Petri nets ii
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Petri Nets II. Monday, October 24, 2005. Review. Petri Net C = ( P, T, I, O) marking µ : instantaneous state of the Petri net Consists of places and transitions, connected by arcs. Token can be placed in places and fired. Properties: Sequential Execution Synchronization Merging

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Petri Nets II

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Petri nets ii

Petri Nets II

Monday, October 24, 2005


Review

Review

  • Petri Net

    • C = ( P, T, I, O)

    • marking µ : instantaneous state of the Petri net

    • Consists of places and transitions, connected by arcs. Token can be placed in places and fired.

    • Properties:

      • Sequential Execution

      • Synchronization

      • Merging

      • Concurrency

      • Conflict


Time in petri net

Time in Petri Net

Original model of Petri Net was timeless. Time was not explicitly considered since

  • measurements of time in distributed systems implies synchronization via a global clock

  • independency describes a form of parallelism(concurrency) without time

  • without time the modeling capabilities of petri nets are larger than with time and modeling is consistent with the laws of modern physics


Time in petri net continued

Time in Petri Net-continued

Even though there are arguments against the introduction of time, there are several applications that require notion of time.

First attempt was made by Ramchandani at MIT in 1974, and since then there have been many different approaches of extending petri net by the integration of time, however not a systemic introduction.


Timed petri net overview

Timed Petri Net - Overview

  • General approach:

    • Transition is associated with a time for which no event/firing of a token can occur until this delay time has elapsed.

    • This delay time can be deterministic or probabilistic.

    • Number of servers should be specified. Different outcomes resulted from plural/single server.


Modeling of time

Modeling of Time

  • Constant times

    • Transition occurs at pre-determined times (deterministic)

  • Stochastic times

    • Time is determined by some random variable (probabilistic)

    • Stochastic Petri Nets(SPN)


Timed petri net w different server options

Timed Petri Net w/ Different Server Options

  • Multi-Server / Infinite Server

    • There are no capacity restrictions to a transition.

    • Multiple tokens can be reserved to be fired.

  • Single Server

    • Capacity of a transition is 1.

    • Only one token can be reserved at the same time.

      *reserved: if a token is ready to fire but scheduled to fire after a delay time, the token is reserved for the transition


Timed petri net with multi server infinite server

Timed Petri Net with Multi-Server / Infinite Server

Di = Ai + σ

i = index of token (by order of arrival)

Ai : arrival time of the token i (i.e. input time)

Di : departure time of the token i (i.e. firing time)

σ : time delay


Timed petri net with a single server

Timed Petri Net with a Single Server

* Use the same algorithm from a single-server queue.

Di = max(Di-1, Ai) + σ

i = index of token

D0 = 0

Ai : arrival time of the token i (i.e. input time)

Di : departure time of the token i (i.e. firing time)

σ : time delay


Examples of timed petri nets

Examples of Timed Petri Nets

Figure 4.39 Petri net with input for times 08, σi = 3

[Multiple Server option]


Examples of timed petri nets1

Examples of Timed Petri Nets

Petri net with input for times 08, σi = 3

[Single Server option]


Petri nets ii 1368183

State Trajectories of Timed Petri Net with input for times 0  8, σi = 3


References

References

  • Fishwick, Paul(1995) – Simulation Model Design and Execution

  • Petri Nets World

  • Kemper, Peter(2004) – Lectures on Petri-Net


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