Test of Discrete Event Systems - 03.11.2015
Exercise 1
Consider the stochastic timed automaton (E, X , Γ, p, x0, F ) depicted in the figure, where p(1|0, b) = 1/3 and p(1|1, b) = 1/2. The lifetimes of event a are deterministic and equal to ta = 2 minutes, whereas those of event b follow an exponential distribution with expected value 90 seconds.
0 1
2 3
a
a a
a b
b b
b
b
1. Compute P (X2 = x) for all x = 0, 1, 2, 3, where X2 denotes the state after the second event.
Exercise 2
A manufacturing cell is composed by two parallel machines M1 and M2, preceded by a one-place buffer, as illustrated in the figure.
M1
M2
Machine M1 is obsolete and subject to frequent breakdowns, with probability of breakdown during a job equal to p = 1/10. In case of breakdown of M1, processing of the part in M1 continues as soon as M1 is repaired. Machine M2 is new, and can be assumed free from breakdowns. Raw parts arrive as generated by a Poisson process with rate λ = 0.8 arrivals/hour, and are rejected if the manufacturing cell is full. When both machines are idle, the next arriving part is routed to M2. Processing times in M1 and M2 follow exponential distributions with expected values 60 and 45 minutes, respectively. It is assumed that times to breakdown are exponentially distributed for M1. Times to repair are also exponentially distributed for M1, with expected value 3 hours.
1. Model the manufacturing cell through a stochastic timed automaton.
2. Assume that M1 is down, M2 is working, and the buffer is empty. Compute the probability that processing of the two parts in the system is completed before a new part arrives and M1
breaks down again.
3. Assume that M1 and M2 are working, and the buffer is full. Compute the probability that processing of the three parts in the system is completed before a new raw part is accepted and M1 breaks down.
4. Compute the average holding time in a state where M1is down, M2 is working, and the buffer is full.
