Locality of Reference and the Use of Sojourn Time Variance for Measuring Queue Fairness

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Locality of Referenceand the Use of Sojourn Time
Variance for Measuring Queue Fairness

• David Raz
• School of Computer Science, Tel Aviv University
• Jointly with
• Hanoch Levy, Tel Aviv University
• Benjamin Avi-Itzhak, RUTGERS University
• EURANDOM, September 2005

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Motivation which system is more fair?
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So what is unfairness about?

• Personal measureof discrimination
• Unfairness of ascenario/sample path
• Unfairness of a system

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Research Methodology
Propose a Measure
Basic Analysis Show that the measure fits common
intuition in simple cases
Use measure in complicatedcases where there is
nocommon intuition. Practical!
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Wasnt the problem already solved? Kingman, 1962

• The purpose of this present note is to consider
  the variance of waiting time, and we shall prove
  that this is minimum when the customers are
  served in the order of arrival. Thus this is, in
  a sense, the fairest queue discipline
• (J. F. C. Kingman, The effect of queue discipline
  on waiting time variance, Proceedings of the
  Cambridge Philosophical Society 58 (1962)
  163-164)
• Note non-preemptive disciplines

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So what does this imply?The common approach to
unfairness

• Measure single customer discrimination by
  measuring waiting/sojourn time
• Measure system unfairness by measuring variance
• Inequalities ? Unfairness

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So what is wrong with this?Consider the
following

• Arrivals only in 6 unit intervals
• All customers require one unit of service
• Either 2 or 4 customers arrive simultaneously
  (50 chance), all served PS

6 units
Time
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Observations

• All customers are treated fairly. Non is
  discriminated
• Some wait 1 unit, some wait 3
• The variance is 0.9
• Why?

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Consider II

• Same scenario, customers are served FCFS
• Mean waiting time is 1.167
• Is the second customer in a busy period with two
  customers really discriminated positively?
• Can a customer even tell?

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So why is the variance wrong

• Req 1 Measurements should compare customers
  whose service (prioritization) can affect each
  other ? customers in the same busy period
• Lemma customers can affect each other iff they
  are in the same BP
• Lemma Variance Between BP Within BP
• Should ignore between BP
• Req 2 Customers should be able to tell if they
  are positively or negatively discriminated ? know
  the mean in advance ? mean of all BPs should be
  equal

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Equivalence of requirements

• Observation
• Req 1 and Req 2 are equivalent!
• If the mean is equal for all busy periods, there
  is no variance between busy periods, only within
  them.

LOCALLY MEASURED
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Approach for finding measure

• Keep the idea
• Measure discrimination
• Use variance
• Lets examine the newly proposed RAQFM (and other
  methods) in this light

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RAQFM Philosophy
Equal Share of Resources
? Fairness
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RAQFM - How to Apply the Philosophy Individual
Discrimination

• At every epoch t with N(t) customers in the
  system, each customer should get 1/N(t)
• Warranted service
• Granted service
• Compare the warranted service with the granted
  service discrimination

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Observations on RAQFM

• Theorem Mean for every busy period is zero
• Customers can tell easily if they are positively
  or negatively discriminated
• If mean is zero there is no variance between busy
  periods

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Uniqueness

• Theorem
• RAQFM (and some close relatives) is unique in
  this property, within a large group of
  measurements
• Proof outline other measures cannot achieve
  equal means. If one achieves equal mean for some
  scenario we can build a scenario where it does
  not

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Some example of built busy periods
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Intermediate Conclusion

• The variance of waiting/sojourn time is not
  locally measured
• Being locally measured is import ant
• variances between and within busy periods
• Customers can tell if they are positively or
  negatively discriminated
• RAQFM is locally measured
• And uniquely so

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Research Methodology
Propose a Measure
Basic Analysis Show that the measure fits common
intuition in simple cases
Use measure in complicatedcases where there is
nocommon intuition. Practical!
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Property Reaction to Seniority

• Theorem If customers have equal service
  requirements
• For each pair of customers, it is more fair to
  serve the senior first
• ?FCFS is the most fair
• ?LCFS is the least fair
• (Proof sketch compare scenarios)

RAQFM Reacts Well to Seniority
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Property Reaction to Size

• Theorem If customers arrive together
• For each pair of customers, it is more fair to
  serve the shorter first
• ?SJF is the most fair
• ?LJF is the least fair
• (Proof sketch prove
  )

RAQFM Reacts Well to Size
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Is Short Job Prioritization Justified?

• Theorem Let s and s be two alternate service
  requirements of C. For a large group of service
  policies
• Proof sketch discrimination can be broken down
  to waiting service. The first is identical, the
  second is larger for larger service requirement.
• Theorem Let S and S be random variables
  representing two alternate service requirements
  of C. For a large group of service policies
• (holds for G/G/m)

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• Conclusion If no prioritization is done
• short jobs are negatively discriminated.
• large jobs are positively discriminated.

Prioritization of smaller jobs is justified
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But not always!
Prioritizing much smaller jobs is fair
Prioritizing larger jobs is unfair
Unfairness
Service time ratio
Prioritizing smaller jobs is not always fair!
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Result Combining Servers

• For every customer mix combining servers is less
  fair than splitting them
• Reason more resource sharing

Unfairness
Service time ratio
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Results Multiple Servers

• When service times are unknown, Global FCFS is
  most fair
• Combining Queues is Fair

Fair!
Not Fair!
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Conclusion

• Locality of Measurement is important
• RAQFM is uniquely locally measured
• RAQFM has many properties that agree with
  intuition (other measures do not)
• It is not hard to derive results for cases where
  there is less intuition (your idea?)