Some analytical aspects of regenerative simulation of fluid models - - PowerPoint PPT Presentation

Some analytical aspects of regenerative simulation of fluid models

Ruslana Goricheva

Institute of Applied Mathematical Research Karelian Research Centre, RAS. (joint work with Oleg Lukashenko, Dr. Evsey Morozov,

• Dr. Michele Pagano)

Fluid model

Input traffic: ( ) ( ) ( ) A t m t m X t m X t   

• mean input rate
• random centred Gaussian process with

covariance

( , ) [ ( ) ( )] t s X t X s   E

Service rate: Constant service rate C

Queue with Brownian input

Input traffic:

( ) ( ) A t m t m B t  

Queue length (discrete scale):

 

 

 

( ) min ( 1) ( ) ( 1) , , t 0,1,...

b b

Q t Q t C m m B t B t b



       

( ) ( ) lim

n l t

A L t EL P A t E

 

 

( )

t n

Q EL EA L t

• workload of a system at time t;
• mean lost work per cycle;
• mean workload arrived per cycle;
• lost work in [0; t] ( n – buffer size);

Regenerative approach

1

m in { : 0} ,

k k t

t Q   



   

Regeneration points: By regeneration theory:

Delta-method for confident estimation.

• random variables;
• 1

2 1 2

, : , : ; : ( , ); X Y z E X z E Y Z z z    ( ). f Z

Confidence interval for By CLT:

( ) 1 1 2

, 1, 2 , ; ( , ); ( )

R k i k i

z z i R N R Z z z f Z

     

   



• point estimator of

( ). f Z

2

( ( ) ( )) (0 , ), , R f Z f Z N a s R 



   

Where and is a covariance matrix.

2

( ) ( ( )) T f Z f Z      

Ratio estimator.

1 2

( ) .

l

z EL f Z P z EA   

[ ; ],

l l l

z z P P P R R

   

  

1( ).

2 z  



 

Confidence interval for loss probability: where

Confidence interval for in BM/D/1/n

l

P

Confidence interval for different frequency of regeneration points

Discretization step

 

 

 

1 ( ) min ( ) ( ) ( ) , , h h .

b b N

Q t h Q t Ch A t h A t b N



       

Comparison with BI/D/1

• overflow probability

( ) exp 2 C m P Q b b m          

Assumption:

( ) . ( )

b l

P Q b const P b



  

Simulation of BM

where where

 

1

, 1,..., ,

h h h n k

B t B B k n     

(0, )

h nB

N h  

1/ ,

N

h h N  

Error of linear interpolation: Error of linear interpolation:

1 1/2

( ) ( ) / , / 32

h

B t B t dt c N c    



E

Conclusions

• Analysis of queue with Brownian input via

regeneration method, based on i.i.d. property of increments of BM.

• Confidence estimation of loss probability via

delta-method.

• Estimation for different frequency of regeneration

points.

• Analysis of value of discretization step.
• Comparison of observed results with explicit

expression on continuous time for BI/D/1 .

References

• Asmussen S. Applied probability and Queues. Springer, 2003.
• Asmussen S., Glynn P. Stochastic Simulation: algorithms and
• analysis. Springer, 2007.
• Law A. M., Kelton W. D. Simulation modeling and analysis/

New York: McGraw- Hill, 1991. 2nd edt.

• Mandjes M. Large Deviations of Gaussian Queues. - Chichester:

Wiley, 2007.

• Norros I. A storage model with self-similar input, Queueing

Systems, vol. 16, pp. 387-396, 1994.

Thank you.