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Midterm Exam - Discrete Event Systems - 16.11.2015

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Midterm Exam - Discrete Event Systems - 16.11.2015

Exercise 1

A machine can work in two operating modes, namely mode 1 and mode 2. In both modes, the machine works continuously (i.e. parts to be processed are always available). Reconfiguration of the machine is needed to switch from one mode to the other. Reconfiguration may start only at the end of a job. During reconfiguration, the machine is idle.

Assume that reconfigurations are decided by an external signal. If the signal arrives while the machine is working, first the ongoing job is terminated, and then reconfiguration is started.

1. Given that the machine is initially in mode 1; jobs in mode 1 take 14, 8, 20, and 16 minutes; jobs in mode 2 take 12, 6, 15, 9, and 25 minutes; the reconfiguration signal arrives at times 35, 78, and 102 minutes; and each reconfiguration takes 10 minutes, determine the average machine throughput (number of jobs per hour) in the time interval [0,150] minutes.

Now assume that reconfiguration of the machine is decided at the end of a job with probability q = 1/10. Moreover, jobs in mode 1 have durations following a uniform distribution over the interval [30, 50] minutes, whereas jobs in mode 2 have durations following a uniform distribution over the interval [20, 45] minutes. Reconfigurations take 10 minutes, and the machine is in mode 1 at initial time t = 0.

2. Model the state of the machine through a stochastic timed automaton (E, X , Γ, p, x

0

, F ).

3. Compute the probability that, after the fourth event, the machine is in mode 1.

4. Assume that the machine enters reconfiguration mode from mode 1. Compute the probability that the machine returns in mode 1 with the minimum number of events and within T = 60 minutes.

5. Compute the probability that the machine is still in mode 1 at time t = 45 minutes.

Exercise 2

A manufacturing cell is composed by a single machine M , preceded by a one-place buffer B.

Arrivals of raw parts are generated by a Poisson process with rate 5 arrivals/hour. Raw parts arriving when the cell is full, are rejected. In M , raw parts are first inspected. They turn out to be defective with probability q = 1/20. Defective parts are removed from M , whereas nondefective parts are processed in M . Inspections have random durations following an exponential distribution with expected value 1 minute. Processing of a part takes a random time following an exponential distribution with expected value 10 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 the machine is processing a part, and the buffer is full. Compute the probability that the manufacturing cell is emptied before a new raw part is accepted.

3. Compute the average holding time in a state where the machine is inspecting a part, and the buffer is full.

4. Assume that the machine is inspecting a part, and the buffer is empty. Compute the

probability that the manufacturing cell is emptied within T = 30 minutes, and no

arrivals of raw parts occur.

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