1. Reconsider the continuous-time Markov chain example discussed in class. Assume now that the machines are not serially connected, and if one of them fails, the other continues to function, and therefore may also fail while the ftrst machine is being repaired.
(a) Develop the rate diagram for this Markov chain.
(b) Write down time-dependent ordinary differential equations for this Markov chain.
(c) Construct the steady-state equations.
(d) Solve the steady-state equations to determine the the steady-state probabilities.
2. A certain shop has three identical machines that are operated continuously except when they are broken down. Because they break down fairly frequently, the top-priority assignment for a full-time maintenance person is to repair them whenever needed.
The mean failure rate of each machine is 1 (once per day), and the repair person is able to repair the broken down machines at a rate of 2 per day, on average.
(a) Develop the rate diagram for this Markov chain.
(b) Write down time-dependent ordinary differential equations for this Markov chain.
(c) Construct the steady-state equations.
(d) Determine the the steady-state probabilities.
3. The state of a particular continuous time Markov chain is deftned as the number of jobs currently at a certain work center, where a maximum of two jobs are allowed. Jobs arrive individually. Whenever fewer than two jobs are present, the next arrival occurs at a mean rate of one in two days. Jobs are processed at the work center one at a time, at a mean rate of one per day, and then leave immediately.
(a) Develop the rate diagram for this Markov chain.
(b) Write down time-dependent ordinary differential equations for this Markov chain.
(c) Construct the steady-state equations.
(d) Determine the the steady-state probabilities.