Q: When do I use the jump chain?

A: Whenever the result you are interested in deriving does not depend on time. 
For example, the probability of absorption does not depend on time.


Q: How do I think about the jump chain? 

A: The way I think about it is this: draw the path of the process. If the process is a 
discrete time process, your process switches states every half inch, say, on your graph
(note that some processes can stay in the same state for several consecutive periods).
Say that you have a friend on the telephone, and that you want your friend to draw the same
graph as you have. All you have to tell your friend is the sequence of states your process
visited, and tell him (her) to draw equal length steps (1/2 inch if that what you have used).

Now imagine that the process is a continuous time process. Then the length of the steps 
between jumps are no longer equal. So you have to tell on your phone friend not only the sequence of 
states the process visits, but also the length of stay in each state visited. You can think
of the sequence of visited states as the jump chain. The length of stay in each state are 
exponentially distributed, as proven in class.