yoni nazarathy gideon weiss university of haifa yoni nazarathy gideon weiss university of haifa on...

Yoni NazarathyGideon Weiss

University of Haifa

Yoni NazarathyGideon Weiss

University of Haifa

On the Variance of Queueing Output Processes

On the Variance of Queueing Output Processes

Haifa Statistics SeminarFebruary 20, 2007

Haifa Statistics SeminarFebruary 20, 2007

With Illustrations and Animations for “Non-Queueists” (Statisticians)

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 2

OutlineOutline

•Background

•A Queueing Phenomenon: BRAVO

•Main Theorem

•More on BRAVO

•Current, parallel and future work


Some Background on Queues

Some Background on Queues


A Bit On Queueing and Queueing Output ProcessesA Bit On Queueing and Queueing Output Processes

A Single Server Queue:A Single Server Queue:

Buffer Server

0 1 2 3 4 5 6 …State:


The Classic Theorem on M/M/1 Outputs:

Burkes Theorem (50’s):Output process of stationary version is Poisson ( ).

A Bit On Queueing and Queueing Output ProcessesA Bit On Queueing and Queueing Output Processes

A Single Server Queue:A Single Server Queue:

Buffer Server

0 1 2 3 4 5 6 …State:

( )D t

t

OutputProcess:

•Poisson Arrivals: M/M/1 Queue:

•Exponential Service times: •State Process is a birth-death CTMC


•Buffer size:

•Poisson arrivals:

•Independent exponential service times:

•Jobs arriving to a full system are a lost.

•Number in system, , is represented by a finite state irreducible birth-death CTMC:

The M/M/1/K QueueThe M/M/1/K Queue

( )

( )

0

1e

K

* (1 )K

{ ( ), 0}Q t t

1

11

1

11

1

iK

i

K

K KFiniteBuffer

Server

M

0,...,i K

“Carried load”


Traffic ProcessesTraffic Processes

Counts of point processes:

• - The arrivals during

• - The entrances into the system during

• - The outputs from the system during

• - The lost jobs during (overflows)

{ ( ), 0}A t t

{ ( ), 0}E t t

{ ( ), 0}D t t

{ ( ), 0}L t t

[0, ]t

1 K

( )A t

( )L t

( )E t

Poisson

K 1K

0 Renewal Renewal

( )D t

( ) ( )D t L t

( )A t Non-Renewal

Poisson

Poisson Poisson Poisson

Non-Renewal

Renewal

( / /1)M M

[0, ]t

[0, ]t

[0, ]t

K

( )D t

( )L t

( )E t( )A t

M/M/1/KM/M/1/K

Renewal

( ) ( ) ( )

( ) ( ) ( )

A t L t E t

E t Q t D t

Book: Traffic Processes in Queueing Networks, Disney, Kiessler 1987.


•Some Attributes: (Disney, Kiessler, Farrell, de Morias 70’s)

•Not a renewal process (but a Markov Renewal Process).

•Expressions for .

•Transition probability kernel of Markov Renewal Process.

•A Markovian Arrival Process (MAP) (Neuts 1980’s).

•What about ?

D(t) – The Output process:D(t) – The Output process:

1( , )n nCov D D

( )Var D t

( )Var D t

t

V

( ) ( )Var D t V t o t

Asymptotic Variance Rate: V


Asymptotic Variance Rate of Outputs:Asymptotic Variance Rate of Outputs:

What values do we expect for ?V

?

( )V

V

( , )K fixed


?

( )V

K / / 1( )M M

( , )K fixed

Asymptotic Variance Rate of Outputs:Asymptotic Variance Rate of Outputs:V


Work in progress by Ward Whitt


?

( )V 40K

* (1 ) ???KV Similar to Poisson:

( , )K fixed




Asymptotic Variance Rate of Outputs:Asymptotic Variance Rate of Outputs:


?

( )V

V

( , )K fixed

40K


( )V

( )fixed40K

2

3

M

Balancing

Reduces

Asymptotic

Variance of

Outputs




Asymptotic Variance of M/M/1/K:Asymptotic Variance of M/M/1/K:

2

2

1 2 1

1 3

21

3 6 3

(1 )(1 (1 2 ) (1 ) )1

(1 )

K K K

K

K K

K KV

K

2lim

3KV


Calculating Using MAPs

Calculating Using MAPs

V


C D Represented as a MAP (Markovian Arrival Process) (Neuts, Lucantoni et. al.) Represented as a MAP (Markovian Arrival Process) (Neuts, Lucantoni et. al.)

( )D t

* * 2 * 2 3 2( ) 2( ) 2 2( ) 2 ( )r btVar D 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 0

0 0

0

0K

K

* De *[ ( )]E D t t

0 0

1 1 1

1 1 1

0 ( )

0 ( )

0K K K

K

Generator Transitions without events Transitions with events

1( )e

r0b

Asymptotic Variance Rate

Birth-Death Process


Attempting to evaluate directly…Attempting to evaluate directly…

* * 2 12( ) 2 ( )V D e De

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

10K

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

40K

1 50 100 150 201

1

50

100

150

201

1 50 100 150 201

1

50

100

150

201

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 j

K K

ijr

But This doesn’t get us far…

V


Main TheoremMain Theorem

Paper submitted to Queueing Systems Journal, Jan, 2008:The Asymptotic Variance Rate of the Output Process of Finite Capacity Birth-Death Queues.


1*

0

K

ii

V v

2

2 ii i

i

Mv M

d

*1i i iM D P

1

i

i jj

P

0

i

i jj

D d

Main Theorem:

i i id

Part (i):

Part (ii):

0iv

1 2 ... K

0 1 1... K

*1

V

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)


Proof OutlineProof Outline


Use the Transition Counting ProcessUse the Transition Counting Process

( ) ( ) ( )M t E t D t

( ) ( )Var M t M t o t

Lemma: 4M V

Proof:

( ) 2 ( ) ( )M t D t Q t

( ) 4 ( ) ( ) 4 ( ), ( )Var M t Var D t Var Q t Cov D t Q t

( ), ( )1

( ) ( )

Cov D t Q t

Var D t Var Q t

( ) (1)Var Q t O ( ) ( )Var D t O t

( ), ( )Cov D t Q t O t

Q.E.D

- Counts the number of transitions in the state space in [0,t]

Asymptotic Variance Rate of M(t): M

Births Deaths


Idea of Proof of part (i):Idea of Proof of part (i):1

*

0

K

ii

V v

Whitt: Book: Stochastic Process Limits, 2001.

Paper: 1992 –Asymptotic Formulas for Markov Processes…

1) Lemma: Look at M(t) instead of D(t).

2) Proposition: The “Fully Counting” MAP of M(t) has an associated MMPP with same variance.

2) Results of Ward Whitt: An explicit expression for the asymptotic variance rate of MMPP with birth-death structure.

0iv Proof of part (ii), is technical.Proof of part (ii), is technical.


t

1( )N t

3 2 1

2 4 2

1 0 1

a b c

a

Q b

c

Proposition (relating Fully Counting MAPs to MMPPs)Proposition (relating Fully Counting MAPs to MMPPs)

11, ( )N t Fully Counting MAP

MMPP (Markov Modulated Poisson Process)

Example:

00, ( )N t

tabc

( )Q t

1( )N t

rate 4Poisson Process




rate 1Poisson Process rate 4

Poisson Process rate 3

Poisson Process





0 ( )N t

0 ( )N t

0 1

0 1

( ( )) ( ( ))

[ ( )] [ ( )]

Var N t Var N t

E N t E N t

The Proposition:


More On BRAVO More On

BRAVOBalancing

Reduces

Asymptotic

Variance of

Outputs


0 1 KK-1

Some intuition for M/M/1/K:Some intuition for M/M/1/K:


Intuition for M/M/1/K doesn’t carry over to M/M/c/K…Intuition for M/M/1/K doesn’t carry over to M/M/c/K…

But BRAVO does…But BRAVO does…V

c

1 c

M/M/40/40

M/M/K/K

K=30 K=20K=10

M/M/c/40 c=1c=20

c=30


BRAVO also occurs in GI/G/1/K…BRAVO also occurs in GI/G/1/K…

V

1 MAP is used to evaluate Var Rate for PH/PH/1/40 queue with Erlang and Hyper-Exp

1

1

2


The “2/3 property” seems to hold for GI/G/1/K!!!The “2/3 property” seems to hold for GI/G/1/K!!!

V

K

V1 and increase K for different CVs

2 2CV

2 3/ 2CV

2 6 / 5CV

2 1/ 2CV

1

3

4

5

1

4

3


Other Phenomena at Other Phenomena at 1


Asymptotic Correlation Between Outputs and OverflowsAsymptotic Correlation Between Outputs and Overflows,

2 3

,

1

11

lim ( ( ), ( )) 15 5 3

4 12 2 4

1

K

t

K

R

KCorr D t L t

K K K

R

1 1

2,

1 2 1 2 2 1

(1 )(1 3 ) (1 )(3 )

(1 )(1 (2 1)(1 ) )((1 )(1 ) 4( 1)(1 ) )

K K K K

KK K K K K

KR

K

0.139772 1

1lim ( ( ), ( )) 1

41

12

tCorr D t L t

M

For Large K:

( )D t

( )L t

M/M/1/KM/M/1/K


Proposition: If , then:

The y-intercept of the Linear Asymptote of Var(D(t))The y-intercept of the Linear Asymptote of Var(D(t))

4 3 2

2

7 28 37 18

180 360 180D

K K K KB

K K

M/M/1/K1

* * 2 * 2 3 2( ) 2( ) 2 2( ) 2 ( )

D

r bt

BV

Var D t D De t De O t e r 0b

1


The variance function in the short rangeThe variance function in the short range

/ /1/ 40M M


The “kick-in” time for the BRAVO effectThe “kick-in” time for the BRAVO effect

1inf{ 0 | ( ( )) ( ( ))}T t Var D t Var D t

( )D t Departures from M/M/1/K with 2

Yet another singularity


How we got here… and where are we going?

How we got here… and where are we going?


A Novel Queueing Network: Push-Pull System (Weiss, Kopzon 2002,2006)A Novel Queueing Network: Push-Pull System (Weiss, Kopzon 2002,2006)

1 1

22

Server 2Server 1

PUSH

PULL

PULL

PUSH

1 1 2 2, 1 1 2 2, “Inherently Stable” “Inherently Unstable”

0,0 1,0 2,0 3,0 4,0 5,0

0,1

0,2

0,3

0,4

n1

n2

1 1 1 1 1 1

1 1 1 1 1 1

2

2

2

2

2 2

2

2

2

2

0,0

1,3

2,0 3,0 4,0 5,0

0,1

0,2

0,3

0,4

n1

n2

11 1 1 1 1

1 1

1 1 1 1

2

2

2

2

2

2,1

2,2

2,3

2

2

22

2

2

3,1

3,2

3,3

2

2

2

2

2

2

4,1

4,2

4,3

2

2

2

2

2

2

5,1

5,2

5,3

2

2

22

2

2

1 1

1,0

1,4

1

1 1

2,4

0,5

2

1,5

1

1 1

2,5

2

2

2

1

1 1

4,5

2For Both Cases,

Positive Recurrent Policies Exist

* 1 1 2 1 2 21

1 2 1 2

( )

* 2 2 1 2 1 12

1 2 1 2

( )

1 2 1 Require:

Low variance of the output processes?

PROBABLY

NOT WITH THESE

POLICIES!!!


Some Queue Size Realizations:Some Queue Size Realizations:

200 400 600 800 1000

50

100

150

200

250

50 100 150 200 250 300

5

10

500 1000 1500 2000

10

20

30

40

50

BURSTY OUTPUTS

BURSTY OUTPUTS

BURSTY OUTPUTS


Work in progress with regards to the Push-Pull system:Work in progress with regards to the Push-Pull system:

1 1

22

Server 2Server 1

PUSH

PULL

PULL

PUSH

•Can we calculate ?

•Is asymptotic variance rate really the right measure of burstines?

•Which policies are “good” in terms of burstiness?

1 2,V V*1 1,V

*2 2,V


Future work (or current work by colleagues):Future work (or current work by colleagues):

View BRAVO through a Heavy Traffic Perspective, using heavy

traffic limits and scaling.

View BRAVO through a Heavy Traffic Perspective, using heavy

traffic limits and scaling.


“Fresh” in Progress work by Ward Whitt:“Fresh” in Progress work by Ward Whitt:

Question: What about the null recurrent M/M/1( ) ?

Some Guessing:

1V 2

3V 4

2 0.727V

1

1 20 1inf { ( ) (1 )}t

L B t B t

( )

( )n

D nt ntD t

n

Iglehart and Whitt 1970:Iglehart and Whitt 1970:

n DD B

2 1 20

( ) ( ) inf { ( ) ( )}Ds t

B t B t B t B s

1 2,B B Standard independent Brownian

motions.Standard independent Brownian motions.

(1)d

DL B2008 (1 week in progress by Whitt): 2008 (1 week in progress by Whitt):

? 4( ) 2 2 ( (0,1) )V Var L Var N

Uniform Integrability

SimulationResults

0.65V

To be continued…


ThankYou

ThankYou

yoni nazarathy gideon weiss university of haifa yoni nazarathy gideon weiss university of haifa on...

Documents

asymptotic variance

state process

asymptotic formulas

queueing networks

stochastic process limits

birthdeath structure

poisson arrivals

queueing systems journal