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Test of Discrete Event Systems - 31.10.2019

Exercise 1

Consider the stochastic timed automaton in the figure. The initial state is drawn from a discrete uniform distribution over the set {1, 3, 4}. Lifetimes of event a are uniformly distributed over the interval [6, 10] min, while lifetimes of event b follow an exponential distribution with expected value 5.5 min. The value of probability q is 0.4.

1 2

4 3

a a

b

b b

a

[1 − q]

a

[q]

, b

1. Compute P (X

2

= 3).

2. Compute the cumulative distribution function of the state holding time of state 1.

Exercise 2

Consider a queueing system with one server and total storage capacity equal to three. Interarrival times follow a uniform distribution over the interval [5, 20] minutes, while service times are all equal to T = 12 minutes. Assume that the queueing system is initially empty.

1. Compute the probability that only one customer is in the system when the third arrival occurs.

2. Compute the cumulative distribution function of the waiting time of the second customer.

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