Simulation Modeling: Quiz 1, 3.11.1999

1. a) Assume that you are to estimate the volume of a body A inside a box B: [0,10] x [5,15] x [20,30]. You have sampled 1000 points uniformly in B and found that 250 of these was also inside A. What is the volume of A.
   b) Like in a) above, you have the bodies A and B and you should find an estimate of the the integral of the function f(x) over the region A. Let the 250 points falling in A be x1, x2 ,..., x250. Give an expression that estimates the integral. (8 p)

2. Describe briefly a) the seven steps in performing a simulation study as described in Törn: Simulation Modelling. Describe b) the step "Designing the simulation model" in five sub-steps. (6 p)

3. Given the Petri Net structure C = (P,T,I,O,mu)

           P = {C, Cday, Del, Low, Nord, S}
           T = {Cre, Noto, Ord, Skip, Wait}
           mu = (1 0 0 0 1 20)
   
          I(Cre) = {C}                   O(Cre) = {C, Cday}
          I(Noto)= {Cday,S}              O(Noto)= {S}
          I(Ord) = {Cday, S(inhibit)}    O(Ord) = {Low}
          I(Skip)= {Low, Nord(inhibit)}  O(Skip)= {}
          I(Wait)= {Low, Nord}           O(Wait)= {Del}
   

   a) give the corresponding matrices D^- and D⁺ and
   b) draw the corresponding Petri Net Graph (let the places S, Nord and Del be located under the rest of the graph and be ordered from left to right). (4 p)

4. An assembly operation requires 1 unit of A and 2 units of B before a worker can assemble them. Storage exists for 5 units of A and 10 units of B. Part A arrives every three minutes and B arrives every 1.5 minutes. The assembly operation takes 2.6 minutes on average (normal distribution with variance of 1 minute). Arriving units continue on if no storage space is available (i.e., balking). Assume 3 units of A and 4 units of B are on hand when the operation starts. Simulate 100 completed assemblies. Pay special attention to a correct modeling of how the units, after a completed assembly, (independently) are shifted from the storages to the assembly point for the next assembly. We assume that the time needed for the shift from a storage to the assembly point can be neglected.

   a) Construct the Simulation Net for the system above. Make the model hierarchical.
   b) Execute the net by starting from a set of initial states so that the correct working of the model can be deduced. (12 p)

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