Inverse Characterization of Hyperexponential Map(2)s

This paper presents closed-form expressions to exactly match three moments and a correlation parameter into a Markovian arrival process of second order (MAP(2)), whose marginal distribution is a mixture of two exponentials. Mixtures of exponential distributions are popular in distribution fitting. In the two-dimensional setting, we study to which extent correlations can be introduced in a sequence of intervals following such a distribution. Besides the known moment bounds, exhaustive bounds for the single correlation parameter are given explicitly (in terms of the first three moments) for the first time. This allows one to quickly check the feasibility of the correlation parameter before model construction or to modify an infeasible parameter set in order to obtain a valid (but approximate) MAP(2) representation. Due to its properties (and despite its obvious limitations), the resulting compact model may be efficiently used both for analysis (e.g., matrix-analytic techniques, Markovian models) and Monte-Carlo simulation – especially in large models, where several – desirably small – correlated traffic and service models have to be specified. Particularly, with respect to fixing the correlation parameter, the (hyperexponential) MAP(2) reveals astonishing flexibility.