Question: 4. Consider one machine maintained by a single machine repair robot. The machine has an exponentially distributed lifetime with mean 4 weeks (i.e. before it

4. Consider one machine maintained by a single machine repair robot. The machine has an exponentially distributed lifetime with mean 4 weeks (i.e. before it fails). The machine repair robot itself has an exponentially distributed lifetime with mean 104 weeks. If the machine repair robot is functioning when the machine fails, it begins its repairs immediately and completes its repair in an exponentially distributed amount of time with mean 1 week. If the machine repair robot fails, it ceases work and is itself repaired in an exponentially distributed amount of time with mean 26 weeks. Initially, both the machine and the machine repair robot are working. (a) Formulate a continuous-time Markov chain model for the problem, specifying the state space E, the initial distribution (0), and the Qmatrix Q. [2] (b) Draw the corresponding transition rate diagram. [1] (c) What is the long run probability that both the machine and the machine repair robot are under repair? [1] (d) Suppose the machine has failed and is under repair. What is the probability that the machine repair robot xes the machine before it itself fails? [I]

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