Simulation of the fluid system with long-range dependent input Oleg - - PowerPoint PPT Presentation

Simulation of the fluid system with long-range dependent input

Oleg Lukashenko

Institute of Applied Mathematical Research KarSC RAS, Russia

Mikhail Nasadkin

Petrozavodsk State University, Russia

Gaussian traffic

, where

Each source is described by ON/OFF process

        period OFF t period ON t t

W

m

, , 1 ) (

) (

 

), (

) (

 t t

W

m

2 1 , ~ ) (  







ON ON

x F

ON

x 2 1 , ~ ) (  







OFF OFF

x F

OFF

x Cumulative traffic of M sources on [0, tT]

 





tT M m m

du u tT W

W

1 ) (

) ( ) (

Convergence to FBM

Interest is in the behavior of this process when M,T are large(Taqqu, 1997).

, ) ( ) (

lim lim

                    



   

t t B c T M TMt tT W

H d H OFF ON ON M T

  

here

1 2 ) , min( 3 2 1    

 

OFF ON

H

It means that

) ( ) ( t cB T M TMt tT W

H H OFF ON ON

  

  

System with finite buffer

Input:

) ( ) ( t B am mt t A

H

 

Service: constant service rate C

Stationary overflow probability:

 

           



b ct t A P b Q P

t

) ( ) (

sup

System with finite buffer

 

) 1 ( ) ( ) 1 ( ) (        t B t B am m c t Q t Q

H H b

) (  t Q

) (  t Qb

b t Q  ) (

b t Qb  ) (

 

 

            

 b

t B t B am m C t Q t Q

H H b b

, ) 1 ( ) ( ) 1 ( min ) (

Buffer size 3 Infinite buffer

System with finite buffer

Overflow and loss probability

 

N b Q I b Q P

N k k





  

1

) ( Overflow probability (N – sample size): Loss probability on [0,T]:

 

 

) ( ) 1 ( ) ( ) 1 (

1

) , (

T A b C t B t B am m t Q Loss

T k H H b

T b P  

 

      

  T

 

 

m b C t B t B am m Q E Loss

H H b

b P



     



) 1 ( ) (

) (

[Kim & Shroff, 2001]

  



b const

b Q P b P

Loss

,

) ( ) (

Then

) ( ) ( ) ( ) ( b Q P Q P P b P

Loss Loss

  

Relative error

M – number of samples

Loss

p

~

• estimation of overflow probability

, 1 ~

~ ~ ~

                         

Loss Loss Loss Loss Loss

p M p E Var RE

p p p

It means that number of samples M must be sufficiently large.

Overflow probability – dependence on sample size

H=0.9

Overflow probability – theoretical results

[Duffield N., O’Connell N.,1995] In the case of fractional Brownian motion input

                         

 H H

H C H b b Q P

2 2 2

1 2 1 exp ) (

Overflow probability – comparison with theoretical results

H=0.9 H=0.9 Simulation Theory

Loss probability – dependence on sample size

H=0.9

Loss probability – dependence on H

Loss probability – dependence on buffer size

References

• Asmussen S., Glynn P. Stochastic Simulation: algorithms and
• analysis. Springer, 2007.
• Norros I. A storage model with self-similar input, Queuing

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

• Han S. Kim, Ness B. Shroff. On the asymptotic relationship

between the overflow probability and the loss ratio, Adv. in Appl.

• Probab. Volume 33, Number 4 (2001), 836-863.
• Taqqu M., Willinger W., Sherman R. Proof of a fundamental

result in self-similar traffic modeling, Computer Communication Review, 27 (1997) 5-23.

• Duffield N., O’Connell N. Large deviations and overflow

probabilities for general single-server queue, with applications.

• Math. Proc. Camb. Phil. Soc. 118 (1995), 363–374. [ p. 69 ]

Thank you