B MB ExamofDiscreteEventSystems-24.02.2016

Exam of Discrete Event Systems - 24.02.2016

Exercise 1

A machine can be in one of three states, namely idle, working and down. From state idle, the machine is changed to state working upon the arrival of a START signal. In state working, the machine processes one part at a time. After processing of one part, either processing of another part starts, or the machine is changed to state idle. While working, the machine may occasionally break down. After the repair, the machine is put in state idle. The initial state is idle.

1. Assume that state holding times in state idle are equal to 10 minutes, and processing times of the first ten parts are equal to 7.0, 4.5, 3.8, 8.2, 3.4, 7.8, 4.6, 5.4, 4.2 e 4.9 minutes, respectively. Moreover, assume that, when the machine is working, it is changed to state idle after processing of any part taking more than 5.0 minutes. Finally, the machine breaks down after a total time of 40 minutes spent working. Determine the throughput of the machine (#parts/minute) in the time interval before the first breakdown.

Now assume that state holding times in state idle follow a uniform distribution over the interval [8, 12] minutes. Processing times are deterministic and equal to 6 minutes. After processing of one part, processing of another part starts immediately with probability p = 2/3, otherwise the machine state is changed to idle. The machine breaks down after 15 minutes of continuous operation.

Repair of the machine takes a random time following a uniform distribution over the interval [10, 20] minutes.

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

, F ).

3. Compute the probability that, after the third event, the machine is idle.

4. Assume that the machine receives the START signal. Compute the probability that the machine returns to state idle within T = 30 minutes.

5. Compute the probability that the machine is idle at time t = 15 minutes.

Exercise 2

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

and B

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

M B

B

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

, whereas type 2 parts are stored in buffer B

. 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 follow 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

, F ).

2. Assume that M is working and both buffers are full. Compute the probability that the manufacturing cell is emptied before a new arriving part is accepted.

Only first part

3. Compute the average state holding time when B

is full, B

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 3

Consider the system of Exercise 2.

1. Verify the condition λ

= µ

for the system at steady state.

2. Compute the probability that an arriving part is rejected at steady state.

3. Compute the average time spent by a type 1 part in B

.

Exercise 4

A cart moves over a railroad network. One-way tracks connect nodes A, B, C and D as shown in the figure. At nodes B and C, the cart is routed to the central node D with probability p = 2/5.

The cart is initially at node A.

A

B

C D

1. Model the movement of the cart over the network through a discrete-time homogeneous Markov chain.

2. Compute the probability that the cart visits five times nodes A, B and C without visiting node D.

3. Compute the average recurrence time of node A.