,ba 1243 aabbba Exercisesonstochastictimedautomata

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Exercises on stochastic timed automata

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 have 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 have 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.

Exercise 3

Consider a queueing system formed by a server and a queue with only one place. Customers arrive according to a Poisson process with rate λ = 1 arrivals/min, while services have constant duration equal to t

s

= 3 min. The queueing system is initially empty.

1. Compute the probabiliy that the queueing system is empty at time t = 4 min.

Exercise 4

Consider the queueing system in the figure, characterized by deterministic interarrival times equal

to T = 20 min, and service times having an exponential distribution with expected value 15 min.

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1. Assume that the queueing system is initially empty. Compute the probability that the third customer has not to wait for service.

2. Assume that one customer is initially in the queueing system. Compute the cdf of the waiting time of the next customer.

3. Assume that the initial number of customers in the system has a uniform distribution over the

set {0, 1, 2, 3}. Compute the probability that at least two places are available in the system

when the next arrival occurs.

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