Consider a system with a two components. We observe the state of the system every hour. A given component operating at time $n$ has probability $p$ of failing before the next observation at time $n + 1$. A component that was in failed condition at time $n$ has a probability $r$ of being repaired by time $n+1$,independent of how long the component has been in a failed state. The component failures and repairs are mutually independent events. Let $X_n$ be the number of components in operation at time n. The process $(X_n, n=0,1,...)$ is a discrete time homogeneous Markov chain with state space $I= 0,1,2.$
a) I want to determine its transition matrix, draw a state diagram
and
b) Obtain the steady state probability vector, if it exists.
Solutions: I am not understanding what are the component's state here? first one is operating component state and second one is failing component state. What is the third one? and what is meant by "mutually independent events"? Does it have same meaning as "mutually exclusive events" have?
Would any one answer these questions a) and b)?
a)The transition probability matrix is :
$\begin{pmatrix}(1-r)^2 & 2r(1-r) & r^2 \\ p(1-r) & pr +(1-p)(1-r) & r(1-p) \\ p^2 & 2p(1-p) & (1-p)^2\end{pmatrix}$
b) The steady-state probabilities are $\pi_0=[p/(p+r)^2]$
$\pi_1=[2[r/(p+r)][p/(p+r)]$
$\pi_2=[r/(p+r)^2]$