Continuous approximation of PEPA models and Petri nets Vashti - PowerPoint PPT Presentation

Motivation PEPA Continuous Petri nets Transformations Conclusions Continuous approximation of PEPA models and Petri nets Vashti Galpin Laboratory for Foundations of Computer Science School of Informatics University of Edinburgh 27 October 2008 Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions Outline Motivation PEPA Continuous Petri nets Transformations Conclusions Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions Motivation ◮ large systems and state space explosion Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions Motivation ◮ large systems and state space explosion ◮ use approximation to avoid Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions Motivation ◮ large systems and state space explosion ◮ use approximation to avoid ◮ PEPA, continuous approximation using ODEs [Hillston 2005] ◮ many identical components ◮ equations for dN ( D , τ ) / d τ Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions Motivation ◮ large systems and state space explosion ◮ use approximation to avoid ◮ PEPA, continuous approximation using ODEs [Hillston 2005] ◮ many identical components ◮ equations for dN ( D , τ ) / d τ ◮ timed continuous Petri nets [Alla & David, Recalde & Silva] ◮ transitions have rates ◮ markings take values from positive reals ◮ large numbers of clients and servers ◮ equations for dm ( p , τ ) / d τ Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions Motivation ◮ large systems and state space explosion ◮ use approximation to avoid ◮ PEPA, continuous approximation using ODEs [Hillston 2005] ◮ many identical components ◮ equations for dN ( D , τ ) / d τ ◮ timed continuous Petri nets [Alla & David, Recalde & Silva] ◮ transitions have rates ◮ markings take 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 Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions Motivation ◮ large systems and state space explosion ◮ use approximation to avoid ◮ PEPA, continuous approximation using ODEs [Hillston 2005] ◮ many identical components ◮ equations for dN ( D , τ ) / d τ ◮ timed continuous Petri nets [Alla & David, Recalde & Silva] ◮ transitions have rates ◮ markings take values from positive reals ◮ large numbers of clients and servers ◮ equations for dm ( p , τ ) / d τ ◮ how do these compare? ◮ what are the server semantics of PEPA? Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions PEPA ◮ Performance Evaluation Process Algebra [Hillston 1996] ◮ stochastic, action durations from exponential distribution ◮ syntax, structured operational semantics ◮ continuous time Markov chain (CTMC) to describe dynamic behaviour Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions PEPA ◮ Performance Evaluation Process Algebra [Hillston 1996] ◮ stochastic, action durations from exponential distribution ◮ syntax, structured operational semantics ◮ continuous time Markov chain (CTMC) to describe dynamic behaviour ◮ restricted PEPA syntax Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions PEPA ◮ Performance Evaluation Process Algebra [Hillston 1996] ◮ stochastic, action durations from exponential distribution ◮ syntax, structured operational semantics ◮ continuous time Markov chain (CTMC) to describe dynamic behaviour ◮ restricted PEPA syntax ◮ sequential component S ::= ( α, r ) . S | S + S | C s def ◮ sequential constant C s = S Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions PEPA ◮ Performance Evaluation Process Algebra [Hillston 1996] ◮ stochastic, action durations from exponential distribution ◮ syntax, structured operational semantics ◮ continuous time Markov chain (CTMC) to describe dynamic behaviour ◮ restricted PEPA syntax ◮ sequential component S ::= ( α, r ) . S | S + S | C s def ◮ sequential constant C s = S ◮ parallel cooperation with multiway synchronisation C 1 [ n 1 ] ⊲ ⊳ L 1 C 2 [ n 2 ] ⊲ ⊳ L 2 . . . ⊲ Lm − 1 C m [ n m ] ⊳ ◮ C i ’s do not synchronise, C i ’s and C j ’s must synchronise Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions PEPA ◮ Performance Evaluation Process Algebra [Hillston 1996] ◮ stochastic, action durations from exponential distribution ◮ syntax, structured operational semantics ◮ continuous time Markov chain (CTMC) to describe dynamic behaviour ◮ restricted PEPA syntax ◮ sequential component S ::= ( α, r ) . S | S + S | C s def ◮ sequential constant C s = S ◮ parallel cooperation with multiway synchronisation C 1 [ n 1 ] ⊲ ⊳ L 1 C 2 [ n 2 ] ⊲ ⊳ L 2 . . . ⊲ Lm − 1 C m [ n m ] ⊳ ◮ C i ’s do not synchronise, C i ’s and C j ’s must synchronise ◮ identical rates for shared activities Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions ODE semantics of PEPA ◮ many identical sequential components Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions ODE semantics of PEPA ◮ many identical sequential components ◮ each sequential component may have a number of derivatives def def def = ( a 1 , r 1 ) . B + ( a 2 , r 2 ) . C = ( b , s ) . A = ( c , t ) . A A B C Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions ODE semantics of PEPA ◮ many identical sequential components ◮ each sequential component may have a number of derivatives def def def = ( a 1 , r 1 ) . B + ( a 2 , r 2 ) . C = ( b , s ) . A = ( c , t ) . A A B C ◮ express states in numerical vector form ( n 1 , . . . n m ) Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions ODE semantics of PEPA ◮ many identical sequential components ◮ each sequential component may have a number of derivatives def def def = ( a 1 , r 1 ) . B + ( a 2 , r 2 ) . C = ( b , s ) . A = ( c , t ) . A A B C ◮ express states in numerical vector form ( n 1 , . . . n m ) ◮ number of copies of each component/derivative Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions ODE semantics of PEPA ◮ many identical sequential components ◮ each sequential component may have a number of derivatives def def def = ( a 1 , r 1 ) . B + ( a 2 , r 2 ) . C = ( b , s ) . A = ( c , t ) . A A B C ◮ express states in numerical vector form ( n 1 , . . . n m ) ◮ number of copies of each component/derivative ◮ transitions update the vector ( a 1 , r 1 ) − − − − → A B ( a 1 , r 1 ) � � � � N ( A ) , N ( B ) , N ( C ) − − − − → N ( A ) − 1 , N ( B ) + 1 , N ( C ) Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions ODE semantics of PEPA (continued) ◮ continuous approximation of changes in numbers ◮ consider what actions lead to change in numbers C 1 ( α, s ) E 1 ( α, s ) D 1 C 2 E 2 Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

Motivation PEPA Continuous Petri nets Transformations Conclusions ODE semantics of PEPA (continued) ◮ continuous approximation of changes in numbers ◮ consider what actions lead to change in numbers entry activity C 1 ( α, s ) E 1 ( α, s ) D 1 C 2 E 2 Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008