Yoni Nazarathy Gideon Weiss University of Haifa

Yoni NazarathyGideon Weiss

University of Haifa

On the Asymptotic Variance Rate of the

Output Process of Finite Capacity Queues

Queueing Analysis, Control and GamesDecember 20,2007

Technion, 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 birth-death CTMC:

The M/M/1/K Queue

( )

( )

01e

K

* (1 )k

{ ( ), 0}Q t t

1

1 11

1 11

iK

i

K

K

kBuffer Server

M

0,...,i K


Traffic Processes

Counts of point processes:

• - The arrivals during

• - The entrances into the system during

• - The outputs 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/K

Renewal

( ) ( ) ( )( ) ( ) ( )A t L t E tE 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:

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:

What values do we expect for ?V

?( )V

V

( , )K fixed


?( )V

K / / 1( )M M

( , )K fixed

Asymptotic Variance Rate of Outputs:V



?( )V 40K

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

( , )K fixed




Asymptotic Variance Rate of Outputs:


?( )V

V

( , )K fixed

40K


( )V

( )fixed40K

23

M

Balancing Reduces Asymptotic Variance of Outputs




Some Results


Results for M/M/1/K:

2

2

1 2 1

1 3

2 13 6 3(1 )(1 (1 2 ) (1 ) ) 1

(1 )

K K K

K

K KK KV

K

2lim3K

V

Other M/M/1/K results:

•Asymptotic correlation between outputs and overflows.

•Formula for y-intercept of linear asymptote when . 1


Calculating Using MAPs

V


C D 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 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

r0b

Asymptotic Variance Rate


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

1 0

10K

1 1 0 2 0 30 4 0

1

1 0

2 0

3 0

4 0

1 1 0 2 0 30 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

But This doesn’t get us far…

V


Main Theorem


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

Main Theorem:

i i id

Part (i):

Part (ii):

0iv

1 2 ... K i

0 1 1... K i

* 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

and

or

If:

Then:

Calculation of : iv

(Asymptotic Variance Rate of Output Process)


Proof Outline


Use 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):1

*

0

K

ii

V v

Whitt: Book: Stochastic Process Limits, 2001.

Paper: 1992 –Asymptotic Formulas for Markov Processes…

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

2) The MAP of M(t) has an associated MMPP with same variance.

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

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


More BRAVOBalancing Reduces Asymptotic Variance of Outputs


0 1 KK-1

Trying to understand what is going on….

M/M/1/K:


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

Vc

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…

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!!!

V

K

V 1 and increase K for different CVs

2 2CV

2 3/ 2CV

2 6 / 5CV

2 1/ 2CV

13

45

1

43


ThankYou

Yoni Nazarathy Gideon Weiss University of Haifa

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