V ariance of O utputs Yoni Nazarathy EURANDOM Eindhoven University of Technology The Netherlands Based on some joint works with Ahmad Al Hanbali Michel Mandjes Gideon Weiss and Ward Whitt ID: 809248
Download The PPT/PDF document "B alancing R educes A symptotic" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Balancing Reduces Asymptotic Variance of Outputs
Yoni Nazarathy*EURANDOM, Eindhoven University of Technology,The Netherlands.Based on some joint works with Ahmad Al Hanbali, Michel Mandjes,Gideon Weiss and Ward Whitt
QTNA 2010, Beijing,July 26, 2010.
*Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber
Slide2OverviewGI/G/1/K Queue (with or ) number of customers served duringAsymptotic variance Surprising results when
Balancing Reduces Asymptotic Variance of O
utputs
Slide3The GI/G/1/K Queue
overflows
* Load:
* Squared coefficient of variation:
*
Assume
Slide4Variance 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:
Slide5Asymptotic Variance for (simple)
After finite time, server busy forever… is approximately the same as when or
Slide6Intermediate Summary
GI/G/1
GI/G/1/K
M/M/1
M/M/1/K
?
?
?
?
Slide7Balancing Reduces Asymptotic Variance of OutputsTheorem (Al
Hanbali, Mandjes, N. , Whitt 2010):For the GI/G/1 queue with , under some further technical conditions:Theorem (N. , Weiss 2008): For the M/M/1/K queue with :
Conjecture (N. , 2009):
For the GI/G/1/K queue
with , under further
technical
conditions :
Slide8BRAVO Summary for GI/G/1/KFor GI/G/1/K with :
Proven: : M/M/1/K : *
M/M/1
* Assuming finite forth moments:
*
M/G/1
*
GI/NWU/1
(includes
GI/M/1)
*Any GI/G/1 with
Numerically Conjectured
:
GI/G/1/K with light tails
Slide9Numerical Illustration: M/M/1/K
Slide10Numerical Illustration: M/M/1
(finite T)
Slide110
1K
K-1
Some (partial) intuition for M/M/1/K
Easy to see:
Slide12ReferencesYoni Nazarathy and Gideon Weiss, The asymptotic variance rate of the output process of finite capacity birth-death queues. Queueing
Systems, 59(2):135-156, 2008.Yoni Nazarathy, 2009, The variance of departure processes: Puzzling behavior and open problems. Preprint, EURANDOM Technical Report Series, 2009-045.Ahmad Al-Hanbali, Michel Mandjes, Yoni Nazarathy and
Ward Whitt. Preprint. The asymptotic variance of departures in critically loaded queues. Preprint, EURANDOM Technical Report Series, 2010-001.