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The Asymptotic Variance of Departures in Critically Loaded
QueuesYoni Nazarathy*
EURANDOM, Eindhoven University of Technology,The Netherlands.
(As of Dec 1: Swinburne University of Technology, Melbourne)
Joint work with Ahmad Al-Hanbali, Michel Mandjes and Ward Whitt.
MASCOS Seminar, Melbourne, July 30, 2010.
*Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber
Overview
• GI/G/1 Queue with • number of served customers during• Asymptotic variance:• Balancing Reduces Asymptotic Variance of Outputs• Main Result:
( )D t [0, ]t Var ( )
limt
D tV
t
1
2
2
2
2 2( ) 1s
a
s
ac c
c
V
c
The GI/G/1/K Queue
2, ac ( )D t2, sc
K
overflows
2 22
variance,
meana sc c
Load:
Squared coefficients of variation:
Assume: (0) 0Q
or K K
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 8
tVar 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 t
t
Simple Examples:
Notes:
Asymptotic Variance for (simple) 1
( ) ( ) ( )
( ) ( ) ( ) ( ), ( ) 2
D t A t Q t
Var 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
**
* *
VV
V V
M/M/1/K: Reduction of Variance when 1
Summary of known BRAVO Results
Balancing Reduces Asymptotic Variance of Outputs
Theorem (N. , Weiss 2008): For the M/M/1/K queue with :
2
2 3 2
3 3( 1)
KV
K
Conjecture (N. 2009):For the GI/G/1/K queue with :
2 2
(1)3
a sK
c cV o
1
1
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
1
Focus of this talk
BRAVO Effect (illustration for M/M/1)
2 221 ( )a sV c c
Assume GI/G/1 with and finite second moments
2 221 ( )a sV c c
The remainder of the talks outlinesthe proof and conditions for:
1
Theorem 1: Assume that is UI,
then , with
2
0
( ),
Q tt t
t
Q
VarV D
Theorem 2: 2 2 2
Var ( ) 1a sD c c
Theorem 3: Assume finite 4’th moments,then, Q is UI under the following cases:(i) Whenever and L(.) bounded (ii) M/G/1(iii) GI/NWU/1 (includes GI/M/1)(iv) D/G/1 with services bounded away from 0
3 Steps for
1/2( ) ( )P B x L x x
2 21 20 1
inf ( ) (1 )a stD c B t c B t
2 221 ( )a sV c c
Proof Outlinefor Theorems 1,2,3
( )D t tD
t
2 2
Var ( ) Var ( )lim lim
E ( ) E ( )lim lim Var( )
t t
t t
D t D t tV
t t
D t t D t tD
t t
D.L. Iglehart and W. Whitt. Multiple Channel Queues in Heavy Traffic. I. Advances in Applied Probability, 2(1):150-177, 1970.
Proof:
2
2( )D t tD
t
so also,
( )lim [ ] , 1,2
kk
t
D t tE ED k
t
If, then,
Theorem 1: Assume that is UI,
then , with
2
0
( ),
Q tt t
t
Q
VarV D 2 21 20 1
inf ( ) (1 )a stD c B t c B t
Theorem 1 (cont.)We now show:
( ) ( ) ( )D t t A t t Q t
t t t
( ) ( ) ( )D t A t Q t
22 2( )( ) ( )2
A t tD t t Q t
t tt
2
0
( ),
A t tt t
t
is UI since A(.) is renewal
2
0
( ),
Q tt t
t
Q is UI by assumption
( )lim [ ] , 1,2
kk
t
D t tE ED k
t
Theorem 2Theorem 2:
2 2 2Var ( ) 1a sD c c
Proof Outline:
2 22 2 1 2 1 1 2 2
0 11 1 2 2
1 1 2 2
inf ( ) min( , )| (1) , (1)
1 min( , )
tP b c c c B t x x b c b c
P D x B b B bx b c b c
1 1 2 2
2 21 2
( ) ( ) (1)b c b c
B t B t t Bc c
2 21 20 1
inf ( ) (1 )a stD c B t c B t
Brownian Bridge:
Theorem 2 (cont.)
1 1 2 2
2 20 11 2
sup ( ) exp 2t
b c b cP B t y y y
c c
2 22 2 1 2 1 1 2 2
0 11 1 2 2
1 1 2 2
inf ( ) min( , )| (1) , (1)
1 min( , )
tP b c c c B t x x b c b c
P D x B b B bx b c b c
Now use (e.g. Mandjes 2007),
Manipulate + use symmetry of Brownian bridge and uncondition….
( , )
1 2 1 2 0
1( , )
2L u xx x x x
P D x e M u x duc c c c
( , )L u x Quadratic expression in u
( , )M u x Linear expression in u
Now compute the variance.
Theorem 3: Proving is UI for some cases2
0
( ),
Q tt t
t
Q
0
'( ) ( ) ( ) inf ( ) ( )s t
Q t A t S t A t S t
4 4 2
0 0[sup ( ) ], [sup ( ) ] ( )
s t s tE A s s E S s s O t
After some manipulation…
0 0
14 2 2sup [ '( ) / ] sup [ '( ) / ]t t t tE Q t t E Q t t
So Q’ is UI
Assume
Now some questions:1) What is the relation between Q’(t) and Q(t)?2) When does (*) hold?
(*)
Some answers:1) Well known for GI/M/1: Q’(.) and Q(.) have the same distribution2) For M/M/1 use Doob’s maximum inequality:
4
4 4 2 2 2
0 0
4[sup ( ) ], [sup ( ) ] 3 ( )
3s t s tE A s s E S s s t t O t
4 2[ '( ) ] ( )E Q t O t
Lemma: For renewal processes with finite fourth moment, (*) holds.
Ideas of proof: Find related martingale, relate it to a stopped martingale, thenUse Wald’s identity to look at the order of growth of the moments.
Going beyond the GI/M/1 queueProposition: (i) For the GI/NWU/1 case:
(ii) For the general GI/G/1 case:
( ) '( )stQ t Q t
4 4 4( ) 8 [ '( ) ] [ ( ) ]EQ t E Q t E C t
C(t) counts the number of busy cycles up to time t
Question: How fast does grow?4[ ( ) ]E C t
Lemma (Due to Andreas Lopker): For renewal process with 1 ( ) ( ) [0,1)F x L x x
E[C(t) ] ( ( ) )m m mO L x t
Zwart 2001: For M/G/1:1/2( ) P B x k x
So, Q is UI under the following cases:(i) Whenever and L(.) bounded (ii) M/G/1(iii) GI/NWU/1 (includes GI/M/1)(iv) D/G/1 with services bounded away from 0
1/2( ) ( )P B x L x x
Summary
• Critically loaded GI/G/1 Queue:
• UI of in critical case is challenging
• Many open questions related to BRAVO,both technical and practical
2 2Var ( ) 2lim ( ) 1a st
D tV c c
t
2
0
( ),
Q tt t
t
Q
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.