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GSPN



Generalized
   Stochastic
      Petri Net

            introduced by:
                      Umar Alharaky, B.Sc.
            supervised by:
                    Khalil Ajami, Ph.D.
Petri Nets




A PN comprises Places, Transitions,
Arcs, and Tokens, which define its
structural component.
Places




Places are used to describe possible
local system states (named
conditions or situations)
Transitions




Transitions are used to describe
events that may modify the system
state.
Arcs




Arcs specify the relation between local
states (Places) and events (Transitions)
in two ways:
• indicates the local state in which the
   event can occur,
• and the local state transformations
   induced by the event.
Tokens




Tokens are indistinguishable
markers that reside in places, and are
used to specify the PN state (usually
called the PN marking).
Enabling Rule




A transition can fire (an event takes
place) if all the transition input places
contain at least one token. In this
case the transition is said to be
enabled.
Firing Rule




The firing of an enabled transition
removes one token from all of its
input places, and generates one
token in each of its output places.
Transition Firing




Typically, the firing of a transition
describes the result of either
• a logical condition becoming
  true in the system, or
• the completion of an activity.
Time Specification




Time is introduced in PN to model
the interaction among several
activities considering their starting
and completion time.
Timed Places

• Time may be associated with places (TPPN):
    tokens generated in an output place
    become available to fire a transition only
    after a delay has elapsed; the delay is an
    attribute of the place.
Timed Tokens

• Time may be associated with tokens:
    tokens carry a time stamp that indicates
    when they are available to fire a transition;
    this time stamp can be incremented at
    each transition firing.
Timed Arcs

• Time may be associated with arcs:
    a travelling delay is associated with each
    arc; tokens are available for firing only
    when they reach a transition
Timed Transitions

• Time associated with transitions (TTPN):
    transitions represent activities
    • activity start corresponds to transition
      enabling,
    • activity end corresponds to transition
      firing.
Firing Policy



• Three-phase firing:
1. tokens are consumed from input places
   when the transition is enabled
2. the delay elapses
3. tokens are generated in output places
Firing Policy



• Atomic firing
  tokens remain in input places for the
  transition delay; they are consumed from
  input places and generated in output
  places when the transition fires
Firing Policy


• We shall consider TTPN with atomic firing.
• TTPN with atomic firing can preserve the
  basic behavior of the underlying untimed
  model.
• It is thus possible to qualitatively study TTPN
  with atomic firing exploiting the theory
  developed for untimed PN (reachability set,
  invariants, etc.).
Internal Timer


We can explain the behavior of one timed
transition with atomic firing by assuming that
it incorporates a timer.
• When the transition is enabled, its timer is
   set to the current delay value
• Then, the timer is decremented at constant
   speed, until it reaches the value zero
• At this point the transition fires
Conflict

When more than one timed transition with
atomic firing is enabled, the behavior is
similar, but a problem arises:
• Which one of the enabled transitions is
  going to fire?
Selection Rule

Two alternative selection rules:
• Preselection:
 the enabled transition that will fire is
 chosen when the marking is entered,
 according to some metric (priority,
 probability, ...)
• Race:
 the enabled transition that will fire is the
 one whose firing delay is minimum
Memory Policy

When a timed transition is disabled by a
conflicting transition, a problem arises:
• How is the transition timer set when
  the transition will again become
  enabled?
• How does the transition keep memory
  of its past enabling time?
Basic Mechanism

Two basic mechanisms can be defined:
• Continue:
  the timer associated with the transition holds
  the present value and will continue later on
  the countdown
• Restart:
  the timer associated with the transition is
  restarted, i.e., its present value is discarded
  and a new value will be generated when
  needed
Transition Memory Policy

• Resampling:
  • At each and every transition firing, the
    timers of all timed transitions in the timed
    PN system are discarded (restart
    mechanism).
  • No memory of the past is recorded.
  • After discarding all timers, new values of
    the timers are set for the transitions that
    are enabled in the new marking.
Transition Memory Policy

• Enabling memory:
  • At each transition firing, the timers of all timed
    transitions that become disabled are restarted,
    whereas the timers of all timed transitions that
    remain enabled hold their present value (continue
    mechanism).
  • The memory of the past is recorded with an
    enabling memory variable associated with each
    transition.
  • The enabling memory variable accounts for the
    work performed by the activity associated with the
    transition since the last instant of time when its
    timer was set.
Transition Memory Policy

• Age memory:
  • At each transition firing, the timers of all timed
    transitions hold their present values (continue
    mechanism).
  • The memory of the past is recorded with an
    age memory variable associated with each
    timed transition.
  • The age memory variable accounts for the
    work performed by the activity associated
    with the transition since the time of its last
    firing.
Immediate Transition


• Immediate transitions fire as soon as they
  become enabled (with a null delay),
• thus acquiring a sort of precedence over
  timed transitions,
GSPN


Two classes of transitions exist in GSPNs:
• timed transitions, whose delays are
  exponentially distributed random
  variables
• immediate transitions, whose delays are
  deterministically zero
Immediate transitions have priority over
timed transitions.
GSPN



• Formally, a GSPN is an 8-tuple:
   GSPN= (P, T, Π(.), I(.),O(.),H(.),W(.),M0)
  where
• PNπ = (P, T, Π(.), I(.),O(.), H(.), M0) is the
  marked PN with priority underlying the
  GSPN
GSPN


• W : T → IR is a function defined on the set of
  transitions

• The quantity W(tk) = wk is called
   the “rate” of transition tk if tk is timed
   the “weight” of transition tk if tk is n-
    immediate
GSPN


• The rate (λ) is the constant of negative
  exponential probability density
  function (pdf) used to specify the random
  delays.
            fX(x) = λ e−λx   (x ≥ 0)

• The value of rate is the inverse of average
  transition time.
GSPN


• The weight are used for the probabilistic
  resolution of Extended Conflict Set (ECS)
  of immediate transitions.
           P{ ti | M } = wi / WI(M)
  where
GSPN Example
GSPN Example
Thank
 You

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Free and Effective: Making Flows Publicly Accessible, Yumi Ibrahimzade
 

Generalized Stochastic Petri Nets

  • 1. GSPN Generalized Stochastic Petri Net introduced by: Umar Alharaky, B.Sc. supervised by: Khalil Ajami, Ph.D.
  • 2. Petri Nets A PN comprises Places, Transitions, Arcs, and Tokens, which define its structural component.
  • 3. Places Places are used to describe possible local system states (named conditions or situations)
  • 4. Transitions Transitions are used to describe events that may modify the system state.
  • 5. Arcs Arcs specify the relation between local states (Places) and events (Transitions) in two ways: • indicates the local state in which the event can occur, • and the local state transformations induced by the event.
  • 6. Tokens Tokens are indistinguishable markers that reside in places, and are used to specify the PN state (usually called the PN marking).
  • 7. Enabling Rule A transition can fire (an event takes place) if all the transition input places contain at least one token. In this case the transition is said to be enabled.
  • 8. Firing Rule The firing of an enabled transition removes one token from all of its input places, and generates one token in each of its output places.
  • 9. Transition Firing Typically, the firing of a transition describes the result of either • a logical condition becoming true in the system, or • the completion of an activity.
  • 10. Time Specification Time is introduced in PN to model the interaction among several activities considering their starting and completion time.
  • 11. Timed Places • Time may be associated with places (TPPN): tokens generated in an output place become available to fire a transition only after a delay has elapsed; the delay is an attribute of the place.
  • 12. Timed Tokens • Time may be associated with tokens: tokens carry a time stamp that indicates when they are available to fire a transition; this time stamp can be incremented at each transition firing.
  • 13. Timed Arcs • Time may be associated with arcs: a travelling delay is associated with each arc; tokens are available for firing only when they reach a transition
  • 14. Timed Transitions • Time associated with transitions (TTPN): transitions represent activities • activity start corresponds to transition enabling, • activity end corresponds to transition firing.
  • 15. Firing Policy • Three-phase firing: 1. tokens are consumed from input places when the transition is enabled 2. the delay elapses 3. tokens are generated in output places
  • 16. Firing Policy • Atomic firing tokens remain in input places for the transition delay; they are consumed from input places and generated in output places when the transition fires
  • 17. Firing Policy • We shall consider TTPN with atomic firing. • TTPN with atomic firing can preserve the basic behavior of the underlying untimed model. • It is thus possible to qualitatively study TTPN with atomic firing exploiting the theory developed for untimed PN (reachability set, invariants, etc.).
  • 18. Internal Timer We can explain the behavior of one timed transition with atomic firing by assuming that it incorporates a timer. • When the transition is enabled, its timer is set to the current delay value • Then, the timer is decremented at constant speed, until it reaches the value zero • At this point the transition fires
  • 19. Conflict When more than one timed transition with atomic firing is enabled, the behavior is similar, but a problem arises: • Which one of the enabled transitions is going to fire?
  • 20. Selection Rule Two alternative selection rules: • Preselection: the enabled transition that will fire is chosen when the marking is entered, according to some metric (priority, probability, ...) • Race: the enabled transition that will fire is the one whose firing delay is minimum
  • 21. Memory Policy When a timed transition is disabled by a conflicting transition, a problem arises: • How is the transition timer set when the transition will again become enabled? • How does the transition keep memory of its past enabling time?
  • 22. Basic Mechanism Two basic mechanisms can be defined: • Continue: the timer associated with the transition holds the present value and will continue later on the countdown • Restart: the timer associated with the transition is restarted, i.e., its present value is discarded and a new value will be generated when needed
  • 23. Transition Memory Policy • Resampling: • At each and every transition firing, the timers of all timed transitions in the timed PN system are discarded (restart mechanism). • No memory of the past is recorded. • After discarding all timers, new values of the timers are set for the transitions that are enabled in the new marking.
  • 24. Transition Memory Policy • Enabling memory: • At each transition firing, the timers of all timed transitions that become disabled are restarted, whereas the timers of all timed transitions that remain enabled hold their present value (continue mechanism). • The memory of the past is recorded with an enabling memory variable associated with each transition. • The enabling memory variable accounts for the work performed by the activity associated with the transition since the last instant of time when its timer was set.
  • 25. Transition Memory Policy • Age memory: • At each transition firing, the timers of all timed transitions hold their present values (continue mechanism). • The memory of the past is recorded with an age memory variable associated with each timed transition. • The age memory variable accounts for the work performed by the activity associated with the transition since the time of its last firing.
  • 26. Immediate Transition • Immediate transitions fire as soon as they become enabled (with a null delay), • thus acquiring a sort of precedence over timed transitions,
  • 27. GSPN Two classes of transitions exist in GSPNs: • timed transitions, whose delays are exponentially distributed random variables • immediate transitions, whose delays are deterministically zero Immediate transitions have priority over timed transitions.
  • 28. GSPN • Formally, a GSPN is an 8-tuple: GSPN= (P, T, Π(.), I(.),O(.),H(.),W(.),M0) where • PNπ = (P, T, Π(.), I(.),O(.), H(.), M0) is the marked PN with priority underlying the GSPN
  • 29. GSPN • W : T → IR is a function defined on the set of transitions • The quantity W(tk) = wk is called  the “rate” of transition tk if tk is timed  the “weight” of transition tk if tk is n- immediate
  • 30. GSPN • The rate (λ) is the constant of negative exponential probability density function (pdf) used to specify the random delays. fX(x) = λ e−λx (x ≥ 0) • The value of rate is the inverse of average transition time.
  • 31. GSPN • The weight are used for the probabilistic resolution of Extended Conflict Set (ECS) of immediate transitions. P{ ti | M } = wi / WI(M) where