PN simulation, token games

1
PN simulation, token games

• dr. András Pataricza pataric_at_mit.bme.hu
• dr. Tamás Bartha bartha_at_mit.bme.hu
• Department of Measurement and Information Systems
  of BME

2
Approaches for PN analysis

• Simulation
• Exploration of state space reachability graph
  analysis
• Structural and dynamical properties
• Invariant analysis
• Semi-decision (abstraction)

3
Simulation

• Goal sound modeling of the real system
• sound validated (VV)
• Investigating the trajectories in the system
• State of the PN token distribution (marking)
• State transition firing
• Trajectories in the state space firing
  sequences
• PN is non-deterministic
• Non-determinism must be modeled as well
• Randomization
• Exploration (state space) interactive simulation
  (decision)

4
Simple simulation algorithm

• while (true) do
• Evaluation of fireable transitions (1)
• if There is fireable transition
• then Pick one to fire (non-det.)
• else End of simulation.
• Firing
• end while

(2)
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(1) Evaluation of fireable transitions

• function collect_fireable_transitions(M )
• // Set of fireable transitions
• Lfireable ? ?
• for all t ? T do
• if enabled(t, M ) then Lfireable ? Lfireable ?
  t
• return Lfireable
• end function

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(2) Firing

• // Initialization
• M ? M0
• Lfireable ? collect_fireable_transitions(M )
• while Lfireable ? ? do
• // Firing
• t ? rnd(Lfireable)
• M ? M W?et
• Lfireable ? collect_fireable_transitions(M )
• M ? M
• end while

7
Redundancy

• Problem
• Why do we evaluate the fireability of every
  transition (T evaluation), if only the
  neighbourhood (?t ? t?) has been changed?

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Disabled transitions

• If t transition fires
• t transition may become disabled, if its input
  is connected to ?t
• If it is in conflict with t ?t ? ?t ? ?
• The numerical definition
• of tokens taken M- W-?et
• (Relevant) predecessors of t (input places) ?t
  p ? P M-(p) gt 0
• transitions in conflict with t T (?t )?

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Enabled transitions

• If t fires
• Transition t may become enabled, if its input is
  connected to t?
• t may enable t ?t ? t? ? ?
• The numerical definition
• of tokens added M W?et
• (Relevant) successors of t (output places) t?
  p ? P M(p) gt 0
• Transitions enabled by t T (t?)?
• T ARE SUFFICIENT TO EVALUATE!

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Efficient algorithm Initialization (1)

• Initialization is the same
• // Initialization
• M ? M0
• Lfireable ? ?
• // Initial set of fireable transitions
• for all t ? T do
• if enabled(t, M0) then Lfireable ? Lfireable ? t

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Efficient algorithm Firing phase (2)

• while Lfireable ? ? do
• // Firing
• t ? rnd(Lfireable)
• M ? M W?et
• // Omitting disabled transitions
• for all t ? (?t )? do
• if not(enabled(t, M )) then Lfireable ?
  Lfireable \ t
• // Adding enabled transitions
• for all t ? (t?)? do
• if enabled(t, M ) then Lfireable ? Lfireable ?
  t
• M ? M
• end while

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Priority

• Firing condition is extended t is fireable iff
• Enabled AND
• There exists no priority greater then its
  priority ?(t )
• Conclusion
• Lfireable is not a set, but a vector of sets
  ordered by ? ? ? priorities Lfireable ?
• If firing we pick one transition (non-det.) from
  the set (Lfireable ?) of the greatest priority
  (?)

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Algorithm with priority Initialization (1)

• // Initialization
• M ? M0
• for all ? ? ? do
• Lfireable? ? ?
• // Initial set of fireable transitions
• for all t ? T do
• if enabled(t, M0) then Lfireable?(t ) ?
  Lfireable?(t ) ? t

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Algorithm with priority Firing phase (2)

• while do
• for ? ?max to ?min step -1 do // Firing
• if Lfireable? ? ? then
• t ? rnd(Lfireable?)
• M ? M W?et
• exit for
• end if
• for all ? ? ? do // Enabled transitions
• for all t ? (?t )? do
• if not(enabled(t, M )) then Lfireable?(t) ?
  Lfireable?(t) \ t
• for all t ? (t?)? do
• if enabled(t, M ) then Lfireable?(t) ?
  Lfireable?(t) ? t
• end for
• M ? M
• end while

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Timeliness

• Introducing logical timeliness
• Actions (firings) takes a given amount of time
• Possibilities
• Non-timed transitions
• Timed transitions
• Deterministic timing
• Stochastic timing firing time is a stochastic
  variable
• Ex. Exponential distribution SPN (Stochastic
  Petri Net)

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Timed Petri Nets

• Firing phases
• Firing condition
• Timing
• Stochastic transitions draw a value of timing
• One of the transitions (non-det.) of the smallest
  timing value fires
• Firing
• Stepping the logical time

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Problems to solve

• Timed vs. Non-timed transitions?
• Solution non-timed transitions with high
  priority
• immediate transition
• What if timed transitions are in conflict (one
  fires and disables another one)?
• Different semantics
• Timing starts again
• Reward saving
• For SPN doesnt matter (for the statistical point
  of view)
• Due to the memorylessness of the exponential
  distr.
• P(? ? xy ? ? x) P(? ? y)

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Algorithm timing starts again (aux.
definition)

• procedure maintenance(t, M )
• // Immediate transitions
• for all ? ? ? do
• for all t ? (?t )? ? Timmediate do
• if not(enabled(t, M ) then Limmediate?(t) ?
  Limmediate?(t) \ t
• for all t ? (t?)? ? Timmediate do
• if enabled(t, M ) then Limmediate?(t) ?
  Limmediate?(t) ? t
• end for

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Algorithm timing starts again (aux.
definition) contd

• // omitting transitions already started
• for all ? gt ?simulation do
• for all t ? (?t )? ? Ttimed ? Ltimed? do
• if not(enabled(t, M ) then Ltimed? ? Ltimed?
  \ t
• // adding new enabled transitions
• for all t ? (t?)? ? Ttimed do
• if enabled(t, M ) then Ltimedtime(t) ?
  Ltimedtime(t) ? t
• end for
• end procedure

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Algorithm Initialization (1)

• // Initialization
• M ? M0
• ?simulation ? 0
• for all ? ? ? do
• Limmadiate? ? ?
• // Initial set of fireable transitions
• for all t ? Timmediate do
• if enabled(t, M0) then Limmediate?(t ) ?
  Limmediate?(t ) ? t

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Algorithm Non-timed transitions (2)

• MAIN while do
• for ? ?max to ?min step -1 do
• if Limmediate? ? ? then
• // Firing immediate transition
• t ? rnd(Limmediate?)
• M ? M W?et
• maintenance(t, M )
• M ? M
• exit for
• end if
• end for
• end while

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Algorithm Timed transitions (2)

• if ?? ? ?simulation Ltimed? ? ? then
• // Stepping the time (if necessary)
• ?simulation ? min(? Ltimed? ? ?)
• // firing timed transaction
• t ? rnd(Ltimed?simulation)
• M ? M W?et
• maintenance(t, M )
• M ? M
• goto MAIN
• end if
• end.