Characterizations of Generalized Hyperexponential Distributions.
VIRGINIA UNIV CHARLOTTESVILLE DEPT OF SYSTEMS ENGINEERING
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Generalized hyperexponential GH distributions are linear combinations of exponential CDFs with mixing parameters positive and negative that sum to unity. The denseness of the class GH with respect to the class of all CDFs defined on 0, infinity is established by showing that a GH distribution can be found that is as close as desired, with respect to a suitably defined metric, to a given CDF. The metric induces the usual topology of weak convergence so that, equivalently, there exits a sequence of GH CDFs that converges weakly to any CDF. The result follows from a similar well-known result for weak convergence of Erlang mixtures. Various set inclusion relations are also obtained relating the GH distributions to other commonly used classes of approximating distributions including generalized Erlang, mixed generalized Erlang, those with reciprocal polynomial Laplace transforms, those with rational Laplace transforms, and phase-type distributions. A brief survey of the history and use of approximating distributions in queueing theory is also included. Key phrases probability distribution cumulative distribution function approximation convergence in distribution weak convergence denseness Erlang distribution generalized hyperexponential distribution method of stages.
- Statistics and Probability