What order to define ( time-homogeneous ) Continuous time Markov chains Hi when defining and explaining continuous time Markov Chains is there a preference in explaining what the condition probability time step matrix P(t) is first or is it more natural and intuitive to start by defining the rates?
Edit:
( In other words does Q completely determine P(t)? and does P(t) completely determine Q ? where Q is the rate matrix)
 A: $P$ and $Q$ both uniquely determine the distributional properties of the chain. $P$ is uniquely determined from $Q$ by $P(t)=e^{\int_0^t Q(s) ds}$. $Q(t)$ is "basically" determined by $Q(t)=P'(t)P(t)^{-1}$ except you need to be a little bit careful because actually $P$ is not necessarily invertible. Still you can make sense of it with some care.
In practice we usually begin with $Q$ simply because it is easier for us to come up with a $Q$ in modeling or similar situations.
Note that a CTMC also has an associated jump chain, which has a transition probability matrix that defines a discrete time Markov chain (DTMC). This transition probability matrix (also confusingly called $P$ some of the time) does not uniquely determine the distributional properties of the chain because it fails to specify the distribution of "holding times" at each state.
Upon request, I elaborate a little bit on these two different $P$'s. The "Markov semigroup operator" $P(t)$ gives you $\mathrm{Pr}(X(t)=j \mid X(0)=i)$ from its $(i,j)$ entry. $P(t)$ fully specifies the chain by itself as I said before. I am unfortunately not sure what the standard elementary term is anymore. I'm too accustomed to it being called a Markov operator or a Markov semigroup operator.
The transition probability matrix $P$ for the jump chain, also called the embedded discrete time chain, tells you the probability that if you are currently at $i$, your next jump (whenever it will be) will be to $j$, from its $(i,j)$ entry. Even in the time-homogeneous case, $P$ does not fully specify the chain because it does not specify the holding time distributions. 
But splitting a CTMC into its jump chain and holding times can still be useful, especially in the time homogeneous case. For example, in Monte Carlo simulation, it is more efficient and more accurate to sample your next destination and then sample the holding time separately, rather than to repeatedly sample where your next position is going to be. This is faster because your next position will very frequently be your starting point. It is more accurate in the sense that it more accurately reproduces the holding time distribution: sampling in time instead of sampling jumps discretizes the holding time distribution, since each holding time can only be an integer multiple of the time step.
