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On scale functions for Lévy processes with negative phase-type jumps

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We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

Original languageEnglish
JournalQueueing Systems
Pages (from-to)3-19
Number of pages17
Publication statusPublished - Mar 2021

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

Copyright 2021 Elsevier B.V., All rights reserved.

    Research areas

  • Fluid flow model, Iterative scheme, Phase-type distribution, Rational transform, Scale function

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