Consider the following Petri Net with the initial state as shown in the figure.

Question

Consider the following Petri Net with the initial state as shown in the figure. Note that this Petri Net has four places. The state of the Petri Net can be described as [N, N2, Na, N4], where N, represents the number of kens in place p1Sis4. The following initial state can therefore be represented as [1, 0, 0,2) P4 2 2 P2 P3 (1) After the Petri Net fires four times, find a state such that all transitions are dead. Write down the resulting state of [Ni, N2,N, N]. (2 points) initial state of the Petri Net. We work out the following corresponding sequence. (4 points) (a) Can we file the sequence once? If so, what is the resulting state of N1, N2,N3, N4)? (b) Can we file the sequence twice? If so, what is the resulting state of N1, N2,N, NJ? (c) Can we file the sequence three times? If so, what is the resulting state of IN, Na, N NJ (d) Conclusion: Can the sequence be repeated infinitely? If yes, explain why. If not, indicate where it (2) We would like to find out if we can repeat the firing seqeinfinitely often from the fails to repeat.

Explanation / Answer

So when the Perti Net fires we have the following states:-

Fire1:- When it strikes at t2 and p4 = 2

Fire2:- When it strikes at t3, p1 will become as p4 as the same p4 is travelling. hence p1 = 2.

Fire3:- When it strikes at t1, p2 becomes p1*2 ie p2 = 4 as indicated in the diagram

Fire4:- When it strikes at t4, p3 will be same as p2 since p2 is travelling towards p3 as indicated in the diagram. Hence p3 = 4.

So the Firing Sequence is <t2, t3, t1, t4> and <p1,p2,p3,p4> = <2,4,4,2>, which is the dead state of Petri Net

<N1, N2, N3, N4> = <2,4,4,2>

2)

a) Let us start from the initial position of the Petri. From p1 it strikes t1 and then bounces back to p1 and then strikes t1 again, so we get <t1,t1>. From t1 we get p2 = 2 and now these 2 strikes <t4,t4> individually. So total sequence is <t1,t1,t4,t4>. So p3 = 2 and when this strikes individually <t2,t2>, we get total sequence as:- <t1,t1,t4,t4,t2,t2,...> which violates our firing sequence <t1,t1,t4,t4,t2,t3,t3> as there's a single t2 in this. Hence we cant file the sequence once.

b) Based on a we can never get t2 once in firing sequence as from p2 = 2 we have 2 striking t2, which gives <t2,t2>. Hence we cant file the sequence twice.

c) We cant file the sequence three times based on a and b

d) No the sequence cannot be repeated infinitely, because from point t4, we have p2 = 2 which when strikes t2 individually produces <t2,t2> and not single t2. So it fails to repeat the firing sequence mentioned above from point t2 onwards.

Please let me know in case of any clarifications required. Thanks!

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