A \(BMAP| G| 1\)-analysis based on convolution calculus (Q1288931)

It is well-known that the batch Markovian arrival processes (BMAPs) play a key role in modelling of the modern communication systems where mixtures of traffic and/or service processes have to be considered. In a previous paper (1996), the author succeeds to express the closed-form solution to an infinite system of differential-difference Chapman-Kolmogorov equations in terms of an infinite series of convolutions. This paper presents a new approach to the \(\text{BMAP}| G|- 1\) queue analysis, based on the convolution calculus for sequences of square matrices. Reformulating BMAPs as generalized compound Poisson processes (GCPPs), the main results of the paper are: (1) the unified representation of Poisson, compound Poisson, and BMAPs by the so-called convolutional exponential distributions, and (2) the computation of the fundamental-period matrix \(G\) via the so-called semi-convolutions. The resulting algorithms are similar to the existing ones [cf. \textit{D. M. Lucantoni}, Commun. Stat. Stochastic Models 7, No. 1, 1-46 (1991; Zbl 0733.60115)] and they have, in essence, the same computational complexity. However, the proposed algorithms allow the direct computation of the step matrices such as the arrival block matrices, avoiding the numerical solutions of the Chapman-Kolmogorov equations. Another potential advantage is 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 could have formally similar properties, as known from the scalar case.