The Asymptotic Variance of Departures in Critically Loaded Queues

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 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

**

* *

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 23 3( 1)

KVK

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 t t tt

Q

VarV D

Theorem 2: 2 2 2Var ( ) 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 t Dt

2 2

Var ( ) Var ( )lim lim

E ( ) E ( )lim lim Var( )

t t

t t

D t D t tVt t

D t t D t t Dt 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 [ ] , 1,2k

k

t

D t tE ED kt

If, then,

Theorem 1: Assume that is UI,

then , with

2

0( ) ,Q t t tt

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 tt t t

( ) ( ) ( )D t A t Q t

22 2( )( ) ( )2A t tD t t Q t

t tt

2

0

( ),

A t tt t

t

is UI since A(.) is renewal

2

0( ) ,Q t t tt

Q is UI by assumption

( )lim [ ] , 1,2k

k

t

D t tE ED kt

Theorem 2Theorem 2:

2 2 2Var ( ) 1a sD c c

Proof Outline:

2 22 2 1 2 1 1 2 20 1

1 1 2 2

1 1 2 2

inf ( ) min( , )| (1) , (1)

1 min( , )t

P b c c c B t x x b c b cP D x B b B b

x b c b c

1 1 2 22 2

1 2

( ) ( ) (1) b c b cB 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 22 20 1

1 2

sup ( ) exp 2t

b c b cP B t y y yc c

2 22 2 1 2 1 1 2 20 1

1 1 2 2

1 1 2 2

inf ( ) min( , )| (1) , (1)

1 min( , )t

P b c c c B t x x b c b cP D x B b B b

x b c b c

Now use (e.g. Mandjes 2007),

Manipulate + use symmetry of Brownian bridge and uncondition….

( , )

1 2 1 2 0

1 ( , )2

L u xx x x xP D x e M u x duc c c c

( , )L u x Quadratic expression in u

( , )M u x Linear expression in uNow compute the variance.

Theorem 3: Proving is UI for some cases2

0( ) ,Q t t tt

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 t

E 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 t t tt

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.