# Stochastic process of a sequence of machine operations in a given sequence

Say an automobile part needs three operations performed by three machines in a given sequence. The part is fed to the first machine, where the operation takes an $$Exp(\mu_1)$$ amount of time. Then it moves to machine 2, where the operation takes an $$Exp(\mu_2)$$ amount of time. Then it moves to machine 3, where the operation takes an $$Exp(\mu_3)$$ amount of time.

Note that if machine 2 is working, then the part from machine 1 can't be removed even if the operation at machine 1 complete. We say that machine 1 is blocked in this case.

There is an ample supply of unprocessed parts available so that machine 1 is always working or blocked.

I want to model this system as a CTMC. I've already defined the state $$X(t)=(X_1(t),X_2(t),X_3(t))$$ where $$X_1(t),X_2(t),X_3(t)$$ represent the status of each machine. Let $$X_i(t)=0$$ denote "working", $$X_i(t)=1$$ "blocked" and $$X_i(t)=2$$ "idle"

So there can be 8 states in total: $$X(t)=\{(0,0,0),(0,0,2),(0,1,0),(0,2,0),(0,2,2),(1,0,0),(1,0,2),(1,1,0)\}$$

How can I find the rate of transferring from one state to another?

$$\mathbb{P}[E(\mu_i)>E(\mu_j)] = \frac{\mu_j}{\mu_i + \mu_j}$$
$$\mathbb{P}[\min(E(\mu_i),E(\mu_j))>x] = e^{-(\mu_i + \mu_j)x}$$