Yoni Nazarathy Gideon Weiss University of Haifa

Yoni NazarathyGideon Weiss

University of Haifa

Yoni NazarathyGideon Weiss

University of Haifa

The Asymptotic Variance of the

Output Process of Finite Capacity Queues

The Asymptotic Variance of the

Output Process of Finite Capacity Queues

EURANDOM QPA SeminarApril 4, 2008

EURANDOM QPA SeminarApril 4, 2008

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

OutlineOutline

• Background

• A Queueing Phenomenon: BRAVO

• Main Theorem

• More on BRAVO

• Some open questions


• 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.

•Assume is stationary.

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

0,...,i K

“Carried load”

{ ( ), 0}Q t t


Counts of point processes:

• - Arrivals during

• - Entrances

• - Outputs

• - Lost jobs

Traffic ProcessesTraffic Processes

{ ( ), 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-RenewalPoisson

Poisson Poisson Poisson

Non-Renewal Renewal

( / /1)M M

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 80’s)

• What about ?

The Output processThe Output process

1Cov( , )n nT T

Var ( )D t Var ( )D t

t

V

Var ( ) ( )D t V t o t

Asymptotic Variance Rate: V


What values do we expect for ?V

?

( )V

Keep and fixed.K


?

( )V

K / / 1( )M M

Work in progress by Ward Whitt

What values do we expect for ?VKeep and fixed.K


?

( )V 40K

?* (1 )KV

Similar to Poisson:



?

( )V

40K



( )V

40K

2

3

Balancing

Reduces

Asymptotic

Variance of

Outputs





Explicit Formula for M/M/1/KExplicit Formula for 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 DMAP (Markovian Arrival Process)(Neuts, Lucantoni et al.)MAP (Markovian Arrival Process)(Neuts, Lucantoni et al.)

* * 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 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

, 0r b

Asymptotic Variance Rate

Birth-Death Process


Attempting to evaluate directlyAttempting 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


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 Outline(of part i)

Proof Outline(of part i)

1*

0

K

ii

V v


Define The Transition Counting ProcessDefine 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

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

Cov ( ), ( )1

Var ( ) Var ( )

D t Q t

D t Q t

Var ( ) (1)Q t O

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

Q.E.D

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

Asymptotic Variance Rate of M(t): ,M

Births Deaths

MAP of M(t) is “Fully Counting” – all transitions result in counts of events.


Proof OutlineProof Outline 1*

0

K

ii

V v

Whitt: Book: 2001 - Stochastic Process Limits,.

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 associated MMPP with same variance.

3) Results of Ward Whitt: An explicit expression of asymptotic variance rate of birth-death MMPP.


t

3 2 1

2 4 2

1 1 2

a b c

a

b

c

Fully Counting MAP and associated MMPPFully Counting MAP and associated MMPP

MMPP (Markov Modulated Poisson Process)

Example:

0 ( )N t

tabc

( )Q t

rate 4Poisson Process

rate 2

rate 3

rate 4

rate 2

rate 4

rate 3

rate 2

rate 3

rate 4

rate 2

1 0

1 0

E[ ( )] E[ ( )]

Var( ( )) Var( ( ))

N t N t

N t N t

Proposition

3 0 0 0 2 1

0 4 0 2 0 2

0 0 2 1 1 0

6 2 1 3 0 0

2 8 2 0 4 0

1 1 4 0 0 2

Transitions without events Transitions with events

1( )N tFully Counting MAP

1( ),N t

( )Q t

0 ( )N t


More On BRAVO More On

BRAVOBalancing

Reduces

Asymptotic

Variance of

Outputs


0 1 KK – 1

Some intuition for M/M/1/KSome intuition for M/M/1/K

…


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

cBut BRAVO doesBut BRAVO does

c

M/M/40/40

M/M/10/10

M/M/1/40

1

K=20K=30

c=30

c=20


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

V

1

MAP used for PH/PH/1/40 with Erlang and Hyper-Exp distributions

1

2


The “2/3 property”The “2/3 property”

K

• GI/G/1/K

• SCV of arrival = SCV of service

• 1 V2 42 33

3 21

2 3

6 2 455 3

1 2 132 3


Other Phenomena at Other Phenomena at 1


Asymptotic Correlation Between Outputs and OverflowsAsymptotic Correlation Between Outputs and Overflows

,

2 3

,

1

11

lim Corr( ( ), ( )) 15 5 3

4 12 2 4

1

K

t

K

R

KD 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 Corr( ( ), ( )) 1

41

12

tD t L t

For Large K

( )D t

( )L t

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


Proposition: For ,

The y-intercept of the Linear AsymptoteThe y-intercept of the Linear Asymptote

4 3 2

2

7 28 37 18

180 360 180D

K K K KB

K K

M/M/1/K1

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

D

r bt

BV

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

1


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

/ /1/ 40M M


Why lookedat asymptotic variance

rate?

Why lookedat asymptotic variance

rate?


Require:

•

• Stable Queues

Push-Pull Queueing Network(Weiss, Kopzon 2002,2006)Push-Pull Queueing Network(Weiss, Kopzon 2002,2006)

1 1

22

Server 2Server 1

PUSH

PULL

PULL

PUSH

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

2

Positive Recurrent Policies Exist!!!

* 1 1 2 21

1 2 1 2

( )

* 2 2 1 12

1 2 1 2

( )

1 2 1

Low variance of the output processes?

PROBABLY

NOT WITH THESE

POLICIES!!!

i i i i


Queue Size RealizationsQueue Size Realizations

i i

50 100 150 200 250 300

5

10

500 1000 1500 2000

10

20

30

40

50

BURSTY OUTPUTS

BURSTY OUTPUTS

i i


• Can we calculate ?

• Diffusion Approximations of the Outputs.

• Is the right measure of burstines?

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

Work in progressWork in progress

1 1

22

Server 2Server 1

PUSH

PULL

PULL

PUSH

iV

*1 1,V

*2 2,V

iV


• Heavy Traffic Scaling, Whitt. Prove the 2/3 Property for GI/G/1/K.

• BRAVO - What is going on?

• M/M/1 with .

• Formulas for asymptotic variance of outputs from other systems.

Other QuestionsOther Questions


In Progress by Ward WhittIn Progress by Ward WhittQuestion: 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

( )( ) , 1, 2,...n

D nt ntD t n

n

1970, Iglehart and Whitt1970, Iglehart and Whitt

w

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)

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

Uniform Integrability

SimulationResults

0.65V


M/M/1+ Impatient Customers - SimulationM/M/1+ Impatient Customers - Simulation

( 1, 1)

( )D t

( )L t

( )E t( )A t

V* *,


Thank YouThank You

Documents

Yoni Nazarathy Gideon Weiss University of Haifa