Closed-form expressions for distribution of sum of exponential random variables

In many systems which are composed of components with exponentially distributed lifetimes, the system failure time can be expressed as a sum of exponentially distributed random variables. A previous paper mentions that there seems to be no convenient closed-form expression for all cases of this problem. This is because in one case the expression involves high-order derivatives of products of multiple functions. The authors prove a simple intuitive multi-function generalization of the Leibnitz rule for high-order derivatives of products of two functions and use this to simplify this expression, thus giving a closed form solution for this problem. They similarly simplify the state-occupancy probabilities in general Markov models.