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 • Concurrency • Conflict
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 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 • 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 • 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 • 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 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 * 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 Figure 4.39 Petri net with input for times 08, σi = 3 [Multiple Server option]
Examples of Timed Petri Nets Petri net with input for times 08, σi = 3 [Single Server option]
References • Fishwick, Paul(1995) – Simulation Model Design and Execution • Petri Nets World • Kemper, Peter(2004) – Lectures on Petri-Net