Test of Discrete Event Systems - 16.11.2016
Exercise 1
Consider the manufacturing cell in the figure, composed of two machines M1 and M2 connected in series through a one-place buffer B.
M1 B M2
Raw parts arrive as generated by a Poisson process with average interarrival time equal to 5 min, and require sequential processing, first in M1 and then in M2. If M1 is busy, the arriving part is rerouted to another manufacturing cell. If B is full when processing terminates in M1, the part is kept in M1 (and hence M1 is blocked) until processing in M2 terminates. Processing times in M1 and M2 follow exponential distributions with rates 0.25 and 0.4 services/min. The manufacturing cell is initially empty.
1. Model the system through a stochastic timed automaton (E, X , Γ, f, x0, F).
2. Assume that both machines are working and the buffer is empty. Compute the probability that the manufacturing cell is emptied with the minimum number of events.
Exercise 2
A warehouse contains up to 150 refrigerators. Two trucks take the refrigerators from the warehouse to the points of sale. Truck 1 may load 50 refrigerators, while truck 2 may load 100 refrigerators.
Truck 1 returns to the warehouse according to a Poisson process with average interevent time equal to 36 hours. Truck 2 returns according to a Poisson process with average interevent time equal to 60 hours. The trucks always return empty, and load as much refrigerators as possible from the warehouse. The full capacity of the warehouse is restored according to a Poisson process with average interevent time equal to 48 hours. The warehouse is initially full.
1. Model the dynamics of the number of refrigerators in the warehouse through a stochastic timed automaton (E, X , Γ, f, x0, F).
2. Assume that the warehouse contains 100 regrigerators. Compute the probability that the warehouse is empty when the full capacity is restored.
3. Compute the average time that the warehouse remains full.
4. Compute the probability that exactly two trucks arrive in one day.
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