1. CAP 4800/CAP 5805: Computer Simulation Concepts
Petri Nets II
Monday, October 24, 2005
Unviersity of Florida

2. 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

3. 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

4. Time in Petri Net -continued
CAP 4800/CAP 5805: Computer Simulation Concepts
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.
Such questions can not be answered with untimed petri nets but are common in practice
Unviersity of Florida

5. 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.

6. Modeling of Time Constant times Stochastic times
Transition occurs at pre-determined times (deterministic)
Stochastic times
Time is determined by some random variable (probabilistic)
Stochastic Petri Nets(SPN)

7. 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

8. 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

9. 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

10. Examples of Timed Petri Nets
Figure 4.39 Petri net with input for times 08, σi = 3
[Multiple Server option]

11. Examples of Timed Petri Nets
Petri net with input for times 08, σi = 3
[Single Server option]

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

13. References
Fishwick, Paul(1995) – Simulation Model Design and Execution
Petri Nets World
Kemper, Peter(2004) – Lectures on Petri-Net