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Departure Process Variability Departure Process Variability

Departure Process Variability - PowerPoint Presentation

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Departure Process Variability - PPT Presentation

of Queues and Networks Yoni Nazarathy Swinburne University of Technology Melbourne Based on collaborations with Ahmad Al Hanbali Daryl Daley Michel Mandjes Gideon Weiss and Ward Whitt ID: 797185

variance queue queueing 2011 queue variance 2011 queueing process renewal mandjes bravo output symptotic technical conditions whitt questa 2008

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Slide1

Departure Process Variabilityof Queues and Networks

Yoni NazarathySwinburne University of Technology, Melbourne.Based on collaborations with Ahmad Al Hanbali, Daryl Daley, Michel Mandjes,Gideon Weiss and Ward Whitt

IFORS 2011, Melbourne,

July

15, 2011.

Slide2

PLANT

OUTPUTProblem Domain: Queueing Output ProcessesDesired over long term:High ThroughputLow Variability

- Single Server

Queue

- Tandem

Queue

-

Re-Entrant

Line

Our focus:

for large T

Slide3

The GI/G/1/K Queue

overflows

* Load:

* Squared coefficient of variation:

* Assume

Slide4

Variance of Outputs

* Stationary

stable

M/M/1:

D(t) is

PoissonProcess

( ):

* Stationary

M/M/1/1

with

:

D(t) is

RenewalProcess

(

Erlang

(2, )):

* In general, for renewal process with

* The output process of most

queueing

systems is NOT renewal

Asymptotic Variance

Simple Examples:

Notes:

Slide5

Asymptotic Variance for GI/G/1/K

What happens here?

Note: the figure assumes

B

alancing

R

educes

A

symptotic

V

ariance of

O

utputs

Slide6

BRAVO Effect (for M/M/1/K)

Slide7

BRAVO Effect (illustration for M/M/1)

More than a singular theoretic phenomenon

Slide8

Balancing Reduces A

symptotic Variance of OutputsTheorem (Al Hanbali, Mandjes, N. , Whitt AAP 2011):For the GI/G/1 queue with , under some further technical conditions:Theorem (N. , Weiss

QUESTA 2008):

For the M/M/1/K queue with :

Conjecture (N. ,

Daely

, QUESTA To appear):

For the GI/G/1/K queue with , under further

technical conditions :

Slide9

Additional Slides7/15/2011Valuetools 2008

9

Slide10

The Basic Loss-Less Stable Queueing System

Q(t)

Slide11

0

1

K

K-1

Some (partial) intuition for M/M/1/K

Easy to see: