This talk introduces functional calculus techniques as a new method in the tool box of phase-type or matrix-exponential distributions. Matrix-exponential distributions are the distributions with support on the positive reals which have a rational Laplace transform, and the phase-type distributions is a sub-class based on a more probabilistic construction.

In these classes of distributions one is often confronted with the problem of finding representations or computing functionals of the corresponding random variables. Functional calculus techniques may substantially simplify many expressions in terms of functions of matrices and has the potential enable smoother derivations of new results as well.

We will present two examples, where the functional calculus method plays an important role: the derivation of the Mellin transform for matrix-exponentially distributed random variables and the calculation of the time-average variance constant for certain queues.