A method for obtaining Laplace transforms of order statistics of Erlang random variables

We present a path-counting method for deriving Laplace transforms of order statistics of independent but not necessarily identically distributed Erlang random variables, based on a probabilistic interpretation of the Laplace transform. The method is applicable also to generalized Erlang variates having different parameters for the exponential stages. The idea is to provide an intuitive understanding of the Laplace transform, based on the Markovian properties of the stages of the Erlang random variable. Thus the derivation technique is applicable to many other Markovian stochastic models. Motivational examples for queues and reliability are mentioned. Computational considerations are discussed.