Encounters with the BRAVO Effect in Queueing Systems

Yoni NazarathySwinburne University of Technology

Based on some joint papers with Ahmad Al-Hanbali, Daryl Daley, Yoav Kerner,

Michel Mandjes, Gideon Weiss and Ward Whitt

University of Queensland Statistics SeminarDecember 2, 2011

Outline

• Queues and Networks

• Variance of Outputs

• BRAVO Effect

• BRAVO Results (Theorems)

• Summary

Queues and Networks

The GI/G/1/K Queue

2, ac ( )D t2, sc

KOverflows

2 22

variance,meana sc c

Load:

Squared coefficients of variation:

or K K Buffer:

OutputsArrivals

Variance of Outputs

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 when 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

Look now at the Asymptotic Variance of M/M/1/K (for any Value of )

?( )V 40K

?* (1 )KV

Similar to Poisson:

What values do we expect for ?V

M/M/1/K

?( )V

40K

M/M/1/K

What values do we expect for ?V

( )V

40K

23

Balancing Reduces Asymptotic Variance of Outputs

M/M/1/K

What values do we expect for ?V

0 1 KK – 1

Some Intuition…

…

4M V

Do we Have the Same BRAVO Effect in

M/M/1 ?

M/M/1 (Infinite Buffer)

Key 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. , 2011):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 further conditions:

2 2 2( ) 1a sV c c

1

A Bit More on the GI/G/1 Result

Reminder: Uniform Integrability (UI)

{| | }lim sup [| |1 ] 0tt Z MM t

E Z

A family of RVs, , is UI if:{ }tZ

A sufficient condition is:

1sup [| | ] for some ttE Z

If (in distribution) and is UI then: { }tZtZ Z

[ ] [ ]tE Z E Z

Theorem : Assume that is UI,

then , with

2

0( ) ,Q t t tt

Q

VarV D

Theorem : 2 2 2Var ( ) 1a sD c c

Theorem : Assume finite 4’th moments,then, is UI under the following cases:(i) Whenever and L(.) bounded slowly varying. (ii) M/G/1(iii) GI/NWU/1 (includes GI/M/1)(iv) D/G/1 with services bounded away from 0

The GI/G/1 Result:

1/2( ) ( )P B x L x x

2 21 20 1

inf ( ) (1 )a stD c B t c B t

2 2 2( ) 1a sV c c

Q

Summary• BRAVO:

Balancing Reduces Asymptotic Variance of Outputs

• Different “BRAVO Constants”: Finite Buffers:

Infinite Buffers:

• Further probabilistic challenges in establishing full UI conditions

• In future: Applications of BRAVO and related results in system identification (model selection)

21

13

BRAVO References

• Yoni Nazarathy and Gideon Weiss, The asymptotic variance rate of the output process of finite capacity birth-death queues. Queueing Systems, 59, pp135-156, 2008.

• Yoni Nazarathy, The variance of departure processes: Puzzling behavior and open problems. Queueing Systems, 68, pp 385-394, 2011.

• Ahmad Al-Hanbali, Michel Mandjes, Yoni Nazarathy and Ward Whitt. The asymptotic variance of departures in critically loaded queues. Advances in Applied Probability, 43, 243-263, 2011.

• Yonjiang Guo, Erjen Lefeber, Yoni Nazarathy, Gideon Weiss, Hanqin Zhang, Stability and performance for multi-class queueing networks with infinite virtual queues, submitted.

• Daryl Daley, Yoni Nazarathy, The BRAVO effect for M/M/c/K+M systems, in preperation.

• Yoni Nazarathy and Gideon Weiss, Diffusion Parameters of Flows in Stable Queueing Networks, in preparation.

• Yoav Kerner and Yoni Nazarathy, On The Linear Asymptote of the M/G/1 Output Variance Curve, in preparation.

Extra Slides

C DMAP (Markovian Arrival Process)

* * 2 * 2 3 2Var ( ) 2( ) 2 2( ) 2 ( )r btD t D De t De O t e

0 0

1 1 1 1

1 1 1 1

( )

( )K K K K

K K

1

1

0 00 0

00

K

K

* De *E[ ( )]D t t

0 0

1 1 1

1 1 1

0 ( )

0 ( )0

K K K

K

Generator Transitions without events Transitions with events

1( )e

, 0r b

Asymptotic Variance Rate

Birth-Death Process

Attempting to Evaluate Directly* * 2 12( ) 2 ( )V D e De

1 2 3 4 5 6 7 8 9 1 0

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 1 0

1

2

3

4

5

6

7

8

9

1 0

10K

1 1 0 2 0 3 0 4 0

1

10

20

30

40

1 1 0 2 0 3 0 4 0

1

10

20

30

40

40K

1 50 100 150 2 01

1

5 0

10 0

15 0

20 1

1 50 100 150 2 01

1

50

10 0

15 0

20 1

200K

For , there is a nice structure to the inverse.

2 2 3

2 3

( 2 ) ( 2 ) ( 1) 7( 1) ,2( 1) 2( 1)ij

i i K j K j K Kr i jK K

ijr

V

1*

0

K

ii

V v

2

2 ii i

i

Mv Md

*

1i i iM D P

1

i

i jj

P

0

i

i jj

D d

Finite B-D Result

i i id

Part (i)

Part (ii)

0iv

1 2 ... K

0 1 1... K

* 1V

0 0

1 1 1 1

1 1 1 1

( )

( )K K K K

K K

*1KD

Scope: Finite, irreducible, stationary,birth-death CTMC that represents a queue.

0 10

1

ii

i

0 1

0 0 1

1iK

j

i j i

and

If

Then

Calculation of iv

(Asymptotic Variance Rate of Output Process)

Vc

Other Systems

c

M/M/40/40

M/M/10/10

M/M/1/40

1

K=20K=30

c=30

c=20

Using a Brownian BridgeTheorem:

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: