The Twin Measure for System Unfairness and Predictability

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The Twin Measure for System Unfairness and
Predictability

• David Raz
• School of Computer Science, Tel Aviv University
• Jointly with
• Hanoch Levy, Tel Aviv University
• Network Seminar, November 2006

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What is Un-Predictability?
How long will I wait?
How well we can predictHow dispersed is the
real value around our prediction

• Prediction EW
• Un-Predictability VarW

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Importance of Predictability

• Uncertainty about waiting time influences overall
  service satisfaction in a through way (Taylor
  94).
• if a consumer is uncertain about how long they
  will have to wait before accessing the web site
  of interest there will be negative affect, and
  this negative affect will affect the evaluation
  of the web site reading event. (Dellaert Kahn
  1999)
• Many other

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Information level
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Predictability Criterion (Wierman
Harchol-Balter Sigmetrics 05)

• A policy P is always predictable if for every
  load ? every service requirement x
• Motivation
• Growth rate of VarT(x) is O(x) for all work
  conserving policies
• Inequality holds for all work conserving policies
  when x tends to infinity, equality holds for PS,
  SRPT and more

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Higher Information level

• Prediction EWx, state.
• Unpredictability Varx, state
• State can be
• Number of customers in queue
• Workload in the system
• More?

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Problem Black-Box systems

• In a black-box system customers are not aware of
  the system structure or state
• Web servers
• Call centers (if no queue position is given)

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Solution Active Prediction
Web Server
Internet

• Customers initiate probe messages to check the
  system state
• Customer can initiate more than one probe, or may
  send two probes by accident

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Predictability in the Black-Box setting
Web Server
Internet

• Customer perception a customer experiencing
  large difference will feel that the system is
  unpredictable
• Design perspective even with the same system
  state the results are very different, so the
  system cannot be predictable
• We call identical and simultaneously arriving
  jobs twins

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The Twin Measure

• Let C1 and C2 be identical customers, with
  arrival times a and ae and service requirements
  x and xd.
• Random variable Z(x, e, d)d2-d1
• Y(x) lim e?0, d?0 Z(x, e, d)
• For service policy P and job size x
• TP(x)EY(x)

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Details, details

• In fact we define higher moments as well
• The limit must exist. In what way do e, d tend to
  zero?
• Customers must be indistinguishable egt0 ? d can
  be negative.
• All limits must be the same

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Policies whose Twin Measure equals zero

• Processor Sharing (PS)
• Least Attained Service (LAS)
• Longest Remaining Processing Time (LRPT)

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Twin Measure for Some Simple Policies

• FCFS

• x

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NP-LCFS
time
C1
C2
Busy Period
Waiting
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P-LCFS
time
Busy Period
Busy Period
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Twin Measure for Some Simple Policies

• FCFS
• NP-LCFS
• P-LCFS

• x
• x/(1-?)
• x/(1-?)

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Shortest Job First (SJF)

• For service time probability density function
  (pdf) b(x)
• The load of customers of size x
• Then we can follow the same analysis

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Twin Measure for Some Simple Policies

• FCFS
• NP-LCFS
• P-LCFS
• P/NP-SJF
• P/NP-LJF

• x
• x/(1-?)
• x/(1-?)
• x/(1-?(x))
• x/(1-(?-?(x)))

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Shortest Remaining Processing Time (SRPT)

• Y(x)W(x)R(x)
• W(x) Waiting time of a customer of size x, until
  first service
• R(x) Service time of customer of size x
• Observe customers arriving in a period dt where
  C1 has t remaining service
• Customers with service req. ltt create busy period
  of size
• Customers arriving in this period with tltreqltx
  are also served before C2, so the total load is

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Shortest Remaining Processing Time (SRPT)

• This creates a busy period for customers of size
  ltx, so the total waiting created at the interval
  dt is

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• Conclusion
• Well known
• And finally

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Twin Measure for Some Simple Policies

• FCFS
• NP-LCFS
• P-LCFS
• P/NP-SJF
• P/NP-LJF
• SRPT

• x
• x/(1-?)
• x/(1-?)
• x/(1-?(x))
• x/(1-(?-?(x)))
• x/(1-?(x))

FCFS lt SJFSRPT, LJF lt LCFS
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So which policies are predictable?
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So which policies are predictable?
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Conclusion

• Predictability is important
• The Twin Measure is a requirement for Good
  Predictability
• We can easily evaluate for many policies
• Results are important since they do not
  necessarily agree with previous criterion

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So what are the questions?

• What are we measuring?
• Predictability?
• Fairness?
• Something else?
• Where do we go from here?
• Networks of queues
• Connection to measurement trains