Convolving Hyper-Erlang with Hyper-Exponential Distributions Using Linear Algebra

In this paper, the sum of a hyper-Erlang and a hyper-exponential distributed random variables is analyzed. Although tedious, the resulting random variable's probability density function (PDF) can be obtained through convolution of its summands' PDFs. Alternatively, the resulting distribution can be stated directly in terms of a phase-type distribution. However, computing its PDF can still be very costly and this representation gives little insight on the distribution. It can be shown that the sum of both random variable is again hyper-Erlang distributed of incremented order and can therefore be described without requiring the matrix exponential function. We derive a closed form linear algebra expression for the probability weights of the sum's hyper-Erlang distribution, which significantly reduces the computational complexity of evaluating its distribution.