Correlation bounds for second-order MAPs with application to queueing network decomposition

Tools for performance evaluation often require techniques to match moments to continuous distributions or moments and correlation data to correlated processes. With respect to efficiency in applications, one is interested in low-dimensional (matrix) representations. For phase-type distributions (or matrix exponentials) of second order, analytic bounds could be derived earlier, which specify the space of permissible moments. In this paper, we add a correlation parameter to the first three moments of the marginal distribution to construct a Markovian arrival process of second order (MAP(2)). Exploiting the equivalence of correlated matrix-exponential sequences and MAPs in two dimensions, we present an algorithm that decides whether the correlation parameter is permissible with respect to the three moments and – if so – delivers a valid MAP(2) which matches the four parameters.We also investigate the restrictions imposed on the correlation structure of MAP(2)s with hyperexponential marginals. Analytic bounds for the envelope correlation region (i.e., for arbitrary third moment) and for the specific correlation region (i.e., for fixed third moment) are given.When there is no need for a MAP(2) representation (as in linear-algebraic queueing theory), the proposed procedure serves to check the validity of the constructed correlated matrix-exponential sequence.Numerical examples indicate how these results can be used to efficiently decompose queueing networks.