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Test of Discrete Event Systems - 04.11.2015

Exercise 1

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, x

0

, 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.

Exercise 2

The call-center of an insurance company is equipped with 10 phone lines, but only three operators.

This means that calling customers exceeding the number of three, are put on hold. Phone calls arrive at the call-center as generated by a Poisson process with rate equal to 2 calls/minute. Time needed to answer a query is independent of the operator, and follows an exponential distribution with expected value 5 minutes. The call-center is open 24/7.

1. Model the call-center through a stochastic timed automaton (E, X , Γ, f, x

0

, F ), assuming that the three operators are initially idle.

2. Assume that the three operators are idle. Compute the probability that the fourth calling customer is put on hold and, in such a case, the average waiting time to be served.

Exercise 3

Consider the stochastic timed automaton in the figure, where a and b are events whose lifetimes follow generic probability distributions, and the initial state is x

0

= 0.

1. Given P (X

2

= 1) = 5/8 and P (E

2

= a) = 1/3, compute the probability p(1|0, a).

b b

a

0 1

2

a

a a

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