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B alancing R educes A symptotic 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. QTNA 2010, Beijing, July 26, 2010. - PowerPoint PPT Presentation
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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
Overview• GI/G/1/K Queue (with or )
• number of customers served during
• Asymptotic variance
• Surprising results when
K K
( )D t [0, ]t
1
Var ( )limt
D tV
t
Balancing Reduces Asymptotic Variance of Outputs
The GI/G/1/K Queue
2, ac ( )D t2, sc
K
overflows
2 22
variance,meana sc c
* Load:
* Squared coefficient of variation:
* Assume (0) 0Q
Variance of Outputs( )tVt o
t
Var ( )D t
Var ( )D T TV
* Stationary stable M/M/1, D(t) is PoissonProcess( ):
* Stationary M/M/1/1 with . D(t) is RenewalProcess(Erlang(2, )):
21 1 1( )
4 8 8tVar D t t e
( )Var D t t V
4V
2 1 23V m cm
* In general, for renewal process with :
* The output process of most queueing systems is NOT renewal
2,m
Asymptotic Variance
Var ( )limt
VD tt
Simple Examples:
Notes:
Asymptotic Variance for (simple) 1
( ) ( ) ( )
( ) ( ) ( ) ( ), ( ) 2
D t A t Q tVar D t Var A t Var Q t Cov A t Q t
t t t t
2aV c
2sV c
, 1K
After finite time, server busy forever…
is approximately the same as when or 1 K V
, 1K
K
1
Intermediate SummaryV
2ac
2sc
GI/G/1 V
2ac
2sc
GI/G/1/K
M/M/1V
V
M/M/1/K
??
? ?
Balancing Reduces Asymptotic Variance of Outputs
Theorem (Al Hanbali, Mandjes, N. , Whitt 2010):For the GI/G/1 queue with , under some further technical conditions:
2 221 ( )a sV c c
Theorem (N. , Weiss 2008): For the M/M/1/K queue with :
2
2 3 23 3( 1)
KVK
Conjecture (N. , 2009):
For the GI/G/1/K queue with , under furthertechnical conditions :
2 21 ( ) (1)3 a s KV c c o
1
1
1
2 2
2 2
21
1 (1)3
a s
a s K
c c KV
c c o K
BRAVO Summary for GI/G/1/KFor GI/G/1/K with :1
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 1/2( ) ( )P B x L x x
K K
Numerically Conjectured: GI/G/1/K with light tails
**
* *
VV
V V
Numerical Illustration: M/M/1/K
Numerical Illustration: M/M/1 (finite T)
0 1 KK-1
Some (partial) intuition for M/M/1/K
4 MV V
Asymptotic variance of number of transitionsMV
Easy to see:
References• Yoni 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.