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

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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

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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

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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

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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

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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

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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)/dτ Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

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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)/dτ

◮ 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)/dτ Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

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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)/dτ

◮ 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)/dτ

◮ how do these compare?

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

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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)/dτ

◮ 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)/dτ

◮ 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

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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

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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

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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 | Cs ◮ sequential constant Cs def

= S

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

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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 | Cs ◮ sequential constant Cs def

= S

◮ parallel cooperation with multiway synchronisation

C1[n1] ⊲

⊳

L1 C2[n2] ⊲

⊳

L2 . . . ⊲

⊳

Lm−1 Cm[nm]

◮ Ci’s do not synchronise, Ci’s and Cj’s must synchronise Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

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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 | Cs ◮ sequential constant Cs def

= S

◮ parallel cooperation with multiway synchronisation

C1[n1] ⊲

⊳

L1 C2[n2] ⊲

⊳

L2 . . . ⊲

⊳

Lm−1 Cm[nm]

◮ Ci’s do not synchronise, Ci’s and Cj’s must synchronise ◮ identical rates for shared activities Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

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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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

ODE semantics of PEPA

◮ many identical sequential components ◮ each sequential component may have a number of derivatives

A

def

= (a1, r1).B + (a2, r2).C B

def

= (b, s).A C

def

= (c, t).A

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

ODE semantics of PEPA

◮ many identical sequential components ◮ each sequential component may have a number of derivatives

A

def

= (a1, r1).B + (a2, r2).C B

def

= (b, s).A C

def

= (c, t).A

◮ express states in numerical vector form (n1, . . . nm)

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

ODE semantics of PEPA

◮ many identical sequential components ◮ each sequential component may have a number of derivatives

A

def

= (a1, r1).B + (a2, r2).C B

def

= (b, s).A C

def

= (c, t).A

◮ express states in numerical vector form (n1, . . . nm) ◮ number of copies of each component/derivative

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

ODE semantics of PEPA

◮ many identical sequential components ◮ each sequential component may have a number of derivatives

A

def

= (a1, r1).B + (a2, r2).C B

def

= (b, s).A C

def

= (c, t).A

◮ express states in numerical vector form (n1, . . . nm) ◮ number of copies of each component/derivative ◮ transitions update the vector

A

(a1,r1)

− − − − → B

  • N(A), N(B), N(C)
  • (a1,r1)

− − − − →

  • N(A) − 1, N(B) + 1, N(C)
  • Vashti Galpin, LFCS, University of Edinburgh

Continuous approximation of PEPA models and Petri nets ESM 2008

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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

C1 D1 C2 E1 E2

(α,s) (α,s)

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

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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 C1 D1 C2 E1 E2

(α,s) (α,s)

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

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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 C1 D1 C2 E1 E2 C3 D2

(α,s) (α,s) (α,s)

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

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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 exit activity C1 D1 C2 E1 E2 C3 D2

(α,s) (α,s) (α,s)

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

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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 exit activity C1 D1 C2 E1 E2 C3 D2 D3 E3

(α,s) (α,s) (α,s) (α,s)

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Change in number of copies of component D

dN(D, τ) dτ =

  • (α, r)

entry activity

r × min{N(C, τ) | C

(α,r)

− →} −

  • (α, r)

exit activity

r × min{N(C, τ) | C

(α,r)

− →}

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Change in number of copies of component D

dN(D, τ) dτ =

  • (α, r)

entry activity

r × min{N(C, τ) | C

(α,r)

− →} −

  • (α, r)

exit activity

r × min{N(C, τ) | C

(α,r)

− →}

◮ can express ODEs as activity graph and activity matrix

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Activity graph

◮ graph nodes are components and activities

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Activity graph

◮ graph nodes are components and activities ◮ edges are added

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Activity graph

◮ graph nodes are components and activities ◮ edges are added

◮ from a component to an exit activity for that component Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Activity graph

◮ graph nodes are components and activities ◮ edges are added

◮ from a component to an exit activity for that component ◮ from an entry activity for a component to that component Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Activity graph

◮ graph nodes are components and activities ◮ edges are added

◮ from a component to an exit activity for that component ◮ from an entry activity for a component to that component

◮ C (α,r)

− → C ′ C C ′ α

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Timed continuous Petri nets

◮ places P, transitions T, disjoint

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Timed continuous Petri nets

◮ places P, transitions T, disjoint ◮ ordinary Petri nets

Pre : P × T → {0, 1} Post : P × T → {0, 1}

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Timed continuous Petri nets

◮ places P, transitions T, disjoint ◮ ordinary Petri nets

Pre : P × T → {0, 1} Post : P × T → {0, 1}

◮ presets and postsets

Pre(p, t) = 1 ⇔ p ∈ •t t ∈ p• Post(p, t) = 1 ⇔ p ∈ t• t ∈ •p

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Timed continuous Petri nets

◮ places P, transitions T, disjoint ◮ ordinary Petri nets

Pre : P × T → {0, 1} Post : P × T → {0, 1}

◮ presets and postsets

Pre(p, t) = 1 ⇔ p ∈ •t t ∈ p• Post(p, t) = 1 ⇔ p ∈ t• t ∈ •p

◮ cost matrix C = Post − Pre

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Timed continuous Petri nets

◮ places P, transitions T, disjoint ◮ ordinary Petri nets

Pre : P × T → {0, 1} Post : P × T → {0, 1}

◮ presets and postsets

Pre(p, t) = 1 ⇔ p ∈ •t t ∈ p• Post(p, t) = 1 ⇔ p ∈ t• t ∈ •p

◮ cost matrix C = Post − Pre ◮ firing rates λ : T → (0, ∞)

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Dynamic behaviour

◮ marking M : P × Time → [0, ∞)

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Dynamic behaviour

◮ marking M : P × Time → [0, ∞) ◮ t is enabled at τ if places preceding t have nonzero marking

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Dynamic behaviour

◮ marking M : P × Time → [0, ∞) ◮ t is enabled at τ if places preceding t have nonzero marking ◮ enabling degree of t: minimum value of markings at places

preceding t, enab(t, τ) = min

p∈•t

  • m(p, τ)
  • Vashti Galpin, LFCS, University of Edinburgh

Continuous approximation of PEPA models and Petri nets ESM 2008

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Dynamic behaviour

◮ marking M : P × Time → [0, ∞) ◮ t is enabled at τ if places preceding t have nonzero marking ◮ enabling degree of t: minimum value of markings at places

preceding t, enab(t, τ) = min

p∈•t

  • m(p, τ)
  • ◮ infinite server semantics

◮ assume many clients and many servers ◮ t fires enab(t, τ) Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Change in marking at place p

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Change in marking at place p

◮ fundamental equation for Petri nets

m(·, τ + δτ) = m(·, τ) + C · σ(τ)

◮ change in marking of place p

dm(p, τ) dτ =

n

  • j=1

C(p, tj).λ(tj). min

p′∈•tj

  • m(p′, τ)
  • =
  • t∈•p

λ(t). min

p′∈•t{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t{m(p′, τ)}

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

PEPA model to continuous Petri net

◮ translate a PEPA model into a timed continuous Petri net ◮ example – clients and servers

C

def

= (serv1, s1).C ′ + (serv2, s2).C ′ C ′

def

= (do, d).C S1

def

= (serv1, s1).S′

1

S2

def

= (serv2, s2).S′

2

S′

1

def

= (reset1, r1).S1 S′

2

def

= (reset2, r2).S2 Sys

def

=

  • C[100]

⊲ ⊳

{serv1,serv2}

  • S1[50] S2[50]
  • Vashti Galpin, LFCS, University of Edinburgh

Continuous approximation of PEPA models and Petri nets ESM 2008

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Motivation PEPA Continuous Petri nets Transformations Conclusions

ODEs

dN(C, τ) dτ = +d.N(C ′, τ) − s1. min(N(C, τ), N(S1, τ)) − s2. min(N(C, τ), N(S2, τ)) dN(C ′, τ) dτ = −d.N(C ′, τ) + s1. min(N(C, τ), N(S1, τ)) + s2. min(N(C, τ), N(S2, τ))

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

ODEs

dN(C, τ) dτ = +d.N(C ′, τ) − s1. min(N(C, τ), N(S1, τ)) − s2. min(N(C, τ), N(S2, τ)) dN(C ′, τ) dτ = −d.N(C ′, τ) + s1. min(N(C, τ), N(S1, τ)) + s2. min(N(C, τ), N(S2, τ)) dN(Si, τ) dτ = ri.N(S′

i ) − si. min(N(C, τ), N(Si, τ))

i = 1, 2 dN(S′

i , τ)

dτ =

  • si. min(N(C, τ), N(Si, τ)) − ri.N(S′

i )

i = 1, 2

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Activity graph

◮ activities and components reset1 S1 serv1 S′

1

serv2 S2 reset2 S′

2

C do C ′

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Petri net

◮ activities become transitions and components become places 50 50 100

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Petri net (continued)

◮ activities become transitions and components become places

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Petri net (continued)

◮ activities become transitions and components become places ◮ rate of transition is rate of activity

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Petri net (continued)

◮ activities become transitions and components become places ◮ rate of transition is rate of activity ◮ Post(p, t) = 1 ⇔ t ∈ •p ⇔ t is an entry activity of p

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

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SLIDE 50

Motivation PEPA Continuous Petri nets Transformations Conclusions

Petri net (continued)

◮ activities become transitions and components become places ◮ rate of transition is rate of activity ◮ Post(p, t) = 1 ⇔ t ∈ •p ⇔ t is an entry activity of p ◮ Pre(p, t) = 1 ⇔ t ∈ p• ⇔ t is an exit activity of p

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Petri net (continued)

◮ activities become transitions and components become places ◮ rate of transition is rate of activity ◮ Post(p, t) = 1 ⇔ t ∈ •p ⇔ t is an entry activity of p ◮ Pre(p, t) = 1 ⇔ t ∈ p• ⇔ t is an exit activity of p ◮ a marking value of x at p is the same as x copies of p

m(p, τ) = N(p, τ)

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

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Motivation PEPA Continuous Petri nets Transformations Conclusions

Petri net (continued)

◮ activities become transitions and components become places ◮ rate of transition is rate of activity ◮ Post(p, t) = 1 ⇔ t ∈ •p ⇔ t is an entry activity of p ◮ Pre(p, t) = 1 ⇔ t ∈ p• ⇔ t is an exit activity of p ◮ a marking value of x at p is the same as x copies of p

m(p, τ) = N(p, τ)

◮ initial marking m(p, 0) = N(p, 0) for each p

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

slide-53
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Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t{m(p′, τ)}

dN(D, τ) dτ =

  • (α, r)

entry activity

  • r. min{N(C, τ) | C

(α,r)

− →}−

  • (α, r)

exit activity

  • r. min{N(C, τ) | C

(α,r)

− →}

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

slide-54
SLIDE 54

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t{m(p′, τ)}

dN(D, τ) dτ =

  • (α, r)

entry activity

  • r. min{N(C, τ) | C

(α,r)

− →}−

  • (α, r)

exit activity

  • r. min{N(C, τ) | C

(α,r)

− →}

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

slide-55
SLIDE 55

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t{m(p′, τ)}

dN(D, τ) dτ =

  • (α, r)

entry activity

  • r. min{N(C, τ) | C

(α,r)

− →}−

  • (α, r)

exit activity

  • r. min{N(C, τ) | C

(α,r)

− →}

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

slide-56
SLIDE 56

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t{m(p′, τ)}

dN(D, τ) dτ =

  • (α, r)

entry activity

  • r. min{N(C, τ) | C

(α,r)

− →}−

  • (α, r)

exit activity

  • r. min{N(C, τ) | C

(α,r)

− →}

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

slide-57
SLIDE 57

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t{m(p′, τ)}

dN(D, τ) dτ =

  • (α, r)

entry activity

  • r. min{N(C, τ) | C

(α,r)

− →}−

  • (α, r)

exit activity

  • r. min{N(C, τ) | C

(α,r)

− →}

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

slide-58
SLIDE 58

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t{m(p′, τ)}

dN(D, τ) dτ =

  • (α, r)

entry activity

  • r. min{N(C, τ) | C

(α,r)

− →}−

  • (α, r)

exit activity

  • r. min{N(C, τ) | C

(α,r)

− →}

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

slide-59
SLIDE 59

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t{m(p′, τ)}

dN(D, τ) dτ =

  • (α, r)

entry activity

  • r. min{N(C, τ) | C

(α,r)

− →}−

  • (α, r)

exit activity

  • r. min{N(C, τ) | C

(α,r)

− →}

◮ both approaches give the same equations

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

slide-60
SLIDE 60

Motivation PEPA Continuous Petri nets Transformations Conclusions

Continuous Petri net to PEPA model

◮ transformation of bounded net to PEPA model

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

slide-61
SLIDE 61

Motivation PEPA Continuous Petri nets Transformations Conclusions

Continuous Petri net to PEPA model

◮ transformation of bounded net to PEPA model ◮ concept of implicit/complementary place: marking of p does

not change enabling degree of any t m(p, τ) ≥ min

p′∈•t\{p}

  • m(p′, τ)
  • = enab(t, τ)

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

slide-62
SLIDE 62

Motivation PEPA Continuous Petri nets Transformations Conclusions

Continuous Petri net to PEPA model

◮ transformation of bounded net to PEPA model ◮ concept of implicit/complementary place: marking of p does

not change enabling degree of any t m(p, τ) ≥ min

p′∈•t\{p}

  • m(p′, τ)
  • = enab(t, τ)

◮ addition of complementary places: for each p, add new p

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

slide-63
SLIDE 63

Motivation PEPA Continuous Petri nets Transformations Conclusions

Continuous Petri net to PEPA model

◮ transformation of bounded net to PEPA model ◮ concept of implicit/complementary place: marking of p does

not change enabling degree of any t m(p, τ) ≥ min

p′∈•t\{p}

  • m(p′, τ)
  • = enab(t, τ)

◮ addition of complementary places: for each p, add new p

◮ for every arc from p to any t, add an arc from t to p ◮ for every arc from t to any p, add an arc from p to t Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

slide-64
SLIDE 64

Motivation PEPA Continuous Petri nets Transformations Conclusions

Continuous Petri net to PEPA model

◮ transformation of bounded net to PEPA model ◮ concept of implicit/complementary place: marking of p does

not change enabling degree of any t m(p, τ) ≥ min

p′∈•t\{p}

  • m(p′, τ)
  • = enab(t, τ)

◮ addition of complementary places: for each p, add new p

◮ for every arc from p to any t, add an arc from t to p ◮ for every arc from t to any p, add an arc from p to t ◮ m(p, 0) = b(p) − m(p, 0) where b(p) is bound for p Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

slide-65
SLIDE 65

Motivation PEPA Continuous Petri nets Transformations Conclusions

Continuous Petri net to PEPA model

◮ transformation of bounded net to PEPA model ◮ concept of implicit/complementary place: marking of p does

not change enabling degree of any t m(p, τ) ≥ min

p′∈•t\{p}

  • m(p′, τ)
  • = enab(t, τ)

◮ addition of complementary places: for each p, add new p

◮ for every arc from p to any t, add an arc from t to p ◮ for every arc from t to any p, add an arc from p to t ◮ m(p, 0) = b(p) − m(p, 0) where b(p) is bound for p

◮ ODEs remain unchanged by addition of complementary places

min

p′∈•t{m(p′, τ)} =

min

p′∈•t ∩ P{m(p′, τ)}

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

slide-66
SLIDE 66

Motivation PEPA Continuous Petri nets Transformations Conclusions

Complementation

◮ example 200 100

prepare finish serve reset fail repair p1 p2 p3 p4 p5 p6

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

slide-67
SLIDE 67

Motivation PEPA Continuous Petri nets Transformations Conclusions

Complementation

◮ example 200 100 200

prepare finish serve reset fail repair p1 p2 p3 p4 p5 p6 p1 p2

200

p3 p4

100

p5

100

p6

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

slide-68
SLIDE 68

Motivation PEPA Continuous Petri nets Transformations Conclusions

Construction of PEPA model

◮ use algorithm by Hillston, Recalde, Ribaudo and Silva [2001] ◮ converts stochastic Petri net to PEPA model

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

slide-69
SLIDE 69

Motivation PEPA Continuous Petri nets Transformations Conclusions

Construction of PEPA model

◮ use algorithm by Hillston, Recalde, Ribaudo and Silva [2001] ◮ converts stochastic Petri net to PEPA model

◮ convert each pair p and p into a sequential component

Cp

def

=

  • t∈p•

(t, λ(t)).Cp Cp

def

=

  • t∈•p

(t, λ(t)).Cp

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

slide-70
SLIDE 70

Motivation PEPA Continuous Petri nets Transformations Conclusions

Construction of PEPA model

◮ use algorithm by Hillston, Recalde, Ribaudo and Silva [2001] ◮ converts stochastic Petri net to PEPA model

◮ convert each pair p and p into a sequential component

Cp

def

=

  • t∈p•

(t, λ(t)).Cp Cp

def

=

  • t∈•p

(t, λ(t)).Cp

◮ for each p define model component

Mp

def

= Cp|| . . . ||Cp||Cp|| . . . ||Cp with m(p, 0) copies of Cp and m(p, 0) copies of Cp.

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

slide-71
SLIDE 71

Motivation PEPA Continuous Petri nets Transformations Conclusions

Construction of PEPA model

◮ use algorithm by Hillston, Recalde, Ribaudo and Silva [2001] ◮ converts stochastic Petri net to PEPA model

◮ convert each pair p and p into a sequential component

Cp

def

=

  • t∈p•

(t, λ(t)).Cp Cp

def

=

  • t∈•p

(t, λ(t)).Cp

◮ for each p define model component

Mp

def

= Cp|| . . . ||Cp||Cp|| . . . ||Cp with m(p, 0) copies of Cp and m(p, 0) copies of Cp.

◮ recursively build up synchronisation sets and system equation Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

slide-72
SLIDE 72

Motivation PEPA Continuous Petri nets Transformations Conclusions

Construction of PEPA model (cont.)

◮ example

Cp1

def

= (prepare, p).Cp1 Cp1

def

= (finish, f ).Cp1

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

slide-73
SLIDE 73

Motivation PEPA Continuous Petri nets Transformations Conclusions

Construction of PEPA model (cont.)

◮ example

Cp1

def

= (prepare, p).Cp1 Cp1

def

= (finish, f ).Cp1 Cp2

def

= (serve, s).Cp2 Cp2

def

= (prepare, p).Cp2

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

slide-74
SLIDE 74

Motivation PEPA Continuous Petri nets Transformations Conclusions

Construction of PEPA model (cont.)

◮ example

Cp1

def

= (prepare, p).Cp1 Cp1

def

= (finish, f ).Cp1 Cp2

def

= (serve, s).Cp2 Cp2

def

= (prepare, p).Cp2 Cp3

def

= (finish, f ).Cp3 Cp3

def

= (serve, s).Cp3 Cp4

def

= (serve, s).Cp4 Cp4

def

= (reset, r).Cp4 + (repair, e).Cp4 Cp5

def

= (reset, r).Cp5 + (fail, a).Cp5 Cp5

def

= (serve, s).Cp5 Cp6

def

= (repair, e).Cp6 Cp6

def

= (fail, a).Cp6

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

slide-75
SLIDE 75

Motivation PEPA Continuous Petri nets Transformations Conclusions

Construction of PEPA model (cont.)

◮ example

Cp1

def

= (prepare, p).Cp1 Cp1

def

= (finish, f ).Cp1 Cp2

def

= (serve, s).Cp2 Cp2

def

= (prepare, p).Cp2 Cp3

def

= (finish, f ).Cp3 Cp3

def

= (serve, s).Cp3 Cp4

def

= (serve, s).Cp4 Cp4

def

= (reset, r).Cp4 + (repair, e).Cp4 Cp5

def

= (reset, r).Cp5 + (fail, a).Cp5 Cp5

def

= (serve, s).Cp5 Cp6

def

= (repair, e).Cp6 Cp6

def

= (fail, a).Cp6 M

def

= Cp1[200]

⊲ ⊳

{prepare,finish} Cp2[200] ⊲

⊳

{serve} Cp3[200] ⊲

⊳

{serve}

Cp4[100]

⊲ ⊳

{serve,reset,repair} Cp5[100] ⊲

⊳

{fail} Cp6[100]

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

slide-76
SLIDE 76

Motivation PEPA Continuous Petri nets Transformations Conclusions

ODE extraction

◮ view PEPA model as high/low concentration, Cp and Cp

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

slide-77
SLIDE 77

Motivation PEPA Continuous Petri nets Transformations Conclusions

ODE extraction

◮ view PEPA model as high/low concentration, Cp and Cp ◮ construct a high/low activity graph

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

slide-78
SLIDE 78

Motivation PEPA Continuous Petri nets Transformations Conclusions

ODE extraction

◮ view PEPA model as high/low concentration, Cp and Cp ◮ construct a high/low activity graph

◮ node for each component Cp ◮ node for each activity α Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

slide-79
SLIDE 79

Motivation PEPA Continuous Petri nets Transformations Conclusions

ODE extraction

◮ view PEPA model as high/low concentration, Cp and Cp ◮ construct a high/low activity graph

◮ node for each component Cp ◮ node for each activity α ◮ edge from component to activity if exit activity ◮ edge from activity to component if entry activity Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

slide-80
SLIDE 80

Motivation PEPA Continuous Petri nets Transformations Conclusions

ODE extraction

◮ view PEPA model as high/low concentration, Cp and Cp ◮ construct a high/low activity graph

◮ node for each component Cp ◮ node for each activity α ◮ edge from component to activity if exit activity ◮ edge from activity to component if entry activity

◮ extract ODEs

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

slide-81
SLIDE 81

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dN(Cp, τ) dτ =

  • (t, λ(t))

entry activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → }−

  • (t, λ(t))

exit activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → } dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

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

slide-82
SLIDE 82

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dN(Cp, τ) dτ =

  • (t, λ(t))

entry activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → }−

  • (t, λ(t))

exit activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → } dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

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

slide-83
SLIDE 83

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dN(Cp, τ) dτ =

  • (t, λ(t))

entry activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → }−

  • (t, λ(t))

exit activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → } dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

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

slide-84
SLIDE 84

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dN(Cp, τ) dτ =

  • (t, λ(t))

entry activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → }−

  • (t, λ(t))

exit activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → } dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

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

slide-85
SLIDE 85

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dN(Cp, τ) dτ =

  • (t, λ(t))

entry activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → }−

  • (t, λ(t))

exit activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → } dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

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

slide-86
SLIDE 86

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dN(Cp, τ) dτ =

  • (t, λ(t))

entry activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → }−

  • (t, λ(t))

exit activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → } dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

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

slide-87
SLIDE 87

Motivation PEPA Continuous Petri nets Transformations Conclusions

Comparison of ODEs

dN(Cp, τ) dτ =

  • (t, λ(t))

entry activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → }−

  • (t, λ(t))

exit activity

λ(t). min{N(Cp′, τ)|Cp′

(t,λ(t))

− → } dm(p, τ) dτ =

  • t∈•p

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

−

  • t∈p•

λ(t). min

p′∈•t ∩ P{m(p′, τ)}

◮ both approaches give the same equations

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

slide-88
SLIDE 88

Motivation PEPA Continuous Petri nets Transformations Conclusions

Further work and conclusions

◮ further work

◮ different approaches to finite server semantics ◮ robustness of ODEs Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

slide-89
SLIDE 89

Motivation PEPA Continuous Petri nets Transformations Conclusions

Further work and conclusions

◮ further work

◮ different approaches to finite server semantics ◮ robustness of ODEs

◮ PEPA → timed continuous Petri nets

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

slide-90
SLIDE 90

Motivation PEPA Continuous Petri nets Transformations Conclusions

Further work and conclusions

◮ further work

◮ different approaches to finite server semantics ◮ robustness of ODEs

◮ PEPA → timed continuous Petri nets

◮ ODEs are identical Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

slide-91
SLIDE 91

Motivation PEPA Continuous Petri nets Transformations Conclusions

Further work and conclusions

◮ further work

◮ different approaches to finite server semantics ◮ robustness of ODEs

◮ PEPA → timed continuous Petri nets

◮ ODEs are identical

◮ bounded timed continuous Petri nets → PEPA

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

slide-92
SLIDE 92

Motivation PEPA Continuous Petri nets Transformations Conclusions

Further work and conclusions

◮ further work

◮ different approaches to finite server semantics ◮ robustness of ODEs

◮ PEPA → timed continuous Petri nets

◮ ODEs are identical

◮ bounded timed continuous Petri nets → PEPA

◮ ODEs are identical Vashti Galpin, LFCS, University of Edinburgh Continuous approximation of PEPA models and Petri nets ESM 2008

slide-93
SLIDE 93

Motivation PEPA Continuous Petri nets Transformations Conclusions

Further work and conclusions

◮ further work

◮ different approaches to finite server semantics ◮ robustness of ODEs

◮ PEPA → timed continuous Petri nets

◮ ODEs are identical

◮ bounded timed continuous Petri nets → PEPA

◮ ODEs are identical

◮ ODE semantics of PEPA has infinite server semantics

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