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 Each source is described by ON/OFF process      1 , t ON period ( m )  ( m )  W  W ( t ) ( t ), t 0 , where    0 , t OFF period      F x ( x ) ~ , 1 2 ON ON ON      x F ( x ) ~ , 1 2 OFF OFF OFF Cumulative traffic of M sources on [0, tT] tT M    ( m ) W W ( tT ) ( u ) du  m 1 0

Convergence to FBM Interest is in the behavior of this process when M,T are large(Taqqu, 1997).       ON W ( tT ) TMt        d     ON OFF lim lim c B ( t ) , t 0 H H  M T      T M          3 min( , ) 1    ON OFF H 1 here 2 2  It means that   H ON W ( tT ) TMt M T cB ( t )    H ON OFF

System with finite buffer   Input : A ( t ) mt am B ( t ) H Service : constant service rate C Stationary overflow probability:           sup P ( Q b ) P A ( t ) ct b      t 0

System with finite buffer          Q ( t ) Q ( t 1 ) c m am B ( t ) B ( t 1 ) b H H   Q ( t ) 0 Q ( t ) b  Q b ( t ) b  Q b ( t ) 0    b              Q ( t ) min Q ( t 1 ) C m am B ( t ) B ( t 1 ) ,   b b H H

System with finite buffer Buffer size 3 Infinite buffer

Overflow and loss probability N     I Q b k Overflow probability (N – sample    k 1 P ( Q b ) size): N   T            Q ( t 1 ) m am B ( t ) B ( t 1 ) C b b H H   P ( b , T ) k 1 Loss probability on [0,T]: Loss A ( T )             E Q m am B ( t ) B ( t 1 ) C b   P ( b ) b H H T Loss m    P ( b ) const , b Loss [Kim & Shroff, 2001]  P ( Q b ) P ( 0 )   Then Loss P ( b ) P ( Q b )  Loss P ( Q 0 )

Relative error M – number of samples ~ p - estimation of overflow probability Loss   ~   p Var     ~   Loss 1     p RE ~ , p 0     Loss ~   Loss p M   p Loss E     Loss 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    2 2 H 2 H       1 b C          P ( Q b ) exp        2 1 H H  

Overflow probability – comparison with theoretical results Simulation H=0.9 Theory H=0.9

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 ]

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