Yoni Nazarathy

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On the Variance of Queueing Output Processes
With Illustrations and Animations for
Non-Queueists (Statisticians)

• Yoni Nazarathy
• Gideon Weiss
• University of Haifa

Haifa Statistics Seminar February 20, 2007
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Outline

• Background
• A Queueing Phenomenon BRAVO
• Main Theorem
• More on BRAVO
• Current, parallel and future work

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• Some Background on Queues

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A Bit On Queueing and Queueing Output Processes

• A Single Server Queue

Server
Buffer

State
2
3
4
5
0
1
6
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A Bit On Queueing and Queueing Output Processes

• A Single Server Queue

Server
Buffer

State
2
3
4
5
0
1
6
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

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
  (overflows)

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 ?
Work in progress by Ward Whitt
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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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(No Transcript)
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(No Transcript)
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Asymptotic Variance of M/M/1/K
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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
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

Paper submitted to Queueing Systems Journal, Jan,
2008The Asymptotic Variance Rate of the Output
Process of Finite Capacity Birth-Death Queues.
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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)
Calculation of
If
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) Lemma Look at M(t) instead of D(t).
2) Proposition The Fully Counting MAP of M(t)
has an associated MMPP with same variance.
2) Results of Ward Whitt An 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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Proposition (relating Fully Counting MAPs to
MMPPs)
Example
Fully Counting MAP
MMPP (Markov Modulated Poisson Process)
The Proposition
rate 1Poisson Process
rate 1Poisson Process
rate 1Poisson Process
rate 1Poisson Process
rate 4Poisson Process
rate 4Poisson Process
rate 4Poisson Process
rate 4Poisson Process
rate 3Poisson Process
rate 3Poisson Process
rate 3Poisson Process
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• More On BRAVO

Balancing Reduces Asymptotic Variance
of Outputs
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Some intuition for 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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• Other Phenomena at

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Asymptotic Correlation Between Outputs and
Overflows
M/M/1/K
For Large K
M
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The y-intercept of the Linear Asymptote of
Var(D(t))
M/M/1/K
Proposition If , then
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The variance function in the short range
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The kick-in time for the BRAVO effect
Departures from M/M/1/K with
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• How we got here and where are we going?

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A Novel Queueing Network Push-Pull System
(Weiss, Kopzon 2002,2006)
Server 2
Server 1
PULL
PUSH
PROBABLYNOT WITH THESE POLICIES!!!
Low variance of the output processes?
PUSH
PULL
Require
Inherently Unstable
Inherently Stable
For Both Cases,Positive Recurrent Policies Exist
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Some Queue Size Realizations
BURSTY OUTPUTS
BURSTY OUTPUTS
BURSTY OUTPUTS
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Work in progress with regards to the Push-Pull
system
Server 2
Server 1
PULL
PUSH

• Can we calculate ?
• Is asymptotic variance rate really the right
  measure of burstines?
• Which policies are good in terms of burstiness?

PUSH
PULL
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Future work (or current work by colleagues)

• View BRAVO through a Heavy Traffic Perspective,
  using heavy traffic limits and scaling.

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Fresh in Progress work by Ward Whitt
Question What about the null recurrent M/M/1(
) ?
Some Guessing
Iglehart and Whitt 1970
Standard independent Brownian motions.
2008 (1 week in progress by Whitt)
To be continued
Uniform Integrability
SimulationResults
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• ThankYou