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Test of Discrete Event Systems - 09.11.2020

Exercise 1

A manufacturing cell is composed of two one-place buffers B

1

and B

2

and one assembling machi- ne M , as shown in the figure.

M B

1

B

2

Arrivals of raw parts are generated by a Poisson process with rate 10 arrivals/hour. Arriving parts are of type 1 with probability p = 1/2 and of type 2 otherwise. Type 1 parts are stored in buffer B

1

, whereas type 2 parts are stored in buffer B

2

. An arriving part is rejected if the corresponding buffer is full. Machine M assembles one type 1 part and one type 2 part to make a finished product.

Assembling starts instantaneously as soon as parts of both types are available in the buffers and M is ready. Assembling times have an exponential distribution with expected value 5 minutes. The manufacturing cell is initially empty.

1. Model the manufacturing cell through a stochastic timed automaton (E, X , Γ, p, x

0

, F ).

2. Assume that M is working and both buffers are full. Compute the probability that the manufacturing cell is empty

(a) when a new part arrives;

(b) when a new part is accepted.

3. Compute the average state holding time when B

1

is full, B

2

is empty and M is working.

4. Assume that M is working and both buffers are full. Compute the probability that two products are finished within T = 10 minutes, and no arrival of type 1 parts occurs.

Exercise 2

Consider the queueing system in the figure, characterized by deterministic interarrival times equal to T = 20 min, and service times having an exponential distribution with expected value 15 min.

1. Assume that the queueing system is initially empty. Compute the probability that the third customer has not to wait for service.

2. Assume that one customer is initially in the queueing system. Compute the cdf of the waiting time of the next customer.

3. Assume that the initial number of customers in the system has a uniform distribution over the

set {0, 1, 2, 3}. Compute the probability that at least two places are available in the system

when the next arrival occurs.

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