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

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Locality of Referenceand the Use of Waiting 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
• TAU CS Networking Seminar May 2006

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Motivation which system is more fair?
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Motivation which system is more fair?
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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 job 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 jobs require one unit of service
• Either 2 or 4 jobs arrive simultaneously (50
  chance), all served PS

6 units
Time
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Observations

• All jobs are treated fairly. None is
  discriminated
• Some wait 1 unit, some wait 3
• The variance is 0.9
• Why?

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Consider II

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

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So why is the variance wrong I

• Req 1 Measurements should only compare jobs
  whose service (prioritization) can affect each
  other
• What is the correct comparison set?

Locality of Measurement
Time
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Comparison Set

• Claim the correct comparison set is the busy
  period
• Lemma jobs can affect each other iff they are in
  the same BP
• ? Obviously jobs on different busy periods cannot
  influence each other
• ? If jobs are on the same busy period one can
  build another busy period where resources are
  moved from one to the other.

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Inter and Intra Variance

• Semi-Formally
• X RV of some measure with mean
• Y RV eq. average of X in local BP
• Intra-Variance second moment of X around Y
• Inter-Variance second moment of Y around

Y
1
1
3
3
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Inter and Intra Variance

• Lemma Variance Between BP (Inter-Variance)
  Within BP (Intra-Variance)

Req1 A Measure is said to have to have its
variance local if the (global) variance equals
the intra-variance
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So why is the variance wrong II

• Req 2 Job owners should be able to tell if they
  are positively or negatively discriminated
• Proposed way Jobs should know the mean for the
  BP in advance

Req 2 In practice the 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
• Proof Sketch
• note that the momentary sum of discriminations is
  zero.

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Observations on RAQFM

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

Both Properties Hold for RAQFM
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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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Alternative Measure Explicit Evaluation of
Intra-Variance

• We claimed that the intra-variance is the
  important property of a measure
• Use the intra-variance of the waiting time
  explicitly as a measure of fairness
• By definition influenced only by jobs within the
  BP
• Can be quite easily evaluated through simulation.
  Requires only a few more counters

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Alternative Measure - Problems

• Does this address the second problem?
• Does not provide any indication to the job owner
  whether the job is positively or negatively
  treated.
• Can we evaluate it?
• Hard to do analytically. Required evaluating
• RAQFM for example can be evaluated at least for
  M/PH/m

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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