An operational calculus for matrix-exponential disributions, with applicaions to a Brownian (q,Q) inventory model
distribution G on [math not displayed] is called matrix-exponential if the density has the form αeTz s where α is a row vector, T a square matrix and s a column vector. Equivalently, the Laplace transform is rational. For such distributions, we develop an operator calculus, where the key step is manipulation of analytic functions f(z) extended to matrix arguments. The technique is illustrated via