FAIRNESS IN QUEUES Adam Wierman Carnegie Mellon University [e-mail] Joint work with Mor Harchol-Balter

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FAIRNESS IN QUEUESAdam WiermanCarnegie Mellon
Universityacw_at_cs.cmu.eduJoint work with Mor
Harchol-Balter
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SCHEDULING SUCCESS STORIES ARE EVERYWHERE
Biersack, Rai, Urvoy-Keller, Harchol-Balter,
Schroeder, Agrawal, Ganger, Petrou, Misra, Feng,
Hu, Zhang, Mangharam, Sadowsky, Rawat, Dinda,
McWherter, Ailamaki, others
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THE ESSENCE OF A SCHEDULING SUCCESS STORY
web server, router, etc.
Goal Minimize user response times
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SRPT WINS BIG
WHAT POLICY SHOULD WE USE?
SRPT
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ARE POLICIES THAT PRIORITIZE SMALL JOBS FAIR TO
LARGE JOBS?
M/GI/1 Preempt-Resume Queue
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WHAT IS FAIRNESS?
Definition A fair decision is one that makes the
proper balance of conflicting interests

• The proper balance is
• determined by
• consulting the majority,
• employing logic,
• referring to the divine.

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when two jobs of the same size arrive one afer
the other...
B
A
1
B
A
2
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when a small job and a large job arrive at the
same time...
B
1
A
B
2
A
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BUT, THE SETTING MATTERS
OUR SETTING Are the response times of large jobs
unfairly long?
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WHAT IS FAIRNESS?
Definition A fair decision is one that makes the
proper balance of conflicting interests

• The proper balance is
• determined by
• consulting the majority,
• employing logic,
• referring to the divine.

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EMPLOYING LOGIC
Aristotles notion of fairness Like cases should
be treated alike, different cases should be
treated differently, and different cases should
be treated differently in proportion to their
differences.
slowdown
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EMPLOYING LOGIC
Rawls Theory of Social Justice All social
goods should be distributed equally,
unless unequal distribution is to the advantage
of the least favored
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EMPLOYING LOGIC
Min-Max fairness (Pareto Efficiency) All jobs
deserve an equal share of the resources ... but
if some jobs can use more without hurting others,
thats okay
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MIN-MAX FAIRNESS
Wierman and Harchol-Balter 2003
Definition Consider an M/GI/1 queue. A policy P
is min-max fair if
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Theorem In an M/GI/1 with an unbounded, continuou
s service distribution having finite EX2, under
any non-idling policy we have and further
Wierman and Harchol-Balter 2003
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MIN-MAX FAIRNESS
Wierman and Harchol-Balter 2003
Definition Consider an M/GI/1 queue. A policy P
is min-max fair if
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PS FCFS SRPT
Processor Sharing
ET(x) / x
PS
x
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PS FCFS SRPT
First Come First Served
The unfairness can be unbounded
ET(x) / x
PS
FCFS
x
Under a Pareto with ?0.8, this is gt80 of the
jobs
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PS FCFS SRPT
Shortest Remaining Processing Time
SRPT
PS
PS
SRPT
Theorem SRPT is fair for ?lt0.5, regardless of the
job size distribution.
Theorem For every service distribution with
finite EX2, SRPT is unfair for high enough
load.
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OTHER POLICIES
Always Unfair
SRPT
PS
Always Fair
Sometimes Fair
FCFS
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OTHER POLICIES
Always Unfair
SRPT
LRPT
Prioritize Small Jobs
Prioritize Large Jobs
Always Fair
PLJF
PSJF
LCFS
Sometimes Fair
ROS
FCFS
SJF
LJF
FB
Wierman and Harchol-Balter 2003, 2005
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OTHER POLICIES
Remaining size based
Always Unfair
SRPT
LRPT
Preemptive size based
Non-preemptive non-size based
Always Fair
PLJF
PSJF
LCFS
Sometimes Fair
ROS
FCFS
SJF
LJF
Age based
Non-preemptive size based
FB
Wierman and Harchol-Balter 2003, 2005
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FAIRNESS VS. EFFICIENCY
Always Unfair
SRPT
LRPT
Always Fair
PLJF
PSJF
LCFS
Sometimes Fair
ROS
FCFS
SJF
LJF
more circles? better mean response time
FB
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Fair Sojourn Protocol (FSP)
Henderson and Friedman 2004
Guarantees that every job finishes earlier than
it would under PS.
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Fair Sojourn Protocol (FSP)
Do SRPT on the PS remaining times

• Compute the time when each job would finish in a
  virtual PS system.
• Order the jobs according to their virtual
  completion times.
• Devote the full processor to the job with the
  earliest PS completion time

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Fair Sojourn Protocol (FSP)
Do SRPT on the PS remaining times
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Fair Sojourn Protocol (FSP)
Do SRPT on the PS remaining times
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...your answer here...
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MIN-MAX FAIRNESS
Wierman and Harchol-Balter 2003
Definition Consider an M/GI/1 queue. A policy P
is min-max fair if
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Raw moments ET(x)i Central moments
VarT(x), etc Cumulant moments
ET(x) ? ?
X
BEYOND EXPECTATION Higher Moments
X
Wierman and Harchol-Balter 2005
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CUMULANTS
Cumulants are a descriptive statistic, similar to
the moments. They can be found as a function
of the moments or from the log of the moment
generating function
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WHYCUMULANTS?
Cumulants have many nice properties
additivity
homogeneity
1st cumulant is shift-equivariant the rest are
shift-invariant
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MIN-MAX FAIRNESS
Wierman and Harchol-Balter 2005
Definition Consider an M/GI/1 queue. A policy P
is min-max fair if, for all i
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MIN-MAX FAIRNESS IS NOT ALWAYS APPROPRIATE
FCFS is the most fair policy in some settings
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TEMPORAL FAIRNESS
It is unfair to violate the seniority of a job
Definition The politeness experienced by a job
of size x under policy P, Pol(x)P, is the
fraction of the response time during which the
seniority of the job is respected.
Wierman 2004
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Definition The politeness experienced by a job
of size x under policy P, Pol(x)P, is the
fraction of the response time during which the
seniority of the job is respected.
FCFS PLCFS PS SRPT
First Come First Served
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Definition The politeness experienced by a job
of size x under policy P, Pol(x)P, is the
fraction of the response time during which the
seniority of the job is respected.
FCFS PLCFS PS SRPT
Preemptive Last Come First Served
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FCFS PLCFS PS SRPT
Processor Sharing
FCFS
FCFS
1 1-? 0
EPol(x)
PS
PS
PLCFS
PLCFS
x
job size -tile
0
1
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FCFS PLCFS PS SRPT
Shortest Remaining Processing Time
FCFS
FCFS
1 1-? 0
SRPT
EPol(x)
SRPT
PLCFS
PLCFS
x
job size -tile
0
1
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MANY INTERESTING QUESTIONS...

• Can you think of a policy (other than FCFS) that
  avoids being maximally impolite to some job size?
• How polite are scheduling heuristics
  techniques?
• What is the tradeoff between temporal fairness
  and proportional fairness? and efficiency?

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PROPORTIONAL FAIRNESS VSTEMPORAL FAIRNESSVS
EFFICIENCY
min-max fairness
politeness
mean response time
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FCFS
FCFS
SRPT
FSP
PS
PLCFS
FSP
SRPT
PS
LRPT
LRPT
PLCFS
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FCFS
SRPT
FSP
PS
more circles? better mean response time
LRPT
PLCFS
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WRAPUP
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FAIRNESS IS AN AMORPHOUS CONCEPT

• a universal definition is unlikely to be useful
  (or possible)

PROPORTIONAL FAIRNESS min-max
TEMPORAL FAIRNESS politeness
...THIS IS A VERY ACTIVE AREA OF RESEARCH,MANY
OPEN QUESTIONS REMAIN
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AN EXPLOSION OF WORK ON FAIRNESS
since the introduction of min-max fairness in 2003
defining other fairness metrics
analyzing policies
Wierman 2004 Raz, Avi-Itzhak 2004 Levy, Raz,
Avi-Itzhak 04 Sandmann 2005
Many papers by Wierman, Harchol-Balter,
Henderson, Friedman, Biersack, Rai,
Ayesta, Aalto, Nunez-Queija, Misra, Feng,
Vernon, Williamson, Brown, Bansal, and others
extending definition to higher moments
Wierman Harchol-Balter 2005a,bWilliamson,
Gong 2003, 2004 Brown 2006 and others.
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MANY OTHER INTERESTING METRICS
Levy, Raz, Avi-Itzhak 04

• DiscFreq ni cmi
• ni number of jobs that arrived later and
  completed earlier than job i
• mi number of larger jobs (at the arrival of job
  i) that complete earlier than job i

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Adam WiermanCarnegie Mellon Universityacw_at_cs.cmu
.eduPapers are available at
http//www.cs.cmu.edu/acw