A comparison of the ODE semantics of PEPA with timed continuous - PowerPoint PPT Presentation

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions A comparison of the ODE semantics of PEPA with timed continuous Petri nets Vashti Galpin LFCS University of Edinburgh 25 July 2007 Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Outline Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions PEPA ◮ Performance Evaluation Process Algebra [Hillston 1996] ◮ syntax, structured operational semantics ◮ equivalence semantics ◮ analysis of dynamic behaviour ◮ stochastic, action durations from exponential distribution Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions PEPA ◮ Performance Evaluation Process Algebra [Hillston 1996] ◮ syntax, structured operational semantics ◮ equivalence semantics ◮ analysis of dynamic behaviour ◮ stochastic, action durations from exponential distribution ◮ syntax ◮ S ::= ( α, r ) . S | S + S | C s , sequential component ◮ P ::= P ⊲ ⊳ L P | P / L | C , model component ◮ C s and C constants ◮ cooperations of sequential components ◮ ergodic continuous time Markov chain (CTMC) Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Structured operational semantics ◮ Prefix and Constant ( α, r ) E − → E ′ def ( A = E ) ( α, r ) ( α, r ) ( α, r ) . E − → E A − → E ′ Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Structured operational semantics ◮ Prefix and Constant ( α, r ) E − → E ′ def ( A = E ) ( α, r ) ( α, r ) ( α, r ) . E − → E A − → E ′ ◮ Choice ( α, r ) ( α, r ) → E ′ → F ′ E − F − ( α, r ) ( α, r ) E + F − → E ′ E + F − → F ′ Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Structured operational semantics ◮ Prefix and Constant ( α, r ) E − → E ′ def ( A = E ) ( α, r ) ( α, r ) ( α, r ) . E − → E A − → E ′ ◮ Choice ( α, r ) ( α, r ) → E ′ → F ′ E − F − ( α, r ) ( α, r ) E + F − → E ′ E + F − → F ′ ◮ Hiding ( α, r ) ( α, r ) − → E ′ − → E ′ E E ( α �∈ L ) ( α ∈ L ) ( α, r ) ( τ, r ) E / L − → E ′ / L E / L − → E ′ / L Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Structured operational semantics (continued) ◮ Cooperation ( α, r ) ( α, r ) E − → E ′ F − → F ′ ( α �∈ L ) ( α �∈ L ) ( α, r ) ( α, r ) → E ′ ⊲ E ⊲ ⊳ L F − ⊳ L F E ⊲ ⊳ L F − → E ⊲ ⊳ L F ′ Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Structured operational semantics (continued) ◮ Cooperation ( α, r ) ( α, r ) E − → E ′ F − → F ′ ( α �∈ L ) ( α �∈ L ) ( α, r ) ( α, r ) → E ′ ⊲ E ⊲ ⊳ L F − ⊳ L F E ⊲ ⊳ L F − → E ⊲ ⊳ L F ′ ( α, r 1 ) ( α, r 2 ) → E ′ → F ′ E − F − ( α ∈ L ) ( α, R ) → E ′ ⊲ E ⊲ ⊳ L F − ⊳ L F ′ r 1 r 2 R = r α ( F ) min( r α ( E ) , r α ( F )) r α ( E ) Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Modelling ◮ operational semantics generate a labelled multi-transition system Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Modelling ◮ operational semantics generate a labelled multi-transition system ◮ equivalence semantics ◮ same behaviour ◮ bisimulation Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Modelling ◮ operational semantics generate a labelled multi-transition system ◮ equivalence semantics ◮ same behaviour ◮ bisimulation ◮ analysis of dynamic behaviour ◮ state transition diagram → continuous time Markov Chain ◮ syntax → activity matrix → ODEs ◮ syntax → rate equations → stochastic simulation Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Motivation and background ◮ hybrid systems Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Motivation and background ◮ hybrid systems ◮ PEPA, continuous approximation using ODEs [Hillston] ◮ many identical components ◮ equations for dN ( D , τ ) / d τ Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Motivation and background ◮ hybrid systems ◮ PEPA, continuous approximation using ODEs [Hillston] ◮ many identical components ◮ equations for dN ( D , τ ) / d τ ◮ timed continuous Petri nets [Alla & David, Recalde & Silva] ◮ transitions have rates ◮ marking values from positive reals ◮ large numbers of clients and servers ◮ equations for dM ( p , τ ) / d τ Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Motivation and background ◮ hybrid systems ◮ PEPA, continuous approximation using ODEs [Hillston] ◮ many identical components ◮ equations for dN ( D , τ ) / d τ ◮ timed continuous Petri nets [Alla & David, Recalde & Silva] ◮ transitions have rates ◮ marking values from positive reals ◮ large numbers of clients and servers ◮ equations for dM ( p , τ ) / d τ ◮ how do these compare? Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions Motivation and background ◮ hybrid systems ◮ PEPA, continuous approximation using ODEs [Hillston] ◮ many identical components ◮ equations for dN ( D , τ ) / d τ ◮ timed continuous Petri nets [Alla & David, Recalde & Silva] ◮ transitions have rates ◮ marking values from positive reals ◮ large numbers of clients and servers ◮ equations for dM ( p , τ ) / d τ ◮ how do these compare? ◮ infinite or finite server semantics? Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions ODE semantics of PEPA ◮ numerical vector form ( n 1 , . . . n m ) ◮ how many copies of each derivative is present in a given state ◮ continuous approximation of changes in numbers C 1 ( α, s ) E 1 ( α, s ) D 1 C 2 E 2 Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions ODE semantics of PEPA ◮ numerical vector form ( n 1 , . . . n m ) ◮ how many copies of each derivative is present in a given state ◮ continuous approximation of changes in numbers entry activity C 1 ( α, s ) E 1 ( α, s ) D 1 C 2 E 2 Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007