Yoni Nazarathy

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The Asymptotic Variance of theOutput Process of
Finite Capacity Queues

• Yoni Nazarathy
• Gideon Weiss
• University of Haifa

ORSIS Conference, Israel April 18-19, 2008
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Queueing Output Process

• A Single Server Queue

Server
Buffer

State
2
3
4
5
0
1
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M/M/1 Queue

• Poisson Arrivals

• Exponential Service times

• State Process is a birth-death CTMC

OutputProcess
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The M/M/1/K Queue
m
Server
Carried load
FiniteBuffer

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

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Traffic Processes

• Counts of point processes
• - Arrivals during
• - Entrances
• - Outputs
• - Lost jobs

M/M/1/K
Poisson
Renewal
Renewal
Renewal
Non-Renewal
Renewal
Non-Renewal
Poisson
Poisson
Poisson
Poisson
Book Traffic Processes in Queueing Networks,
Disney, Kiessler 1987.
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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
  80s)
• What about ?

Asymptotic Variance Rate
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Similar to Poisson
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Balancing Reduces Asymptotic Variance
of Outputs
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• Calculating
• Using MAPs

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MAP (Markovian Arrival Process)(Neuts, Lucantoni
et al.)
Transitions with events
Transitions without events
Generator
Birth-Death Process
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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Main Theorem
Scope Finite, irreducible, stationary,birth-deat
h CTMC that represents a queue.
(Asymptotic Variance Rate of Output Process)
Part (i)
Part (ii)
Calculation of
If
and
Then
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Explicit Formula for M/M/1/K
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• Proof Outline(of part i)

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Define The Transition Counting Process
- Counts the number of transitions in 0,t
Births
Deaths
Asymptotic Variance Rate of M(t) ,
MAP of M(t) is Fully Counting all transitions
result in counts of events.
Lemma
Proof
Q.E.D
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Proof Outline
1) Lemma Look at M(t) instead of D(t).
2) Proposition The Fully Counting MAP of M(t)
has associated MMPP with same variance.
3) Results of Ward Whitt An explicit expression
of asymptotic variance rate of birth-death
MMPP.
Whitt Book 2001 - Stochastic Process Limits,.
Paper 1992 - Asymptotic Formulas for
Markov Processes
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Fully Counting MAP and associated MMPP
Example
Transitions with events
Transitions without events
Fully Counting MAP
MMPP (Markov Modulated Poisson Process)
Proposition
rate 2
rate 2
rate 2
rate 2
rate 4
rate 4
rate 4Poisson Process
rate 4
rate 3
rate 3
rate 3
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• More On BRAVO

Balancing Reduces Asymptotic Variance
of Outputs
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Some intuition for M/M/1/K

0
1
K
K 1
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Intuition for M/M/1/K doesnt carry over to
M/M/c/K
But BRAVO does
M/M/1/40
c20
c30
K20
K30
M/M/10/10
M/M/40/40
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BRAVO also occurs in GI/G/1/K
MAP used for PH/PH/1/40 with Erlang and Hyper-Exp
distributions
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The 2/3 property

• GI/G/1/K
• SCV of arrival SCV of service

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• Thank You