Abstract
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, ∞) is established by showing that a GH distribution can be found that is as close to a given CDF as desired, with respect to a suitably defined metric. The metric induces the usual topology of weak convergence so that, equivalently, there exists a sequence of GH CDFs that converges weakly to a given CDF. This result is established by using a similar 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 (GE), mixed generalized Erlang (MGE), those with reciprocal polynomial Laplace transforms (K n ), those with rational Laplace transforms (R n ), and phase-type (PH) distributions. A brief survey of the history and use of approximating distributions in queueing theory is also included.
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This research was partially supported by the Office of Naval Research under Contract No. N00014-86-K0029. Much of this work is taken from the first-named author's doctoral dissertation, accepted by the faculty at the University of Virginia.
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Botta, R.F., Harris, C.M. Approximation with generalized hyperexponential distributions: Weak convergence results. Queueing Syst 1, 169–190 (1986). https://doi.org/10.1007/BF01536187
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DOI: https://doi.org/10.1007/BF01536187