A salesman flies between Atlanta, Boston, and Chicago as the following rates:
\begin{bmatrix} -4 & 2 & 2 \\ 1 & -4 & 3 \\ 5 & 0 & -5 \end{bmatrix}
(a) If the salesman takes a trip out of Atlanta, what is the probability of it being a trip to Chicago? (b) What is the expected time spent in Atlanta?
I'm not sure how to approach this problem. For (a), how do I obtain the probabilities from the transition rate matrix provided? Or is there another way to calculate this probability? Any hints/advice would be appreciated. Thanks in advance!
Assuming Atlanta means the top row. You see the jump rates to the other two cities are the same. So the probability to transit to either city is $1/2$.
In general just imagine that there are independent exponential times $T_{ij}$ for different possible transitions from state $i$ to state $j$, which have the rates specified in the generator. We wait for the first appearance of those times (i.e. the minimum), and that decide which is the next state to jump. So the transition probability from state $i$ to state $j$ can be thought as the probability that $T_{ij}$ is smallest among the other.
See the formula here: https://en.wikipedia.org/wiki/Markov_chain#Embedded_Markov_chain