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 Page 1/19 J.M. Vincent Laboratoire LIG-IMAG
Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations Context and Motivation 1 Modeling SAN as discrete-event systems 2 Perfect Simulation of SAN 3 Final considerations 4 Page 2/19 J.M. Vincent Laboratoire LIG-IMAG
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 GRAPHICAL MODEL DISCRETE SYSTEM NUMERICAL SOLUTION (STATES + TRANSITIONS) (ALGEBRAIC SOLUTION) A (1) A (2) STATISTICAL ANALYSIS 0 (1) 0 (2) Markovian Probability vector υ Descriptor Iterative Methods e 4 e 5 −Vector−descriptor product −Performance indexes ( Q = � � Q ( i ) ) e 2 e 1 e 2 υQ = 0 e 5 1 (1) 2 (2) 1 (2) e 3 Structured Markov Chain −Stochastic Automata Network Page 3/19 J.M. Vincent Laboratoire LIG-IMAG
Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations Analysis of complex discrete systems GRAPHICAL MODEL DISCRETE SYSTEM NUMERICAL SOLUTION (STATES + TRANSITIONS) A (1) A (2) Markovian Iterative Methods Descriptor −Vector−descriptor product 0 (1) 0 (2) ( Q = � � Q ( i ) ) STATISTICAL e 4 e 5 ANALYSIS e 2 e 1 e 2 (ALGEBRAIC SOLUTION) Probability vector υ e 5 −Performance indexes 1 (1) 2 (2) 1 (2) Discrete−events Table ( e 1 . . . e 5 ) Simulation Methods e 3 Transition Function Structured Markov Chain Φ( s, e ) −Stochastic Automata Network (DIRECT STATE SIMULATION) (BACKWARD SIMULATION) Other Markovian modeling view discrete-event simulation to estimate steady-state distribution on long run trajectories * establishing transition functions * table of events - uniformization Page 4/19 J.M. Vincent Laboratoire LIG-IMAG
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 Forward simulation States Generated state Initial Steady state ? state Biased sample control of the burn in time dependence on initial state . . . e 1 e 2 e 3 e 4 e f Time Stopping rule (empirical) Complexity: related to the warm-up period Page 5/19 J.M. Vincent Laboratoire LIG-IMAG
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 States (˜ s ) τ ∗ s ) Initial States (˜ Trajectories Coupling Generated state -8 -7 -6 -5 -4 -3 -2 -1 0 Time e 8 e 7 e 6 e 5 e 4 e 3 e 2 e 1 Perfect Simulation Complexity: mainly related to the cardinality of X and τ Page 6/19 J.M. Vincent Laboratoire LIG-IMAG
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 = { e 1 , .., e p } global states are tuples of local states ˜ s = ( s 1 , . . . , s K ) transition function: Φ(˜ s, e i ) = ˜ 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 Page 7/19 J.M. Vincent Laboratoire LIG-IMAG
Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations Analysis of complex discrete systems GRAPHICAL MODEL r = Φ(˜ ˜ s, e p ) , e p ∈ ξ s ∈ X R ˜ (STATES + TRANSITIONS) Φ(˜ s, e 1 ) Φ(˜ s, e 2 ) Φ(˜ s, e 3 ) Φ(˜ s, e 4 ) Φ(˜ s, e 5 ) A (1) A (2) { 0;0 } { 1;0 } { 0;0 } { 0;0 } { 0;0 } { 0;0 } 0 (1) 0 (2) { 0;1 } { 1;1 } { 0;1 } { 0;2 } { 0;1 } { 0;1 } e 4 e 5 { 0;2 } { 1;2 } { 0;2 } { 0;2 } { 0;0 } { 0;2 } e 2 e 1 e 2 e 5 { 1;0 } { 1;0 } { 0,2 } { 1;0 } { 1;0 } { 0;1 } 1 (1) 2 (2) 1 (2) { 1;1 } { 1;1 } { 1;1 } { 1;2 } { 1;1 } { 1;1 } { 1;2 } { 1;2 } { 1;2 } { 1;2 } { 1;0 } { 1;2 } e 3 e p ∈ ξ Rates Uniformized Rates e 1 λ 1 λ 1 / ( λ 1 + λ 2 + λ 3 + λ 4 + λ 5 ) e 2 λ 2 λ 2 / ( λ 1 + λ 2 + λ 3 + λ 4 + λ 5 ) e 3 λ 3 λ 3 / ( λ 1 + λ 2 + λ 3 + λ 4 + λ 5 ) e 4 λ 4 λ 4 / ( λ 1 + λ 2 + λ 3 + λ 4 + λ 5 ) e 5 λ 5 λ 5 / ( λ 1 + λ 2 + λ 3 + λ 4 + λ 5 ) Page 8/19 J.M. Vincent Laboratoire LIG-IMAG
Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations SAN Backward coupling simulation s ∈ X R do 1: for all ˜ 2: ω (˜ s ) ← ˜ s { choice of the initial value of vector ω } 3: end for 4: repeat e ← Generate-event( ) { generation of e according the distribution ( λ 1 Λ . . . λ E 5: Λ ) } 6: ˜ ω ← ω { copying vector ω to ˜ ω } s ∈ X R do 7: for all ˜ s at time − τ ∗ } 8: { computing ω (˜ s ) at time 0 of trajectory issued from ˜ 9: ω (˜ s ) ← ˜ ω (Φ(˜ s, e )) 10: end for 11: until All ω (˜ s ) are equal 12: Return ω (˜ s ) Page 9/19 J.M. Vincent Laboratoire LIG-IMAG
Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations Partially Ordered State Spaces Monotonicity property e p ∈ 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 States (˜ s ) τ ∗ ˜ s sup s generated ˜ ˜ s inf -8 -4 -2 -1 0 Time Page 10/19 J.M. Vincent Laboratoire LIG-IMAG
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) Page 11/19 J.M. Vincent Laboratoire LIG-IMAG
Context and Motivation Modeling SAN as discrete-event systems Perfect Simulation of SAN Final considerations Partially Ordered State Spaces Canonical component-wise ordering 00 Queueing Network Model Equivalent MC λ µ ν 10 Event e 1 K 1 K 2 Event e 2 20 01 Equivalent SAN Model Event e 12 A (1) A (2) 11 Type Event Rate 0 0 loc e 1 λ 21 02 syn e 12 µ e 12 e 1 e 2 e 12 loc e 2 ν . . . . . . 12 e 12 e 1 e 2 e 12 K 1 K 2 22 03 e 1 e 12 13 23 Page 12/19 J.M. Vincent Laboratoire LIG-IMAG
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 Page 13/19 J.M. Vincent Laboratoire LIG-IMAG
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) Type Event Rate loc lt i µ P (1) P ( K ) P ( i ) P ( K − 1) syn tr i λ rl 2 tr K − 1 tr i − 1 tr K − 2 syn rl i λ lr K rl 0 rl i +1 tl K T (0) T ( K ) T ( i ) T ( K − 1) loc rt K µ syn tl K λ syn lr K λ tr 1 lt 1 rt K tl K tr i lt i tr K − 1 lt K − 1 lr K tr i − 1 tr K − 2 R (0) L (0) R ( K ) L ( K ) R ( i ) L ( i ) R ( K − 1) L ( K − 1) rl 0 rl 1 lr K rl i rl K − 1 i = 2 . . . ( K − 2) Page 14/19 J.M. Vincent Laboratoire LIG-IMAG
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 000 001 010 100 002 011 101 020 200 012 102 201 Page 15/19 J.M. Vincent Laboratoire LIG-IMAG
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