Stationary Markovian Arrival Processes, Results and Open Problems

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dc.contributor.author Asanjarani, Azam en
dc.contributor.author Nazarathy, Y en
dc.date.accessioned 2020-01-10T03:10:28Z en
dc.date.issued 2019-05-05 en
dc.identifier.citation Arxiv (1905.01736v1). 05 May 2019. 11 pages en
dc.identifier.uri http://hdl.handle.net/2292/49499 en
dc.description.abstract We consider two classes of irreducible Markovian arrival processes specified by the matrices $C$ and $D$. The Markov Modulated Poison Process (MMPP) and the Markovian Switched Poison Process (MSPP). The former exhibits a diagonal $D$ while the latter exhibits a diagonal $C$. For these two classes, we consider the following statements: (I) Overdispersion of the counts process. (II) A non-increasing hazard rate of the stationary inter-event time. (III) The squared coefficient of variation of the event stationary process is greater or equal to unity. (IV) A stochastic order showing that the time stationary inter-arrival time dominates the event-stationary time. For general MSPPs and two-state MMPPs, we show that (I)-(IV) hold. Then for general MMPPs, it is easy to establish (I), while (II) is false due to a counter-example of Miklos Telek and Illes Horvath. For general simple point processes, (III) follows from (IV). For MMPPs we conjecture and numerically test that (IV) and thus (III) hold. Importantly, modeling folklore has often treated MMPPs as ``bursty'' and implicitly assumed that (III) holds. However, this is still an open question. en
dc.relation.ispartof Arxiv en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher. en
dc.rights.uri https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm en
dc.rights.uri http://arxiv.org/licenses/nonexclusive-distrib/1.0/ en
dc.subject math.PR en
dc.subject math.PR en
dc.title Stationary Markovian Arrival Processes, Results and Open Problems en
dc.type Report en
dc.rights.holder Copyright: The authors en
pubs.author-url http://arxiv.org/abs/1905.01736v1 en
dc.rights.accessrights http://purl.org/eprint/accessRights/RestrictedAccess en
pubs.subtype Working Paper en
pubs.elements-id 771893 en
pubs.org-id Science en
pubs.org-id Statistics en
pubs.arxiv-id 1905.01736 en
pubs.number 1905.01736v1 en
pubs.record-created-at-source-date 2020-01-13 en


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