Perfect Simulation of Stochastic Automata Networks P. Fernandes, J. - - PowerPoint PPT Presentation

Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Perfect Simulation of Stochastic Automata Networks

• P. Fernandes, J. M. Vincent∗, T. Webber

PUCRS, Porto Alegre, Brazil. CAPES, CNPq, FINEP Project grant 4284/05. {paulo.fernandes,twebber}@inf.pucrs.br

∗Laboratoire LIG, MESCAL-INRIA, Grenoble, France.

ANR SETIN Checkbound and ANR blanche SMS. Jean-Marc.Vincent@imag.fr

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

1

Context and Motivation

2

Modeling SAN as discrete-event systems

3

Perfect Simulation of SAN

4

Final considerations

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Analysis of complex discrete systems

Markovian modeling Structured representations (e.g. SAN, GSPN, PEPAnets) * model complex dynamics (synchronizations, functions) * multidimensional product state space X Aim: stationary or transient distribution (for statistical analysis) Constraints: deal with state space explosion problem

Markovian Descriptor GRAPHICAL MODEL (STATES + TRANSITIONS) (ALGEBRAIC SOLUTION) DISCRETE SYSTEM NUMERICAL SOLUTION Structured Markov Chain Iterative Methods −Vector−descriptor product STATISTICAL ANALYSIS −Performance indexes −Stochastic Automata Network 0(1) 1(1) e1 e2

A(1)

(Q = Q(i)) υQ = 0 Probability vector υ e3

A(2)

0(2) 2(2) 1(2) e4 e5 e2 e5

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Analysis of complex discrete systems

Structured Markov Chain −Stochastic Automata Network Markovian Descriptor NUMERICAL SOLUTION DISCRETE SYSTEM Discrete−events Table Transition Function −Vector−descriptor product Iterative Methods STATISTICAL ANALYSIS −Performance indexes (STATES + TRANSITIONS) GRAPHICAL MODEL (ALGEBRAIC SOLUTION) Simulation Methods (DIRECT STATE SIMULATION) (BACKWARD SIMULATION) 0(1) 1(1) e1 e2

A(1)

(e1 . . . e5) Φ(s, e) e3

A(2)

0(2) 2(2) 1(2) e4 e5 e2 e5 Probability vector υ (Q = Q(i))

Other Markovian modeling view discrete-event simulation to estimate steady-state distribution

• n long run trajectories

* establishing transition functions * table of events - uniformization

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Analysis of complex discrete systems

Classical Simulation Techniques Advantage: storage of current/initial state Problems: * number of iterations needed to steady-state estimation * biased samples

. . . control of the burn in time dependence on initial state

Biased sample Steady state ?

Initial state Generated state Stopping rule (empirical) States Time

Forward simulation

e1 e2 e3 e4 ef

Complexity: related to the warm-up period

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Analysis of complex discrete systems

Backward Simulation Techniques [Propp and Wilson 1996] Advantages: samples from the steady-state distribution * avoid warm-up period, coupling time τ Constraints: X trajectories in parallel in the worst case

Backward Simulation

Generated state Trajectories Coupling

Perfect Simulation

Time

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• 2
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• 4
• 5
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• 7
• 8

e1 e2 e3 e4 e5 e6 e7 e8 τ ∗

States (˜ s) Initial States (˜ s)

Complexity: mainly related to the cardinality of X and τ

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Adaptation to Structured models

Model Parameters set of uniformized events E = {e1, .., ep} global states are tuples of local states ˜ s = (s1, . . . , sK) transition function: Φ(˜ s, ei) = ˜ r * each ˜ s ∈ X has a set of enabled events and its firing conditions and consequences Constraints Well-formed SAN models needed * exploring the subset X R (Reachable state space) State space explosion still a problem

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Analysis of complex discrete systems

(STATES + TRANSITIONS) GRAPHICAL MODEL 0(1) 1(1) e1 e2

A(1)

e3

A(2)

0(2) 2(2) 1(2) e4 e5 e2 e5

˜ s ∈ X R ˜ r = Φ(˜ s, ep), ep ∈ ξ Φ(˜ s, e1) Φ(˜ s, e2) Φ(˜ s, e3) Φ(˜ s, e4) Φ(˜ s, e5) {0;0} {1;0} {0;0} {0;0} {0;0} {0;0} {0;1} {1;1} {0;1} {0;2} {0;1} {0;1} {0;2} {1;2} {0;2} {0;2} {0;0} {0;2} {1;0} {1;0} {0,2} {1;0} {1;0} {0;1} {1;1} {1;1} {1;1} {1;2} {1;1} {1;1} {1;2} {1;2} {1;2} {1;2} {1;0} {1;2}

ep ∈ ξ Rates Uniformized Rates e1 λ1 λ1/(λ1 + λ2 + λ3 + λ4 + λ5) e2 λ2 λ2/(λ1 + λ2 + λ3 + λ4 + λ5) e3 λ3 λ3/(λ1 + λ2 + λ3 + λ4 + λ5) e4 λ4 λ4/(λ1 + λ2 + λ3 + λ4 + λ5) e5 λ5 λ5/(λ1 + λ2 + λ3 + λ4 + λ5)

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

SAN Backward coupling simulation

1: for all ˜

s ∈ X R do

2:

ω(˜ s) ← ˜ s { choice of the initial value of vector ω}

3: end for 4: repeat 5:

e ← Generate-event( ) { generation of e according the distribution ( λ1

Λ . . . λE Λ )}

6:

˜ ω ← ω { copying vector ω to ˜ ω}

7:

for all ˜ s ∈ X R do

8:

{ computing ω(˜ s) at time 0 of trajectory issued from ˜ s at time −τ ∗}

9:

ω(˜ s) ← ˜ ω(Φ(˜ s, e))

10:

end for

11: until All ω(˜

s) are equal

12: Return ω(˜

s)

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Partially Ordered State Spaces

Monotonicity property ep ∈ E is monotone if it preserves the partial order

∀(x, y) ∈ X x ≤ y = ⇒ Φ(x, e) ≤ Φ(y, e)

Monotone Backward Simulation [Propp and Wilson 1996] Advantages: samples from the steady-state distribution Complexity: related to τ, two trajectories (instead of X)

* If all events in the model are considered monotone

Time

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States (˜ s)

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˜ ssup ˜ sinf ˜ s generated τ ∗

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

New solutions for huge SAN models

Monotonicity and Perfect Simulation Idea Monotonicity property for SAN related to the analysis of structural conditions * component-wise state space formation Families of SAN models SAN models with a natural partial order (canonical) * e.g. derived from Queueing systems models [Vincent 2005] SAN models with a given component-wise partial order (non-lattice)

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Partially Ordered State Spaces

Canonical component-wise ordering

Queueing Network Model Equivalent MC

λ µ K1

. . . Equivalent SAN Model

K1

A(1)

K2

. . . A(2)

K2

e2 e2 e12 e12 e1 e1 e12 e12 e12 e1 ν

Type Event Rate loc e1 λ syn e12 µ loc e2 ν

00 20 10 01 11 21 02 12 22 03 13 23 Event e12 Event e2 Event e1

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Partially Ordered State Spaces

Non-lattice component-wise ordering Find a partial order of X demands a high c.c. Possible to find extremal global states in the underlying chain * |X M| states: more than two extremal states Complexity: related to τ, but also |X M| Extremal states Component-wise formation has ordered state indexes * consider an initial state composing X M * add to X M the states without transitions to states with greater indexes

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Partially Ordered State Spaces

Non-lattice component-wise ordering Resource sharing model with reservation (Dining Philosophers)

T (0) R(0) L(0)

P(1)

rl2 lrK tr1 lt1 rl1 lrK lrK rl0 tlK rtK T (K) trK−1 rl0

P(K)

Type Event Rate loc lti µ syn tri λ syn rli λ loc rtK µ syn tlK λ syn lrK λ

P(K−1)

T (K−1) L(K−1) R(K−1) ltK−1 rlK−1 trK−2 tlK trK−1 trK−2

P(i)

tri−1 tri rli lti rli+1 T (i) L(i) R(i)

i = 2 . . . (K − 2)

tri−1 R(K) L(K) J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Partially Ordered State Spaces

Non-lattice component-wise ordering e.g. three philosophers with resources reservation, graphical model of the underlying transition chain, extremal states identification

101 011 002 200 001 010 100 000 020 012 102 201 J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

SAN Monotone Backward coupling simulation

1: n = 1 2: E[1] ← Generate-event( ) {E stores the backward sequence of events} 3: repeat 4:

n ← 2n {doubling scheme}

5:

for each ˜ s ∈ X M do

6:

ω(˜ s) ← ˜ s {initial states at time −n}

7:

end for

8:

for i = n downto ( n

2 + 1) do

9:

E[i] ← Generate-event( )

10:

end for

11:

for i = n downto 1 do

12:

for each ˜ s ∈ X M do

13:

ω(˜ s) ← Φ(ω(˜ s), E[i])

14:

end for

15:

end for

16: until All ω(˜

s) are equal

17: Return ω(˜

s)

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

SAN Perfect Simulation

Resource sharing model with reservation- K Philosophers

K X X R X M PEPS* (iteration) Perfect PEPS* (sample) 8 6,561 985 43 0.003185 sec. 0.032354 sec. 10 59,049 5,741 111 0.038100 sec. 0.111365 sec. 12 531,441 33,461 289 0.551290 sec. 0.689674 sec. 14 4,782,969 195,025 755 5.712210 sec. 2.686925 sec. 16 43,046,721 1,136,689 1,975 68.704325 sec. 15.793501 sec. 18 387,420,489 6,625,109 5,169 —- 83.287321 sec.

Numerical results 3.2 GHz Intel Xeon processor under Linux, 1 GByte RAM times: for one iteration on PEPS and for one sample generation on Perfect PEPS Remarks: X contraction in |X M| * X limitation 6 × 107 on PEPS overcame

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Analysis of complex discrete systems

Using model structural information model complexity reduction to achieve the numerical solution increase of solution bounds overcoming memory constraints perfect simulation and monotonicity applied to a structured formalism as SAN Future Works comparative study of convergence control deeper understanting of Φ properties evaluation or bounds on the coupling time strategy adaptation to other structured formalisms

J.M. Vincent Laboratoire LIG-IMAG

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Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations

Thank you for your attention!

J.M. Vincent Laboratoire LIG-IMAG

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