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 leonardo.brenner@imag.fr

{paulof, asales, twebber}@inf.pucrs.br.

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

P1 P2 P3 P4 t1 t2 t3 t4 P5 P6 P7 t5

ICATPN 2005: A Framework to Decompose GSPN Models – p.6/23

Decomposing as SGSPN

GSPN model

P1 P2 P3 P4 t1 t2 t3 t4 P5 P6 P7 t5

ICATPN 2005: A Framework to Decompose GSPN Models – p.7/23

Decomposing as SGSPN

GSPN model Decomposed model

P1 P2 P3 P4 t1 t2 t3 t4 P5 P6 P7 t5

SSM(2) SSM(1) t2 t2 t3 t3 t3 t4 t4 t3 t5 t1

ICATPN 2005: A Framework to Decompose GSPN Models – p.8/23

Tensor Algebra

(tensor product)

A =   a11 a12 a21 a22   B =     b11 b12 b13 b21 b22 b23 b31 b32 b33     C = A B C =              a11b11 a11b12 a11b13 a12b11 a12b12 a12b13 a11b21 a11b22 a11b23 a12b21 a12b22 a12b23 a11b31 a11b32 a11b33 a12b31 a12b32 a12b33 a21b11 a21b12 a21b13 a22b11 a22b12 a22b13 a21b21 a21b22 a21b23 a22b21 a22b22 a22b23 a21b31 a21b32 a21b33 a22b31 a22b32 a22b33             

ICATPN 2005: A Framework to Decompose GSPN Models – p.9/23

Tensor Algebra

(tensor sum)

A =   a11 a12 a21 a22   B =     b11 b12 b13 b21 b22 b23 b31 b32 b33     C = A B = (A IB) + (IA B)

C =              a11 + b11 b12 b13 a12 b21 a11 + b22 b23 a12 b31 b32 a11 + b33 a12 a21 a22 + b11 b12 b13 a21 b21 a22 + b22 b23 a21 b31 b32 a22 + b33             

ICATPN 2005: A Framework to Decompose GSPN Models – p.10/23

Generalized Tensor Algebra

(generalized tensor product)

A(B) =   a11(B) a12(B) a21(B) a22(B)   B(A) =     b11(A) b12(A) b13(A) b21(A) b22(A) b23(A) b31(A) b32(A) b33(A)     C = A(B)

• g B(A)

C =             a11(b11)b11(a11) a11(b11)b12(a11) a11(b11)b13(a11) a12(b11)b11(a12) a12(b11)b12(a12) a12(b11)b13(a12) a11(b21)b21(a11) a11(b21)b22(a11) a11(b21)b23(a11) a12(b21)b21(a12) a12(b21)b22(a12) a12(b21)b23(a12) a11(b31)b31(a11) a11(b31)b32(a11) a11(b31)b33(a11) a12(b31)b31(a12) a12(b31)b32(a12) a12(b31)b33(a12) a21(b11)b11(a21) a21(b11)b12(a21) a21(b11)b13(a21) a22(b11)b11(a22) a22(b11)b12(a22) a22(b11)b13(a22) a21(b21)b21(a21) a21(b21)b22(a21) a21(b21)b23(a21) a22(b21)b21(a22) a22(b21)b22(a22) a22(b21)b23(a22) a21(b31)b31(a21) a21(b31)b32(a21) a21(b31)b33(a21) a22(b31)b31(a22) a22(b31)b32(a22) a22(b31)b33(a22)            

ICATPN 2005: A Framework to Decompose GSPN Models – p.11/23

Tensor Algebra Representation

Two Subnets (one synch.)

P1 P2 P3 P4 t1 t2 t3 t4 P5 P6 P7 t5

                                      

• 

                                     

ICATPN 2005: A Framework to Decompose GSPN Models – p.12/23

Tensor Algebra Representation

Two Subnets (one synch.)

SSM(2) SSM(1) t2 t2 t3 t3 t3 t4 t4 t3 t5 t1

    −t2 t2 −t2 t2    

• g

          −t1 t1 −t4 t4 0 −t4 t4 t5 −t5               0 t3 0 0 0 t3 0 0 0    

• g

          0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0               −t3 −t3 0    

• g

          0 0 0 0 0 0 1 0 0 0 0 0 1 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

P1 P2 P3 P4 t1 t2 t3 t4 P5 P6 P7 t5

SSM(2) SSM(1) t2 t2 t3 t3 t3 t4 t4 t3 t5 t1

ICATPN 2005: A Framework to Decompose GSPN Models – p.15/23

Decomposing by P-Invariants

GSPN model

P1 P2 P3 P4 t1 t2 t3 t4 P5 P6 P7 t5

ICATPN 2005: A Framework to Decompose GSPN Models – p.16/23

Decomposing by P-Invariants

GSPN model Decomposed model

P1 P2 P3 P4 t1 t2 t3 t4 P5 P6 P7 t5

SSM(1) SSM(2) SSM(3) t2 t2 t3 t3 t1 t4 t5 t1 t3 t5

ICATPN 2005: A Framework to Decompose GSPN Models – p.17/23

Decomposing by Places

GSPN model

P1 P2 P3 P4 t1 t2 t3 t4 P5 P6 P7 t5

ICATPN 2005: A Framework to Decompose GSPN Models – p.18/23

Decomposing by Places

GSPN model Decomposed model

P1 P2 P3 P4 t1 t2 t3 t4 P5 P6 P7 t5

SSM(1) SSM(3) SSM(4) SSM(5) SSM(6) SSM(7) t1 t5 t3 t3 t2 t2 t3 t1 t4 t1 t2 t2 t3 t3 t5 t3 t5 t4 SSM(2)

1 2 1 1 1 1 2 1 1 ICATPN 2005: A Framework to Decompose GSPN Models – p.19/23

Examples

(Simultaneous Synchronized Tasks)

t0 P0 t1 P1 P5 P9 t5 t2 P2 P6 P7 P10 t3 t6 t9 P3 P8 P11 P13 t10 t7 t4 P4 P12 P14 P16 t11 P15 N t8

ICATPN 2005: A Framework to Decompose GSPN Models – p.20/23

Examples

(Simultaneous Synchronized Tasks)

N approach #SSM SSM sizes pss rss mem 1 Places 17 (2 × · · · × 2 × 2 × 2) 1.31 × 105 98 1 KB P-Inv 6 (5 × 5 × 5 × 5 × 5 × 2) 6.25 × 103 98 1 KB SGSPN 2 (49 × 2) 9.80 × 101 98 1 KB 3 Places 17 (4 × · · · × 4 × 2 × 2) 4.29 × 109 12, 100 2 KB P-Inv 6 (35 × 35 × 35 × 35 × 35 × 2) 1.05 × 108 12, 100 7 KB SGSPN 2 (6, 050 × 2) 1.21 × 104 12, 100 236 KB 9 Places 17 (10 × · · · × 10 × 2 × 2) 4.00 × 1015 22, 391, 512 6 KB P-Inv 6 (715 × · · · × 715 × 2) 3.74 × 1014 22, 391, 512 201 KB SGSPN 2 (11, 195, 756 × 2) 2.24 × 107 22, 391, 512 672 MB 10 Places 17 (11 × · · · × 11 × 2 × 2) 1.67 × 1016 51, 887, 550 6 KB P-Inv 6 (1, 001 × · · · × 1, 001 × 2) 2.01 × 1015 51, 887, 550 288 KB SGSPN 2 (25, 943, 775 × 2) 5.19 × 107 51, 887, 550

• 20

Places 17 (21 × · · · × 21 × 2 × 2) 2.72 × 1020 18, 994, 747, 662 12 KB P-Inv 6 (10, 626 × · · · × 10, 626 × 2) 2.71 × 1020 18, 994, 747, 662 3 MB SGSPN 2 (9, 497, 373, 831 × 2) 1.90 × 1010 18, 994, 747, 662

• 21

Places 17 (22 × · · · × 22 × 2 × 2) 5.48 × 1020 29, 368, 986, 350 13 KB P-Inv 6 (12, 650 × · · · × 12, 650 × 2) 6.48 × 1020 29, 368, 986, 350 4 MB SGSPN 2 (14, 684, 493, 175 × 2) 2.94 × 1010 29, 368, 986, 350

• 27

Places 17 (28 × · · · × 28 × 2 × 2) 2.04 × 1022 286, 448, 238, 746 16 KB P-Inv 6 (31, 465 × · · · × 31, 465 × 2) 6.17 × 1022 286, 448, 238, 746 10 MB SGSPN 2 (143, 224, 119, 373 × 2) 2.86 × 1011 286, 448, 238, 746

• ICATPN 2005: A Framework to Decompose GSPN Models – p.21/23

Examples

(Resource Sharing)

without guards with guards

... ... ... ...

U1 SN UN S1 ta1 tr1 taN trN RS R RU

... ... ... ...

U1 SN UN S1 ta1 tr1 taN trN

Guards

ga1 = ((tk(U1) + ... + tk(UN)) < R) . . . gaN = ((tk(U1) + ... + tk(UN)) < R)

ICATPN 2005: A Framework to Decompose GSPN Models – p.22/23

Examples

(Resource Sharing) Decomposed by P-invariants

R model #SSM SSM sizes pss rss c.c. mem time 1 without guards 17 (2 × · · · × 2 × 2) 131, 072 17 1.34 × 108 6 KB 0.33 s with guards 16 (2 × · · · × 2) 65, 536 17 2.10 × 106 1 KB 0.45 s 3 without guards 17 (2 × · · · × 2 × 4) 262, 144 697 2.73 × 108 8 KB 0.71 s with guards 16 (2 × · · · × 2) 65, 536 697 2.10 × 106 1 KB 0.49 s 9 without guards 17 (2 × · · · × 2 × 10) 655, 360 50, 643 6.88 × 108 14 KB 1.81 s with guards 16 (2 × · · · × 2) 65, 536 50, 643 2.10 × 106 1 KB 0.53 s 15 without guards 17 (2 × · · · × 2 × 16) 1, 048, 576 65, 535 1.10 × 109 20 KB 3.07 s with guards 16 (2 × · · · × 2) 65, 536 65, 535 2.10 × 106 1 KB 0.53 s

ICATPN 2005: A Framework to Decompose GSPN Models – p.23/23

Conclusion

It is a follow up to Ciardo and Trivedi (91), Donatelli (94), Miner (01) Trade-offs among possible decompositions The use of generalized tensor algebra Future Work Actually deal with immediate transitions Offer choices of modular granularity to

• ther formalisms

Integrate in solvers

ICATPN 2005: A Framework to Decompose GSPN Models – p.24/23

Contact

Performance Evaluation Group

www.inf.pucrs.br/peg

ICATPN 2005: A Framework to Decompose GSPN Models – p.25/23