Yoni Nazarathy Gideon Weiss University of Haifa

Yoni NazarathyGideon Weiss

University of Haifa

Yoni NazarathyGideon Weiss

University of Haifa

The Asymptotic Variance Rate of the Departure Process of the M/M/1/K

Queue

The Asymptotic Variance Rate of the Departure Process of the M/M/1/K

Queue

The XXVI International Seminaron Stability Problems for Stochastic Models

October 24, 2007, Nahariya, Israel

The XXVI International Seminaron Stability Problems for Stochastic Models

October 24, 2007, Nahariya, Israel

Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 2

•Poisson arrivals:

•Independent exponential service times:

•Finite buffer size:

•Jobs arriving to a full system are a lost.

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

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

( )

( )

0

1e

K

* (1 )k

{ ( ), 0}Q t t

1

1

1

1

1

iK

i

K

K

kBuffer Server

M

0,...,i K


Traffic ProcessesTraffic Processes

Counts of point processes:

• - The arrivals during

• - The entrances into the system during

• - The departures from the system during

• - The lost jobs during

{ ( ), 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.


•A Markov Renewal Process (Cinlar 1975).

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

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

•Not a renewal process.

•Expressions for .

•Transition probability kernel of the Markov Renewal Process.

•Departures processes of M/G/. Models.

•What about ?

D(t) – The Departure process of M/M/1/KD(t) – The Departure process of M/M/1/K

1( , )n nCov D D

( )Var D t


Asymptotic Variance RateAsymptotic Variance Rate

( ) ( )Var D t V t o t For a given system ( ), what is ?, ,K V

?

( )V

V

( )fixed




?

( )V

V

( )fixed

K / / 1( )M M




?

( )V

V

( )fixed40K

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




( )V

V

( )fixed40K

2

3

OUR MAINRESULT

M


An Explicit FormulaAn Explicit Formula

2 2 1 1 2 2 2 2 1

1 1 2

2

2

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

( )( )

2

3 6 3

K K K K K K K K K K

K K K

K K K K

VK K

K K

Theorem:

Corollary:

2lim

3KV

Corollary:

*

V

Is minimal over all when .,


What is going on?What is going on?

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

( )M t - The number of movements on the state space during [0, ]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

0 1 KK-1

The State Space


What is going on? (continued…)What is going on? (continued…)

0 1 KK-1

Observation: When , is minimal. 0 K

As a result the “modulation” of M(t) is minimal.

Rate of M(t), depends on

current state of Q(t)

And thus the “modulation” of D(t) is minimal.

M


Calculationsand Proof Outline

Calculationsand Proof Outline


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 1( ) 2( ) 2 ( ) ( )

V

Var D t D e De t o t

( )

( )

C D 0 0

0 0

0

0

* (1 )KDe

*[ ( )] ( )E D t t o t

0 ( )

0 ( )

0

Note: We may similarly represent M(t), E(t), L(t) and we may also use similar methods (MMAP) to find cross-covariances.

Generator Transitions without “arrivals” Transitions with “arrivals”


Calculation of : Calculation of :

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

V

Option 1: Invert Numerically

Option 2: For only, we have the explicit structure of the inverse…

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

1 10 20 30 40

1

10

20

30

40

1 10 20 30 40

1

10

20

30

40

1 50 100 150 201

1

50

100

150

201

1 50 100 150 201

1

50

100

150

201

10K

40K

200K

Option 3: Find an associated Markov Modulated Poisson Process (MMPP) to the MAP of (Proof of the explicit formula for any ).

( )M t

2 2 2

2

3 2( 1) ( 1) 2( 1) 7

6( 1)ij

i i j j K j K Kr i j

K

,


Proof Outline:Proof Outline:

Option 3: Find an associated Markov Modulated Poisson Process (MMPP) to the MAP of (Proof of the explicit formula).

( )M t

1) M(t) is “fully counting”: It exactly counts the number of movements in the state-space during [0,t].

2) “Decoupling Theorem” (stated loosely): There exists a MMPP that has the same expectation and variance as a fully counting MAP.

3) Combined results of Ward Whitt (2001 book and 1992 paper) are used to find explicit formulas for the asymptotic variance rate of birth-death type MMPPs.

Note: This technique can be used to find similar explicit formulas for the asymptotic variance rate of departures from M/M/c/K and other structured finite birth and death queues.


Open Questions:Open Questions:


Open Questions:Open Questions:

1) The limiting value of 2/3 also appears in the asymptotic variance rate of the losses (e.g. Whitt 2001). What is the connection?

2) Non-Exponential Queueing systems. Is minimization of the characteristic attribute of the “dip” in the asymptotic variance rate?

3) Asymptotic variance rate of departures from the null-recurrent M/M/1?

4) Variance of departure processes from more complex queueing networks (our initial motivation).

0 K


ThankYou

ThankYou

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Yoni Nazarathy Gideon Weiss University of Haifa