Petri Nets: Quiz 1, March 4, 1998

1. In the following let A, B, C be places, and if X is a place then µ(X) denotes the marking of X. Design Petri Nets that perform the calculations below. Include a place Ready so that µ(Ready)=1 when the calculation is complete. You may assume that µ(C)=0 initially. Check your design for (µ(A) µ(B)) = (2 3).
   1. µ(C) := µ(A)
   2. µ(C) := µ(A)+µ(B)
   3. µ(C) := |µ(A)-µ(B)|
   4. µ(C) := if µ(A) < µ(B)then µ(A)else µ(B)

2. Like in 1. above, let A, B, C be places, and if X is a place then µ(X) denotes the content of X. Design a Petri Net that performs the calculation below. Include a place Ready so that µ(Ready)=1 when the calculation is complete. You may assume that µ(C)=0 initially. Check your design for (µ(A) µ(B)) = (2 3).

   µ(C) := µ(A)*µ(B)

3. Construct a net that tests a 2-bounded place p for its marking 0, 1, and 2, i.e., a net with the places Test, p, p', q0, q1, q2, so that given µ(Test)=1 and µ(p), µ(p') with µ(p)+µ(p')=2, the result vector (µ(q0) µ(q1) µ(q2)) will be (1 0 0) , (0 1 0), and (0 0 1) if µ(p)=0, 1, and 2 respectively. Do not use inhibitor arcs. The markings of p, p' after the test should be the initial. Check your design.

4. Given the incidence matrix below for a Petri Net without selfloops (no double arcs), draw the graph and the reachability tree for the the initial marking (1 0 0 0). (There should be 8 nodes in the resulting tree.)

                          p1  p2  p3  p4
                      t1  -1  +1   0   0
                      t2  -1  +1  +1   0
                      t3  -1   0  +1   0
                      t4   0  -1  +1   0
                      t5   0   0  -2  +1
                      t6  +1   0   0  -1
   

5. Model the system described below:
   A simple public hospital system may be described as follows: Patients arrive to the hospital for medical checks. They will first be interviewed by a nurse and thereafter a medical check will be made by the doctor. Based on the result of the check 50% of the patients will undergo a small operation performed by the doctor assisted by a nurse and then leave the hospital. Those which do not need an operation leave after the check.
   There are two nurses and one doktor. The patient arrivals are negative exponential with mean time 45 minutes. The interview takes 10 minutes, the check from 10 to 15 minutes, and an operation 20-40 minutes to perform (both uniformly distributed).

Write structuredly, clearly, legibly, and briefly! Use proper naming of places and transitions.