A BMAP|G|1-analysis based on convolution calculus

Abstract

Stochastic models of high-speed communication systems typically require the analysis of special Markov additive processes, known thus far as “batch Markovian arrival processes” (BMAPs). In most cases, the solution of nonlinear matrix equations of the form\(G = \sum\nolimits_{n = 0}^\infty {A_n G^n } \) or\(R = \sum\nolimits_{n = 0}^\infty {R^n A_n } \) is necessary, leading, in general, to severe numerical problems. Several efficient algorithms for that purpose have been reported in the literature. In this contribution, we present a new approach (the BMAP|G| 1-analysis), which is based on a convolution calculus for matrix sequences. The main results are the unified representation of Poisson, compound Poisson, and batch Markovian arrival processes by so-called convolutional exponential distributions, and the computation of the fundamental-period matrix G via so-called semi-convolutions. The resulting algorithms, in general, cannot outperform known algorithms as, for instance, those exploiting Horner schemes as proposed by D. M. Lucantoni and other authors, or the highly efficient cyclic reduction algorithm of D. Bini and B. Meini; nevertheless, they allow the direct computation of step matrices such as the “arrival matrices” An with\(A = \sum\nolimits_{n = 0}^\infty {A_n } \) and the component matrices Gn summing to G. Further, there is some hope that the proposed framework may lead to a unification of the theory in the sense that queues with arrival and/or service processes that are characterized by convolutional exponential distributions may possess formally similar properties, as known from the scalar case.