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Given a machine that remains operating for a period of time is exponentially distributed with a mean of $10$ days. When the machine fails, the failure is one of two types: If the failure is type-I, then the time to repair the machine is exponential with a mean of $1$ day. If the failure is type-II, then the time to repair is exponential with a mean of $4$ days. A failure is type-I with probability $0.8$ and type-II with probability $0.2$, independent of all else. A repaired machine is as good as new.

(a) Find the rate transition matrix $Q$.

(b) Give the transition matrix $P$ for the embedded Markov chain.

(c) What is the fraction of time the machine is operating?

(d) Suppose that the machine generates $100$ per day in revenue when working. Suppose that a type-I failure costs $150$ to repair. A type-II failure costs $1000$ to repair. What is the net rate of money (revenue minus cost) generated by the machine?

My attempted solution We define $3$ states for our CTMC: (A) $= 0$ machine fails, (B) $= 1$ machine fails with type-I, (C) $= 1$ machine fails with type-II. Now the rate transition matrix is $$Q =\pmatrix{-0.1 & 0.08 & 0.02\\ 1 & -1 & 0 \\ 0.25 & 0 & -0.25}$$

(b) From part (a), we get: $$P =\pmatrix{0 & 0.8 & 0.2\\ 1 & 0 & 0\\ 1 & 0 & 0}$$

(c) We need to find $\pi_{0}$ for solving the equation: $\pi P=\pi$ with $\sum_{i=0}^{2} \pi_{i} = 1$. Solving this system of equations, we get: $\pi_{0} =\frac{1}{2}$, $\pi_{1} = 0.4$ and $\pi_2 = 0.1$.

(d) We have: Revenue $= 100\times \pi_{0} = 50$ (this is the part I'm skeptical, since the problem said 100 PER DAY).

Cost $= 150\times \pi_{1} + 1000\times \pi_{2} = 160$.

Thus net rate of money $= 50 - 160 = -110$ (the result seems not to make sense, as this would be a money-losing machine)

My question: Could someone please help me review the solution to part (a) + (d) above?

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  • $\begingroup$ How many machines are there? $\endgroup$ – Math1000 Oct 2 '16 at 8:22
  • $\begingroup$ Exactly $1$ machine. $\endgroup$ – user177196 Oct 2 '16 at 8:22
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    $\begingroup$ Then why does your solution suggest there are two machines? $\endgroup$ – Math1000 Oct 2 '16 at 8:23

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