A Framework to Decompose GSPN Models ICATPN 2005 Leonardo Brenner - PowerPoint PPT Presentation

A Framework to Decompose GSPN Models ICATPN 2005 Leonardo Brenner Paulo Fernandes Afonso Sales Thais Webber { paulof, asales, twebber } @inf.pucrs.br. leonardo.brenner@imag.fr PUCRS - ID/INRIA - CAPES - CNPq Porto Alegre, Brazil ICATPN 2005: A Framework to Decompose GSPN Models – p.1/23

Motivation Formalisms to performance and reliability Transient and stationary solutions Large models are modularly conceived Some formalisms are almost mandatorily modular Some suggest a specific modular approach Some can be viewed in different degrees of modularity Modularity may be a luxe to model, but it is a necessity to solve How modular we want our model to be? ICATPN 2005: A Framework to Decompose GSPN Models – p.2/23

Introduction GSPN formalism to model complex systems Parallel and Synchronous behavior Structured vision (storage and solution) Tensor (Kronecker) Algebra Generalized Tensor Algebra (guards) Provide decomposition choices Many (small) or Few (large) subnets Product vs. Reachable State Space Memory usage and Time to Solve ICATPN 2005: A Framework to Decompose GSPN Models – p.3/23

Outline Motivation and Introduction SPN/GSPN/SGSPN ( Tensor Algebra ) Decomposition Choices Two Examples Conclusion and Future Works ICATPN 2005: A Framework to Decompose GSPN Models – p.4/23

SPN/GSPN/SGSPN SPN: P/T net - transitions fired by a stochastic process GSPN: SPN with immediate transitions There is always an equivalent SPN SGSPN: Superposed GSPN A way to decompose GSPN models Superposed transitions Disjoint subsets of places Each subsystem as a Stochastic State Machine (SSM) ICATPN 2005: A Framework to Decompose GSPN Models – p.5/23

Decomposing as SGSPN GSPN model P 1 t 1 P 2 P 3 P 4 t 2 t 3 t 4 P 5 P 6 P 7 t 5 ICATPN 2005: A Framework to Decompose GSPN Models – p.6/23

Decomposing as SGSPN GSPN model P 1 t 1 P 2 P 3 P 4 t 2 t 3 t 4 P 5 P 6 P 7 t 5 ICATPN 2005: A Framework to Decompose GSPN Models – p.7/23

Decomposing as SGSPN GSPN model Decomposed model SSM (2) P 1 t 1 SSM (1) t 1 P 2 P 3 P 4 t 2 t 3 t 3 t 4 t 2 t 3 t 4 P 5 P 6 P 7 t 2 t 3 t 4 t 3 t 5 t 5 ICATPN 2005: A Framework to Decompose GSPN Models – p.8/23

Tensor Algebra (tensor product)   b 11 b 12 b 13    a 11 a 12   A = B = b 21 b 22 b 23    a 21 a 22   b 31 b 32 b 33 C = A � B   a 11 b 11 a 11 b 12 a 11 b 13 a 12 b 11 a 12 b 12 a 12 b 13   a 11 b 21 a 11 b 22 a 11 b 23 a 12 b 21 a 12 b 22 a 12 b 23       a 11 b 31 a 11 b 32 a 11 b 33 a 12 b 31 a 12 b 32 a 12 b 33   C =     a 21 b 11 a 21 b 12 a 21 b 13 a 22 b 11 a 22 b 12 a 22 b 13       a 21 b 21 a 21 b 22 a 21 b 23 a 22 b 21 a 22 b 22 a 22 b 23     a 21 b 31 a 21 b 32 a 21 b 33 a 22 b 31 a 22 b 32 a 22 b 33 ICATPN 2005: A Framework to Decompose GSPN Models – p.9/23

Tensor Algebra (tensor sum)   b 11 b 12 b 13    a 11 a 12   A = B = b 21 b 22 b 23    a 21 a 22   b 31 b 32 b 33 C = A � B = ( A � I B ) + ( I A � B )   a 11 + b 11 b 12 b 13 a 12 0 0   a 11 + b 22 0 0 b 21 b 23 a 12       a 11 + b 33 0 0 b 31 b 32 a 12   C =     0 0 a 22 + b 11 a 21 b 12 b 13       0 0 a 22 + b 22 a 21 b 21 b 23     0 0 a 22 + b 33 a 21 b 31 b 32 ICATPN 2005: A Framework to Decompose GSPN Models – p.10/23

Generalized Tensor Algebra (generalized tensor product)   b 11 ( A ) b 12 ( A ) b 13 ( A )   a 11 ( B ) a 12 ( B )   A ( B ) = B ( A ) = b 21 ( A ) b 22 ( A ) b 23 ( A )     a 21 ( B ) a 22 ( B )   b 31 ( A ) b 32 ( A ) b 33 ( A ) � C = A ( B ) g B ( A )   a 11 ( b 11 ) b 11 ( a 11 ) a 11 ( b 11 ) b 12 ( a 11 ) a 11 ( b 11 ) b 13 ( a 11 ) a 12 ( b 11 ) b 11 ( a 12 ) a 12 ( b 11 ) b 12 ( a 12 ) a 12 ( b 11 ) b 13 ( a 12 )   a 11 ( b 21 ) b 21 ( a 11 ) a 11 ( b 21 ) b 22 ( a 11 ) a 11 ( b 21 ) b 23 ( a 11 ) a 12 ( b 21 ) b 21 ( a 12 ) a 12 ( b 21 ) b 22 ( a 12 ) a 12 ( b 21 ) b 23 ( a 12 )      a 11 ( b 31 ) b 31 ( a 11 ) a 11 ( b 31 ) b 32 ( a 11 ) a 11 ( b 31 ) b 33 ( a 11 ) a 12 ( b 31 ) b 31 ( a 12 ) a 12 ( b 31 ) b 32 ( a 12 ) a 12 ( b 31 ) b 33 ( a 12 )    C =   a 21 ( b 11 ) b 11 ( a 21 ) a 21 ( b 11 ) b 12 ( a 21 ) a 21 ( b 11 ) b 13 ( a 21 ) a 22 ( b 11 ) b 11 ( a 22 ) a 22 ( b 11 ) b 12 ( a 22 ) a 22 ( b 11 ) b 13 ( a 22 )       a 21 ( b 21 ) b 21 ( a 21 ) a 21 ( b 21 ) b 22 ( a 21 ) a 21 ( b 21 ) b 23 ( a 21 ) a 22 ( b 21 ) b 21 ( a 22 ) a 22 ( b 21 ) b 22 ( a 22 ) a 22 ( b 21 ) b 23 ( a 22 )     a 21 ( b 31 ) b 31 ( a 21 ) a 21 ( b 31 ) b 32 ( a 21 ) a 21 ( b 31 ) b 33 ( a 21 ) a 22 ( b 31 ) b 31 ( a 22 ) a 22 ( b 31 ) b 32 ( a 22 ) a 22 ( b 31 ) b 33 ( a 22 ) ICATPN 2005: A Framework to Decompose GSPN Models – p.11/23

Tensor Algebra Representation Two Subnets (one synch.)   • • • • • • • • • • • • • • •     • • • • • • • • • • • • • • • P 1       • • • • • • • • • • • • • • •      • • • • • • • • • • • • • • •  t 1     P 2 P 3 P 4 • • • • • • • • • • • • • • •       • • • • • • • • • • • • • • •       • • • • • • • • • • • • • • •   t 2 t 3 t 4     • • • • • • • • • • • • • • •       • • • • • • • • • • • • • • •     P 5 P 6 P 7   • • • • • • • • • • • • • • •   t 5    • • • • • • • • • • • • • • •      • • • • • • • • • • • • • • •       • • • • • • • • • • • • • • •       • • • • • • • • • • • • • • •     • • • • • • • • • • • • • • • ICATPN 2005: A Framework to Decompose GSPN Models – p.12/23

Tensor Algebra Representation Two Subnets (one synch.)   − t 1 t 1 0 0 0     SSM (2) 0 0 0 0 − t 4 t 4 0 0         � − t 2 t 2 0 0 0 0 0 0     g       0 − t 2 t 2 0 0 0 − t 4 t 4 t 1   SSM (1)   t 5 0 0 0 − t 5   t 2 t 3 t 3 t 4 0 0 0 0 0     0 t 3 0 0 0 0 1 0         � 0 0 t 3 0 0 0 0 1     g t 2 t 3 t 4 t 3       0 0 0 0 0 0 0 0     0 0 0 0 0 t 5   0 0 0 0 0     − t 3 0 0 0 1 0 0 0         � 0 − t 3 0 0 0 1 0 0     g       0 0 0 0 0 0 0 0     0 0 0 0 0 ICATPN 2005: A Framework to Decompose GSPN Models – p.13/23

Decomposition Many ways to decompose SGSPN -like P-invariants Single places Any place decomposition can by made ICATPN 2005: A Framework to Decompose GSPN Models – p.14/23

Decomposing as SGSPN GSPN model Decomposed model SSM (2) P 1 t 1 SSM (1) t 1 P 2 P 3 P 4 t 2 t 3 t 3 t 4 t 2 t 3 t 4 P 5 P 6 P 7 t 2 t 3 t 4 t 3 t 5 t 5 ICATPN 2005: A Framework to Decompose GSPN Models – p.15/23

Decomposing by P-Invariants GSPN model P 1 t 1 P 2 P 3 P 4 t 2 t 3 t 4 P 5 P 6 P 7 t 5 ICATPN 2005: A Framework to Decompose GSPN Models – p.16/23

Decomposing by P-Invariants GSPN model Decomposed model SSM (1) SSM (2) SSM (3) P 1 t 2 t 3 t 1 t 1 t 1 P 2 P 3 P 4 t 2 t 3 t 4 t 2 t 3 t 3 t 4 P 5 P 6 P 7 t 5 t 5 t 5 ICATPN 2005: A Framework to Decompose GSPN Models – p.17/23

Decomposing by Places GSPN model P 1 t 1 P 2 P 3 P 4 t 2 t 3 t 4 P 5 P 6 P 7 t 5 ICATPN 2005: A Framework to Decompose GSPN Models – p.18/23

Decomposing by Places GSPN model Decomposed model SSM (1) P 1 SSM (2) 0 t 1 t 5 0 t 1 t 3 t 2 1 P 2 P 3 P 4 SSM (3) SSM (4) 1 t 3 t 2 0 0 t 3 t 1 t 4 t 1 t 2 t 3 t 4 2 1 1 SSM (5) SSM (6) SSM (7) P 5 P 6 P 7 0 t 5 t 2 t 3 0 0 t 5 t 3 t 5 t 4 1 t 2 t 3 1 1 2 ICATPN 2005: A Framework to Decompose GSPN Models – p.19/23

Examples (Simultaneous Synchronized Tasks) t 0 P 0 N t 1 P 1 P 5 P 9 t 2 t 5 t 8 P 2 P 6 P 7 P 10 t 3 t 6 t 9 P 3 P 8 P 11 P 13 t 4 t 7 t 10 P 4 P 12 P 14 t 11 P 15 P 16 ICATPN 2005: A Framework to Decompose GSPN Models – p.20/23