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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
12
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