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

Meeting of the euro working group on stochastic modeling.Koc university, Istanbul. June 23-25, 2008

Meeting of the euro working group on stochastic modeling.Koc university, Istanbul. June 23-25, 2008

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

Sojourn Times

Queue Sizes

Server Idleness

Lost Job Rates

Variability of Outputs

Typical Queueing Performance MeasuresTypical Queueing Performance Measures


Variability of OutputsVariability of Outputs

Sometimes related to

down-stream queue sizes

Aim for little

variability over [0,T]

Var ( )D t V

Queueing Networks Setting

Manufacturing Setting

( )Vt o t Asymptotic

Variance Rate of Outputs

t


Previous work – Asymptotic Variance Rate of OutputsPrevious work – Asymptotic Variance Rate of Outputs

Baris Tan, Asymptotic variance rate of the output in production lines with finite buffers, Annals of Operations Research, 2000.


In this work…In this work…

Analyze for simple finite queueing systems

• A surprising phenomenon

• Simple formula for

• Extensions

V

V

Results:e.g. M/M/1/K


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

K

* (1 )K

1

11

1

11

1

iK

i

K

KFiniteBuffer

0,...,i K

NOTE: output process D(t) is non-renewal.


What values do we expect for ?V

?

( )V

Keep and fixed.K


?

( )V

K / / 1( )M M

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



**

* *

VV

V V

Output from M/M/1/KOutput from M/M/1/K


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)


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


Idea of Proof Idea of Proof 1

*

0

K

ii

V v


Counts of TransitionsCounts of Transitions 1*

0

K

ii

V v

Book: 2001 - Stochastic Process Limits,.

Paper: 1992 - Asymptotic Formulas for Markov Processes…

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

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

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

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

Var ( ) ( )M t M t o t

Lemma: 4M V

Asymptotic Variance Rate of M(t): ,MBirths Deaths

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

Proof OutlineProof Outline


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

…

Use Lemma: 4M V


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

c

But 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/KMAP used for PH/PH/1/40 with Erlang

and Hyper-Exp distributions

1


The “2/3 property”The “2/3 property”

• GI/G/1/K

• SCV of arrival = SCV of service

1


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


ExtensionsExtensions


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.


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

Asymptotic Variance Rate of the

output processes?

i i i i


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


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


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

yoni nazarathy gideon weiss university of haifa

Documents

asymptotic variance

asymptotic correlation

asymptotic formulas

mm1kthe variance function

map of mt

output processes

death mmpp

associated mmpp mmpp