Mean Delay Optimization for the MG1 Queue with Pareto Type Service Times

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Mean Delay Optimizationfor the M/G/1 Queuewith
Pareto Type Service Times

• Samuli Aalto
• TKK Helsinki University of Technology, Finland
• Urtzi Ayesta
• LAAS-CNRS, France

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Known optimality results for M/G/1

• Among all scheduling disciplines
• SRPT (Shortest-Remaining-Processing-Time) optimal
• minimizing the queue length process thus, also
  the mean delay (i.e. sojourn time)
• Among non-anticipating (i.e. blind) scheduling
  disciplines
• FCFS (First-Come-First-Served) optimal for NBUE
  (New-Better-than-Used-in-Expectation) service
  times
• minimizing the mean delay
• FB (Foreground-Background) optimal for DHR
  (Decreasing-Hazard-Rate) service times
• minimizing the mean delay
• Definitions
• NBUE ES ES xS gt x for all x
• DHR hazard rate h(x) f(x)/(1-F(x)) decreasing
  for all x

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Pareto service times

• Pareto distribution
• has a power-law (thus heavy) tail
• has been used to model e.g. flow sizes in the
  Internet
• Definition (type-1)
• belongs to the class DHR
• thus, FB optimal non-anticipating discipline
• Definition (type-2)
• does not belong to the class DHR
• optimal non-anticipating discipline an open
  question ... until now!

h(x)
h(x)
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CDHR service times

• CDHR(k) distribution class (first-Constant-and-the
  n-Decresing-Hazard-Rate)
• includes type-2 Pareto distributions
• Definition
• A1 hazard rate h(x) constant for all x lt k
• A2 hazard rate h(x) decreasing for all x k
• A3 h(0) lt h(k)
• Examples

h(x)
h(x)
h(x)
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Gittins index

• Function J(a,?) for a job of age a and service
  quota ?
• numerator completion probability payoff
• denominator expected servicing time
  investment
• Gittins index G(a) for a job of age a
• Original framework
• Multiarmed Bandit Problems Gittins (1989)

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Example Pareto distribution

• Type-2 Pareto distribution with k 1 and a 2
• Left Gittins index G(a) as a function of age a
• Right Optimal quota ?(a) as a function of age
  a
• Note
• ?(0) gt k
• G(?(0)) G(0)
• G(a) h(a) for all a gt k

?(0)
G(a)
?(a)
G(0)
k
k
?(0)
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Gittins discipline

• Gittins discipline
• Serve the job with the highest Gittins index if
  multiple, then PS among those jobs
• Known result Gittins (1989), Yashkov (1992)
• Gittins discipline optimal among non-anticipating
  scheduling disciplines
• minimizing the mean delay
• Our New Result
• For CDHR service times (satisfying A1-A3) the
  Gittins discipline (and thus optimal) is
  FCFSFB(?(0))
• give priority for jobs younger than threshold
  ?(0) and apply FCFS among these priority jobs
• if no priority jobs, serve the youngest job in
  the system (according to FB)

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Numerical results Pareto distribution

• Type-2 Pareto distribution with k 1 and a 2
• Depicting the mean delay ratio
• Left Mean delay ratio as a function of threshold
  ?
• Right Minimum mean delay ratio as a function of
  load ?
• Note

? 0.5
max gain 18
? 0.8
?(0)
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Impact of an upper bound Bounded Pareto

• Bounded Pareto distribution
• lower bound k and upper bound p
• Definition
• does not belong to the class CDHR

h(x)
G(a)
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Conclusion and future research

• Optimal non-anticipating scheduling studied for
  M/G/1 by applying the Gittins index approach
• Observation
• Gittins index monotone iff the hazard rate
  monotone
• Main result
• FCFSFB(?(0)) optimal for CDHR service times
• Possible further directions
• To generalize the result for IDHR service times
• To apply the Gittins index approch
• in multi-server systems or networks with the
  non-work-conserving property
• in wireless systems with randomly time-varing
  server capacity
• in G/G/1
• To calculate performance metrics for a given G(a)