DTMC problem： Two machine

Question

In a certain manufacturing system, there are 2 machines $M_1$ and $M_2$: $M_1$ is a fast and high precision machine whereas $M_2$ is a slow and low precision machine. $M_2$ is employed only when $M_1$ is down, and it is assumed that $M_2$ does not fail. Assume that the processing time of parts on $M_1$; the processing time of parts on $M_2$; the time to failure of $M_1$; and the repair time of $M_1$ are independent geometric random variables with parameters $p_1，p_2，f$ and $r$; respectively. Identify a suitable state space for the DTMC model of the above system and compute the TPM. Investigate the steady-state behavior of the DTMC.

The question is: in my point of view, it seems like a CTMC problem so I don't know how to include the $p_1，p_2，f$ and $r$ in the DTMC TPM.

Thanks for your help in advance.

Answer 1

Since you haven't said where you are stuck, I will just work the beginning of the problem.

A geometric random variable is discrete, and each attempt is either a success or a failure. There is always one machine working. In each time interval, either machine one is broken or working, and the item being worked on is either finished or not. At the end of each interval, we need to know whether the item got finished or not, and whether $M_1$ is broken or not. Precision doesn't seem to come into it, except to say that $M_1$ is always used if available. Each attempt takes one time unit, so as soon as $M_1$ is working again, it makes the nexts attempt, with no time wasted. If $M_1$ is working, it has probability $f$ of switching to "failed" each time. If it is broken, it has probability $r$ of being repaired each time. Meanwhile, if $M_1$ is working, the item has a probability $p_1$ of being completed, but if $M_2$ is working instead, it has probability $p_2$ of being completed. Suppose

$A = M_1$ working, item finished,

$B = M_1$ failed, item finished,

$C = M_1$ working, item not finished, or

$D = M_1$ failed, item not finished.

I am assuming that if M1 fails it does so after finishing an attempt.

Those would appear to be the possible states.