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Laplace Transform Inversion and Passage-Time Distributions in Markov Processes

Peter G. Harrison

Journal Article
Journal of Applied Probability
Volume 27
Issue 1
March, 1990
Applied Probability Trust
DOI 10.2307/3214596

Products of the Laplace transforms of exponential distributions with different parameters are inverted to give a mixture of Erlang densities, i.e. an expression for the convolution of exponentials. The formula for these inversions is expressed both as an explicit sum and in terms of a recurrence relation which is better suited to numerical computation. The recurrence for the inversion of certain weighted sums of these transforms is then solved by converting it into a linear first-order partial differential equation. The result may be used to find the density function of passage times between states in a Markov process and it is applied to derive an expression for cycle time distribution in tree-structured Markovian queueing networks.

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