A Maiden Analysis of Longest Wait First

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A Maiden Analysis of Longest Wait First

• Jeff Edmonds
• York University
• Kirk Pruhs
• University of Pittsburgh

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Client-Server System
Requests for page transmission pull
Server
Clients
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Client-Server System
Transmit page
Server scheduling problem How does the server
decide which requests to respond to first?
Server
Clients
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The Big Problem
Movie Distribution
Olympics
Database Replication via Internet
Software Download
Harry Potter Book Download
Pay-Per-View Movies
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The 1-1 communication is not scalable
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Broadcast Common Pages
From www.direcpc.com
From Newsweek magazine
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Time
Requests
Pages
8
Time

• NP-complete EH, 2002

Scheduling Problem
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O(1)-Approximate Algorithms
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O(1)-Approximate Algorithms
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O(1)-Approximate Algorithms
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Resource Augmentation Analysis

• Algorithm is s-speed c-competitive if
• maxI Onlines(I)/Opt1(I) lt c

Time
Time
Requests
Requests
Optimal
Online
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Classic Server QoS Curves
Online
Optimal
Average response time
High load
Low load
Fast processor
Slow Processor
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Scheduling Algorithms

• 2-speed 2-competitive KPV 01, EH 02, GKKW
  02, GKPS 02

Difficult Off-line Linear Programming Algorithms
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Scheduling Algorithms
First In First Out (FIFO)

• 2-competitive for Max-Waitbut bad for Total-Wait.

Time
Requests
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Scheduling Algorithms
Most Requests First (MRF)

• not O(1)-speed O(1)-competitive. KPV 00

Time
Requests
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Scheduling Algorithms
B-Equipoise Proportional to number of requests

• Not 2-speed O(1)-competitive.
• (4e)-speed O(1)-competitivefor any page lengths
  EP 2002

Time
Requests
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Scheduling Algorithms
B-Equipoise-EDF Non-preemptive number of
requests

• (8e)-speed O(1)-competitive for unit sized
  files EP 2002

Time
Requests
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Scheduling Algorithms
Longest Wait First(LWF)

• Best Experimentally AM

• Efficient implementation, KTT 01

• Was hoped to be (1e)-speed O(1)-competitive.

Time
Requests
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LWF is not 1.6-speed O(1)-competitive.
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LWF is not 1.6-speed O(1)-competitive.
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LWF is not 1.6-speed O(1)-competitive.
With s1.6 LWF catches up.
LWF is competitive.
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LWF is not 1.6-speed O(1)-competitive.
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LWF is not 1.6-speed O(1)-competitive.
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LWF is 6-speed O(1)-competitive.

• LWF6(I) lt Opt1(I) x c

Time
Time
Requests
Requests
Optimal
LWF
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LWF is 6-speed O(1)-competitive.

• LWF6(I) lt Opt1(I) x c

xc
Time
Time
Requests
Requests
Optimal
LWF
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LWF is 6-speed O(1)-competitive.
Optimal
LWF
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LWF is 6-speed O(1)-competitive.
Optimal
LWF
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LWF is 6-speed O(1)-competitive.
xc
Optimal
LWF
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LWF is 6-speed O(1)-competitive.
Optimal
LWF
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LWF is 6-speed O(1)-competitive.
LWF
LWF
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LWF is 6-speed O(1)-competitive.
LWF
LWF
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LWF is 6-speed O(1)-competitive.
LWF
LWF
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LWF is 6-speed O(1)-competitive.
Needs to be paid
Able to pay
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LWF is 6-speed O(1)-competitive.
LWF
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LWF is 6-speed O(1)-competitive.
LWF
LWF
LWF
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LWF is 6-speed O(1)-competitive.

LWF
LWF
LWF
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LWF is 6-speed O(1)-competitive.

LWF
LWF
LWF
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LWF is 6-speed O(1)-competitive.


LWF
LWF
LWF
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LWF is 6-speed O(1)-competitive.
LWF6(I) lt Opt1(I) x c
xc
Time
Time
Requests
Requests
Optimal
LWF
Everyone paid enough.
No one pays to much.
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Conclusion
A Maiden Analysis of Longest Wait First

• LWF is best experimentally AM

• Efficient implementation, KTT 01

• Was hoped to be (1e)-speed O(1)-competitive.

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Conclusion
Future
A Maiden Analysis of Longest Wait First

• LWF is best experimentally AM

• Efficient implementation, KTT 01

• Was hoped to be (1e)-speed O(1)-competitive.

• LWF is not 1.6-speed O(1)-competitive.

• LWF is 6-speed O(1)-competitive.

for any file lengths
The End
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Multicast Pull Scheduling When Fairness is Fine

• Jeff Edmonds
• York University
• Kirk Pruhs
• University of Pittsburgh

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Scheduling Algorithms
B-Equipoise Proportional to number of requests

• Not 2-speed O(1)-competitive.
• (4e)-speed O(1)-competitivefor any page lengths
  EP 2002

Time
Requests
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The Power of the Adversary in Multicast Pull

• Basic idea of the proof that there is no
  O(1)-competitive online algorithm
• Immediately after the online algorithm broadcasts
  a document, the adversary requests that document
• The adversary broadcasts the document after the
  second request to the document utilizing the
  power of broadcast
• After a while the online algorithm still has a
  lot of work left while the adversary has little
  work left
• Then a high load stream of work that requires the
  full processing power of the server arrives

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More on the Power of the Adversary in Multicast
Pull

• Hence, the adversary forces the online algorithm
  to labor on sequential work
• Sequential work increasing the processing power
  devoted to the work does not change the rate at
  which the remaining work decreases
• Parallel work doubling the processing power
  devoted to work doubles that rate at which that
  work is completed
• IMHO, the main contribution of this paper is the
  insight that
• Multicast pull scheduling scheduling of jobs
  with arbitrary speed-ups

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Scheduling Jobs with Variable Speed-up Curves In
the Context of Parallel Processing

• Equipoise (Round Robin) Give each job equal
  processing time
• Equipoise is a 3-speed 6-competitive algorithm
  for jobs with arbitrary speed-up curves E, 1999
  
• Formally means that Equipose with a speed 3
  processor has average flow time at most 6 times
  the optimal average flow time for a speed 1
  processor
• Intuitively means that Equipoise will perform
  reasonably well at low loads

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Proof by picture that Bequi is O(1)-speed
O(1)-approximation algorithm
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More proof by picture
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More proof by picture
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More proof by picture
Replace jobs by sequential and parallel work in
such a way that Broadcast-Equipoise is
unaffected, and optimal is not hurt. Then apply
Equipoise analysis.
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Why transformation doesnt hurt optimal
Each reverse L shaped region, which contains the
parallel work that optimal must finish, is
contained within two consecutive BEqui broadcasts
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Scheduling Algorithms
B-Equipoise-EDF Non-preemptive number of
requests

• (8e)-speed O(1)-competitive for unit sized
  files EP 2002

Time
Requests
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BEQUI-EDF Algorithm for Unit Sized Documents (no
preemption)

• B-EQUI-EDF Algorithm
• Simulate BEQUI to get deadlines for jobs
• Run EDF on the jobs using these deadlines
• B-EQUI-EDF is an O(1)-speed O(1)-competitive
  algorithm