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Petri Nets II

Monday, October 24, 2005

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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

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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

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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.

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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.

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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)

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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

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Timed Petri Net with Multi-Server / Infinite Server

DDii = A = Aii + + σσ

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

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Timed Petri Net with a Single Server

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

DDii = max(D = max(Di-1i-1, A, Aii) + ) + σσi = index of tokenD0 = 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

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Examples of Timed Petri Nets

Figure 4.39 Petri net with input for times 08, σi = 3

[Multiple Server option]

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

time

P1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

time

P2

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Examples of Timed Petri Nets

Petri net with input for times 08, σi = 3

[Single Server option]

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

time

P1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

time

P2

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0

2

4

6

8

0 1 2 3 4 5 6 7 8

time

P1

infinite server

single server

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

0

2

4

6

8

0 1 2 3 4 5 6 7 8

time

P2

infinite server

single server

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References

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

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