I would like to calculate the mixing time of a continuous time starting from the **rate matrix** and **not** necessarily assuming that the time in between jumps have rate 1 - all I have is the (finite dimensional) transition rate matrix $Q$ for this system. I am fine with other basic assumptions about the chain (e.g., irreducible). The only references I can find (e.g., [here](http://darkwing.uoregon.edu/~dlevin/MARKOV/markovmixing.pdf)) do not seem to cover this case. Is there a theorem or transformation that covers this case? 


**UPDATE**
It seems that by [uniformization](http://en.wikipedia.org/wiki/Uniformization_(probability_theory)), I can turn my continuous time Markov chain with different jump rates $\gamma\_i$ for each element in my state space $\Omega$ (defined by the generator matrix $Q$) into a continuous time Markov chain with transition matrix $P$ (which is now not a generator matrix) and jump rate $\gamma \geq \sup\_i | \gamma\_i |$. So, my question above is really: 


**How can the methods for finding mixing times with transition times distributed as an exponential random variable of mean 1 be generalized to the case of where the transition time are distributed according to an exponential random variable with arbitrary (positive, real valued) mean $\gamma$**?


Really, I think this just means transforming a CTMC with rate $\gamma$ into a CTMC with rate 1 that has the same dynamics, but I don't really know how to do this. 


**UPDATE**
I figured it out: uniformization let's you choose any value for the rate, larger than the slowest rate, which works out in my case, since my rates are all less than 1. 



Mixing time of a continuous time Markov chain with arbitrary rate matrix
Given a (not necessarily reversible but well-behaved) Markov generator $Q$ with invariant distribution $p$, the corresponding Dirichlet form is 


$\mathcal{E}(f) := \frac{1}{2} \sum\_{j,k} p\_j Q\_{jk} (f\_j-f\_k)^2.$


Write 


$\lambda\_\* := \inf\_{f} 2\mathcal{E}(f)/\mbox{Var}\_{p}(f),$


where the infimum is over $f$ s.t. $\text{Var}\_{p}(f) \ne 0$. Now $\lambda\_\*^{-1}$ is the $L^2$ mixing time.


Another reference than the one you cite is Montenegro and Tetali, "Mathematical Aspects of Mixing Times in Markov Chains", available as of this writing [here (PDF)](http://people.math.gatech.edu/~tetali/PUBLIS/survey.pdf).


