Can the interarrival times of a continuous time markov chain be distributed with 2 parameter (scale,location) exponential distributions?

I'm trying to model data with a time-homogenous CTMC with a number of states with corresponding constant transition rates $$\lambda_{i}$$ when I notice that much of the transition times from one state to another (lets call it $$a \to b$$) don't occur past a certain time, lets call it $$t_0$$.

I'm aware that much of the structure of a Continuous time Markov Chain is dependant on the memorylessness of the exponential distribution but I'm wondering if introducing location parameters into the distribution of first hitting times would interfere with that structure. How would the Kolmogorov equations ($$\frac{dX(t)}{dt}=G X(t)$$)of the process change? Can this be accounted for by making $$\lambda_{a \to b} = \lambda(t)_{a\to b}$$ ie. moving to a time inhomogeneous setting?

Whats the best way to address this without wholesale abandoning the CTMC framework?

• I think your statement destroys the memoryless property of markov chains so trying to incorporate that behavior will then mean that the use of a CTMC is inappropriate since CTMC depends on the memory-less property. But this is not my field at all so others should confirm or correct. – mlofton Jul 31 at 17:01