Yoni Nazarathy

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On the Asymptotic Variance Rate of the Output
Process of Finite Capacity Queues

• Yoni Nazarathy
• Gideon Weiss
• University of Haifa

Queueing Analysis, Control and Games December
20,2007 Technion, Israel
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The M/M/1/K Queue
m
Server
Buffer

• Poisson arrivals
• Independent exponential service times
• Finite buffer size
• Jobs arriving to a full system are a lost.
• Number in system, , is represented
  by a finite state irreducible birth-death CTMC

M
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Traffic Processes
M/M/1/K

• Counts of point processes
• - The arrivals during
• - The entrances into the system
  during
• - The outputs from the system
  during
• - The lost jobs during

Poisson
Renewal
Renewal
Non-Renewal
Renewal
Non-Renewal
Renewal
Poisson
Book Traffic Processes in Queueing Networks,
Disney, Kiessler 1987.
Poisson
Poisson
Poisson
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D(t) The Output process

• Some Attributes (Disney, Kiessler, Farrell, de
  Morias 70s)
• Not a renewal process (but a Markov Renewal
  Process).
• Expressions for .
• Transition probability kernel of Markov Renewal
  Process.
• A Markovian Arrival Process (MAP) (Neuts 1980s).
• What about ?

Asymptotic Variance Rate
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Asymptotic Variance Rate of Outputs
What values do we expect for ?
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Asymptotic Variance Rate of Outputs
What values do we expect for ?
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Asymptotic Variance Rate of Outputs
What values do we expect for ?
Similar to Poisson
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Asymptotic Variance Rate of Outputs
What values do we expect for ?
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Asymptotic Variance Rate of Outputs
What values do we expect for ?
Balancing Reduces Asymptotic Variance
of Outputs
M
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• Some Results

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Results for M/M/1/K
Other M/M/1/K results

• Asymptotic correlation between outputs and
  overflows.
• Formula for y-intercept of linear asymptote when
  .

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• Calculating
• Using MAPs

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Represented as a MAP (Markovian Arrival
Process) (Neuts, Lucantoni et. al.)
Transitions with events
Transitions without events
Generator
Asymptotic Variance Rate
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Attempting to evaluate directly
But This doesnt get us far
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• Main Theorem

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Scope Finite, irreducible, stationary,birth-deat
h CTMC that represents a queue
Main Theorem
(Asymptotic Variance Rate of Output Process)
Part (i)
Part (ii)
If
Calculation of
and
or
and
Then
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• Proof Outline

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Use the Transition Counting Process
- Counts the number of transitions in the state
space in 0,t
Births
Deaths
Asymptotic Variance Rate of M(t)
Lemma
Proof
Q.E.D
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Idea of Proof of part (i)
1) Look at M(t) instead of D(t).
2) The MAP of M(t) has an associated MMPP with
same variance.
2) Results of Ward Whitt allow to obtain explicit
expression for the asymptotic variance rate of
MMPP with birth-death structure.
Whitt Book Stochastic Process Limits, 2001.
Paper 1992 Asymptotic Formulas for
Markov Processes
Proof of part (ii), is technical.
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• More BRAVO

Balancing Reduces Asymptotic Variance
of Outputs
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Trying to understand what is going on.
M/M/1/K
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Intuition for M/M/1/K doesnt carry over to
M/M/c/K
But BRAVO does
c30
c20
M/M/c/40
c1
K30
K20
M/M/40/40
K10
M/M/K/K
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BRAVO also occurs in GI/G/1/K
MAP is used to evaluate Var Rate for PH/PH/1/40
queue with Erlang and Hyper-Exp
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The 2/3 property seems to hold for GI/G/1/K!!!
and increase K for different CVs
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• ThankYou