I am looking for a reference to an explicit expression to the $M/H_2/1$ queue's response time distribution. I.e., when you invert the PK-Formula, I am looking for a reference that gives a "nice" expression for the response time distribution.
Thanks for help in these matters.
Assuming a first come first served service discipline, using parameters $\mu_1$ and $\mu_2$ for the two phases of the hyperexponential distribution and $p$ for the routing probability between these two phases and $\lambda$ for the arrival rate to the queue, I compute $F(t)$, the response time distribution, to be
$$ \frac{(\mu_1 \mu_2+\lambda \mu_1 (p-1)-\lambda \mu_2 p) e^{\frac{1}{2} t (\lambda -\mu_1-\mu_2)}}{\mu_1 \mu_2 \sqrt{R}} \left(\sinh \left(\frac{\sqrt{R} t}{2}\right) (\mu_2 (\lambda +\mu_1-\mu_2)+p (\mu_1-\mu_2) (\lambda -\mu_1-\mu_2))+\sqrt{R} (\mu_2+p (\mu_1-\mu_2)) \cosh \left(\frac{\sqrt{R} t}{2}\right)\right)$$
where $R=\lambda ^2+(\mu_1-\mu_2)^2-2 \lambda (2 p-1) (\mu_1-\mu_2)$ as the inverse and can't see a way to significantly simplify the expression.
See a screenshot of the Mathematica calculation below. (Please excuse typos in the first couple of lines which should obviously read 'transform of service time distribution' and 'mean service time', and not mention waiting times.