(t) supplied power at time t

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Throughput (under max speed constraint)

• Throughput total workload of those
• jobs finished by their deadlines
• Max Throughput NP-hard
• Approx maximizing throughput while Approx
  minimizing energy
• An online algorithm Chan et al. (SODA 2007)
• 14-competitive in throughput
• 68-competitive in energy
• An offline algorithm Li et al. 2007
• 3-approx in throughput 4-approx in energy

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Summary

• Job scheduling for variable speed processor
• Optimal discrete DVS schedule O(n logn)
• Multi-level partition
• Two-level schedule
• Optimal continuous DVS schedule O(n2 logn)
• Find successive approx to optimal speed curve
• Online heuristics

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Conclusion

• Many open problems in DVS scheduling throughput,
  job switches (online offline) etc.
• Algorithmic techniques needed to enable more
  efficient use of energy in various domains
• variable voltage processors
• wireless ad hoc networks
• ? Suitable modeling to capture the essence
• ? Specific problems solutions
• ? Unifying techniques for multiple models
• ? Algorithmic foundations for new paradigms

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Thank you!