I don't know offhand of any non-homogeneous CTMC references, sorry.
Regarding your case, this part of the help section regarding ths inputs of simCTMC.m is relevant:
So if you don't pass in a vector of initial states in the variable instt, the program randomly draws nsim initial states from the stationary distribution of the MC. But when you have an absorbing state, the stationary distribution of the MC has probability 1 associated with the absorbing state, and 0 for every other state. If you draw the initial state randomly from this stationary distribution, you're always going to start with the absorbing state, and then of course you'll stay in that state forever, which obviously isn't very interesting.
Long story short, for cases like these you probably want to pick your own initial vector instt. How to pick it is up to you.