Time until absorption markov chain with weights

Question

I have a markov chain with one absorbing state. In each state i, t_i seconds must elapse before moving to the next state. Starting at t₀ what is the expected time to reach t₃?

t₀ = 5, t₁ = 10, t₂ = 15, t₃ = 20

My transition matrix looks like this:

\begin{bmatrix}3/4&1/4&0&0\\4/5&0&1/5&0\\6/7&0&0&1/7\\0&0&0&1\end{bmatrix}

I know that I need to solve this as a system of linear equations namely each equation dictated by:

$$h_i = \sum_{k=0}^Np_{ik}h_k+1$$

But I'm not sure how the "+ 1" or the time/weight affects it. Any help is appreciated.

Answer 1

The +1 in the equation for state $i$ would be replaced by $t_i$, in which case this system of equations is exactly what you would get using first transition analysis. Intuitively, you can think of it as "expected hitting time from state i is equal to time spent in the state i + expected hitting time from the state you visit from state i (which is given by the summation term)"