This is linked to my previous question Discounted optimization problem
I have difficulties again finding a formula for $F(x)$. We consider a machine at time $t$ which is in state $x$. The machine costs $cx$ to run until time $t+1$. With probability $a=1-b$ the machine is serviced and so goes to state $0$ at time $t+1$. If it is not serviced then the machine will be in states $x$ or $x+1$ at time $t+1$ with respective probabilities $p$ and $q=1-p$. Costs are again discounted by a factor $\beta$ per unit time.
$F(x)$ denotes the expected discounted cost over an infinite future for a machine starting from state $x$. Now I do not see why $F(x)$ should have a linear form $\phi+\theta x$, may you can help me with that.