Convolution Algorithms for BMAP/G/1-Queues

The equilibrium state probabilities for queues with batch Markovian arrival processes are determined in form of matrix expressions, in which the central item to be computed is the so called fundamental-period-matrix G . G appears as the solution of the non-linear matrix equation G = AνG ν Σ , or as the infinite sum over matrices Gν , which in turn are functions of the matrices Aν as has been shown in [1]. In this note we present new convolution algorithms for the computation of the matrices Aν , and the sum G = Gν Σ , which are stable and in all probability much more efficient than any previously known algorithm.