Thank you a million! That really clarifies the intuition. Just one more question: is there any specific rule for applying Laplace transform to summation like this, or is there any decent textbook about this topic where I can find similar problem? (You are welcomed to make it as a complete answer too...)

@MateuszKwaśnicki: I think you may be right. I am just still wondering: with probability $\frac{\mu}{\mu + \theta}$, will $T$ remain to follow $exp(\mu+\theta)$, or would that be the case we KNOW that it would go to state $n+1$ and $T$ should be following $exp(\mu)$ now?