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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,

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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

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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

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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

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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 | Cs, sequential component ◮ P ::= P ⊲

⊳

L P | P/L | C, model component

◮ Cs 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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Structured operational semantics

◮ Prefix and Constant

(α, r).E

(α,r)

− → E E

(α,r)

− → E ′ A

(α,r)

− → E ′ (A

def

= E)

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

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SLIDE 6

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Structured operational semantics

◮ Prefix and Constant

(α, r).E

(α,r)

− → E E

(α,r)

− → E ′ A

(α,r)

− → E ′ (A

def

= E)

◮ Choice

E

(α,r)

− → E ′ E + F

(α,r)

− → E ′ F

(α,r)

− → F ′ E + F

(α,r)

− → F ′

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Structured operational semantics

◮ Prefix and Constant

(α, r).E

(α,r)

− → E E

(α,r)

− → E ′ A

(α,r)

− → E ′ (A

def

= E)

◮ Choice

E

(α,r)

− → E ′ E + F

(α,r)

− → E ′ F

(α,r)

− → F ′ E + F

(α,r)

− → F ′

◮ Hiding

E

(α,r)

− → E ′ E/L

(α,r)

− → E ′/L (α ∈ L) E

(α,r)

− → E ′ E/L

(τ,r)

− → E ′/L (α ∈ L)

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Structured operational semantics (continued)

◮ Cooperation

E

(α,r)

− → E ′ E ⊲

⊳

L F

(α,r)

− → E ′ ⊲

⊳

L F

(α ∈ L) F

(α,r)

− → F ′ E ⊲

⊳

L F

(α,r)

− → E ⊲

⊳

L F ′

(α ∈ L)

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Structured operational semantics (continued)

◮ Cooperation

E

(α,r)

− → E ′ E ⊲

⊳

L F

(α,r)

− → E ′ ⊲

⊳

L F

(α ∈ L) F

(α,r)

− → F ′ E ⊲

⊳

L F

(α,r)

− → E ⊲

⊳

L F ′

(α ∈ L) E

(α,r1)

− → E ′ F

(α,r2)

− → F ′ E ⊲

⊳

L F

(α,R)

− → E ′ ⊲

⊳

L F ′

(α ∈ L) R = r1 rα(E) r2 rα(F) min(rα(E), rα(F))

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

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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

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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

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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

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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

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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)/dτ Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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SLIDE 15

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)/dτ

◮ 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)/dτ Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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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)/dτ

◮ 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)/dτ

◮ 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

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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)/dτ

◮ 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)/dτ

◮ 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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

ODE semantics of PEPA

◮ numerical vector form (n1, . . . nm) ◮ how many copies of each derivative is present in a given state ◮ continuous approximation of changes in numbers

C1 D1 C2 E1 E2

(α,s) (α,s)

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

ODE semantics of PEPA

◮ numerical vector form (n1, . . . nm) ◮ how many copies of each derivative is present in a given state ◮ continuous approximation of changes in numbers

entry activity C1 D1 C2 E1 E2

(α,s) (α,s)

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

ODE semantics of PEPA

◮ numerical vector form (n1, . . . nm) ◮ how many copies of each derivative is present in a given state ◮ continuous approximation of changes in numbers

entry activity C1 D1 C2 E1 E2 C3 D2

(α,s) (α,s) (α,s)

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

ODE semantics of PEPA

◮ numerical vector form (n1, . . . nm) ◮ how many copies of each derivative is present in a given state ◮ continuous approximation of changes 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

ODE semantics of PEPA

◮ numerical vector form (n1, . . . nm) ◮ how many copies of each derivative is present in a given state ◮ continuous approximation of changes 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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SLIDE 27

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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SLIDE 28

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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SLIDE 29

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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)

− →}

◮ create activity graph and matrix from syntax

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Activity graph and activity matrix

◮ graph nodes are activities ◮ C (α,r)

− → C ′ C α C ′

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Activity graph and activity matrix

◮ graph nodes are activities and components and derivatives ◮ C (α,r)

− → C ′ C α C ′

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Activity graph and activity matrix

◮ graph nodes are activities and components and derivatives ◮ edges are added ◮ C (α,r)

− → C ′ C α C ′

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Activity graph and activity matrix

◮ graph nodes are activities and components and derivatives ◮ edges are added

◮ from a derivative to an exit activity for that derivative

◮ C (α,r)

− → C ′ C α C ′

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Activity graph and activity matrix

◮ graph nodes are activities and components and derivatives ◮ edges are added

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

◮ C (α,r)

− → C ′ C α C ′

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

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SLIDE 35

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Activity graph and activity matrix

◮ graph nodes are activities and components and derivatives ◮ edges are added

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

◮ activity matrix, derivatives × activities

◮ (d, a) = −1 if a exit activity for d ◮ (d, a) = +1 if a entry activity for d

◮ C (α,r)

− → C ′ then (C, α) = −1 and (C ′, α) = +1 C α C ′

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Timed continuous Petri nets

◮ places P,

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Timed continuous Petri nets

◮ places P, transitions T, disjoint

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Timed continuous Petri nets

◮ places P, transitions T, disjoint ◮ arcs from places to transitions Pre : P × T → {0, 1}

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

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SLIDE 39

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Timed continuous Petri nets

◮ places P, transitions T, disjoint ◮ arcs from places to transitions Pre : P × T → {0, 1} ◮ arcs from transitions to places Post : P × T → {0, 1}

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Timed continuous Petri nets

◮ places P, transitions T, disjoint ◮ arcs from places to transitions Pre : P × T → {0, 1} ◮ arcs from transitions to places Post : P × T → {0, 1} ◮ cost matrix C = Post − Pre

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Timed continuous Petri nets

◮ places P, transitions T, disjoint ◮ arcs from places to transitions Pre : P × T → {0, 1} ◮ arcs from transitions to places Post : P × T → {0, 1} ◮ cost matrix C = Post − Pre ◮ standard definitions of •p, •t, p•, t•

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Dynamic behaviour

◮ firing rates λ : T → (0, ∞) 3.2 1.5 6.0 2.7

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Dynamic behaviour

◮ firing rates λ : T → (0, ∞) ◮ marking M : P × Time → [0, ∞) 3.2 1.5 6.0 2.7

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Dynamic behaviour

◮ firing rates λ : T → (0, ∞) ◮ marking M : P × Time → [0, ∞) 3.2 1.5 6.0 2.7

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Dynamic behaviour

◮ firing rates λ : T → (0, ∞) ◮ marking M : P × Time → [0, ∞) ◮ t is enabled at τ if places preceding t have nonzero marking 3.2 1.5 6.0 2.7

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Dynamic behaviour

◮ firing rates λ : T → (0, ∞) ◮ 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) = minp∈•t

  • m(p, τ)
  • 3.2

1.5 6.0 2.7

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

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SLIDE 47

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Dynamic behaviour

◮ firing rates λ : T → (0, ∞) ◮ 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) = minp∈•t

  • m(p, τ)
  • ◮ t can fire with any amount up to enab(t, τ)

3.2 1.5 6.0 2.7

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

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SLIDE 48

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Dynamic behaviour

◮ firing rates λ : T → (0, ∞) ◮ 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) = minp∈•t

  • m(p, τ)
  • ◮ t can fire with any amount up to enab(t, τ)

2.7 2.0 0.3 4.8

fires 1.2

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

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SLIDE 49

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Dynamic behaviour

◮ firing rates λ : T → (0, ∞) ◮ 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) = minp∈•t

  • m(p, τ)
  • ◮ t can fire with any amount up to enab(t, τ)

2.0 0.3 4.8 1.2 3.9

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

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SLIDE 50

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Dynamic behaviour

◮ firing rates λ : T → (0, ∞) ◮ 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) = minp∈•t

  • m(p, τ)
  • ◮ t can fire with any amount up to enab(t, τ)

2.0 0.3 4.8 1.2 3.9

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Change in marking at place p

◮ infinite server semantics: many servers, many clients

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Change in marking at place p

◮ infinite server semantics: many servers, many clients ◮ fundamental equation for Petri nets

m(·, τ + δτ) = m(·, τ) + C(·, t) · σ(τ)

◮ change in marking of place p

dm(p, τ) dτ =

n

  • j=1

C(p, tj).λ(t). min

p′∈•t

  • 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Comparison

◮ 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

A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Activity graph

◮ activities and derivatives reset1 S1 serv1 S′

1

serv2 S2 reset2 S′

2

C do C ′

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Petri net

◮ activities become transitions and derivatives become places 50 50 100

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

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SLIDE 56

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Petri net (continued)

◮ Post(p, t) = 1 if t is an entry activity of p ◮ Pre(p, t) = 1 if t is an exit activity of p

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Petri net (continued)

◮ Post(p, t) = 1 if t is an entry activity of p ◮ Pre(p, t) = 1 if t is an exit activity of p ◮ C = Post − Pre, same as activity matrix

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

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Petri net (continued)

◮ Post(p, t) = 1 if t is an entry activity of p ◮ Pre(p, t) = 1 if t is an exit activity of p ◮ C = Post − Pre, same as activity matrix ◮ rate of transition is rate of activity

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

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SLIDE 59

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Petri net (continued)

◮ Post(p, t) = 1 if t is an entry activity of p ◮ Pre(p, t) = 1 if t is an exit activity of p ◮ C = Post − Pre, same as activity matrix ◮ rate of transition is rate of activity ◮ t ∈ •p ⇔ t is an entry activity of p ◮ t ∈ p• ⇔ t is an exit activity of p ◮ p ∈ •t ⇔ t is an exit activity of p

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

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SLIDE 60

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Petri net (continued)

◮ Post(p, t) = 1 if t is an entry activity of p ◮ Pre(p, t) = 1 if t is an exit activity of p ◮ C = Post − Pre, same as activity matrix ◮ rate of transition is rate of activity ◮ t ∈ •p ⇔ t is an entry activity of p ◮ t ∈ p• ⇔ t is an exit activity of p ◮ p ∈ •t ⇔ 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Petri net (continued)

◮ Post(p, t) = 1 if t is an entry activity of p ◮ Pre(p, t) = 1 if t is an exit activity of p ◮ C = Post − Pre, same as activity matrix ◮ rate of transition is rate of activity ◮ t ∈ •p ⇔ t is an entry activity of p ◮ t ∈ p• ⇔ t is an exit activity of p ◮ p ∈ •t ⇔ 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

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SLIDE 62

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

slide-63
SLIDE 63

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

slide-64
SLIDE 64

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

slide-65
SLIDE 65

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

slide-66
SLIDE 66

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

slide-67
SLIDE 67

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

slide-68
SLIDE 68

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

slide-69
SLIDE 69

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison 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 ◮ ODE semantics of PEPA has infinite server semantics.

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

slide-70
SLIDE 70

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Further work

◮ finite server semantics: many clients, few servers

◮ special case of infinite in discrete Petri nets ◮ can apply to PEPA ◮ two definitions for continuous Petri nets Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

slide-71
SLIDE 71

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Further work

◮ finite server semantics: many clients, few servers

◮ special case of infinite in discrete Petri nets ◮ can apply to PEPA ◮ two definitions for continuous Petri nets

◮ timed continuous Petri nets to PEPA model

◮ stochastic Petri net to PEPA model in discrete case ◮ uses addition of complementary places ◮ use a similar approach for continuous case Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

slide-72
SLIDE 72

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Further work

◮ finite server semantics: many clients, few servers

◮ special case of infinite in discrete Petri nets ◮ can apply to PEPA ◮ two definitions for continuous Petri nets

◮ timed continuous Petri nets to PEPA model

◮ stochastic Petri net to PEPA model in discrete case ◮ uses addition of complementary places ◮ use a similar approach for continuous case

◮ robustness of ODEs

◮ what happens with small numbers Vashti Galpin, LFCS, University of Edinburgh A comparison of the ODE semantics of PEPA with timed continuous Petri nets PASTA 2007

slide-73
SLIDE 73

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Conclusions

◮ PEPA → timed continuous Petri nets

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

slide-74
SLIDE 74

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Conclusions

◮ PEPA → timed continuous Petri nets ◮ ODEs are identical

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

slide-75
SLIDE 75

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Conclusions

◮ PEPA → timed continuous Petri nets ◮ ODEs are identical ◮ ODE semantics of PEPA has infinite server semantics

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

slide-76
SLIDE 76

Introduction ODE semantics of PEPA Timed continuous Petri nets Comparison Conclusions

Conclusions

◮ PEPA → timed continuous Petri nets ◮ ODEs are identical ◮ ODE semantics of PEPA has infinite server semantics

Thank you

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