Convolving Hyper-Erlang with Hyper-Exponential Distributions Using Linear Algebra
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Abstract
In this paper, the sum of a hyper-Erlang and a hyper-exponential distributed random variables is analyzed. Although tedious, the resulting random variable’s probability density function (PDF) can be obtained through convolution of its summands’ PDFs. Alternatively, the resulting distribution can be stated directly in terms of a phase-type distribution. However, computing its PDF can still be very costly and this representation gives little insight on the distribution. It can be shown that the sum of both random variable is again hyper-Erlang distributed of incremented order and can therefore be described without requiring the matrix exponential function. We derive a closed form linear algebra expression for the probability weights of the sum’s hyper-Erlang distribution, which significantly reduces the computational complexity of evaluating its distribution.
References
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Qi-Ming He. 2014. Fundamentals of Matrix-Analytic Methods. Springer New York, New York, NY. https://doi.org/10.1007/978-1-4614-7330-5
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Therrar Kadri and Smaili Khaled. 2015. Convolutions of Hyper-Erlang and of Erlang Distributions. International Journal of Pure and Apllied Mathematics 98 (Jan. 2015). https://doi.org/10.12732/ijpam.v98i1.8
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Benjamin Legros and Oualid Jouini. 2015. A Linear Algebraic Approach for the Computation of Sums of Erlang Random Variables. Applied Mathematical Modelling 39, 16 (Aug. 2015), 4971–4977. https://doi.org/10.1016/j.apm.2015.04.013
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- Convolving Hyper-Erlang with Hyper-Exponential Distributions Using Linear Algebra
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Published In
June 2022
137 pages
ISBN:9781450396233
DOI:10.1145/3545839
Copyright © 2022 ACM.
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Association for Computing Machinery
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Published: 10 September 2022
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ICoMS 2022
ICoMS 2022: 2022 5th International Conference on Mathematics and Statistics
June 17 - 19, 2022
Paris, France
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