(Solved): consider this poisson process b) A machine is running continuously except when it is broken. Suppose ...

b) A machine is running continuously except when it is broken. Suppose that while the machine is running, breakages happen according to a Poisson process of rate $$?$$. As soon as the machine breaks an engineer is called. The time it takes for the engineer to arrive follows an $$\mathrm{Exp}(?)$$ distribution. When the engineer has arrived they spend a time with $$\mathrm{Exp}(?)$$ distribution working on the machine. At the end of this time it is either mended and running (this happens with probability $$p$$ ) or permanently broken (this happens with probability $$1?p$$ ). (i) Describe how to model the state of the machine as a continuous-time Markov chain with four states. Write down the generator matrix for the chain. (ii) Suppose that the machine is running at time 0 . What is the probability that it runs continuously without breaking until at least time 1 ? (iii) Why is it natural to model this situation as a continuous-time process.