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8126 results in Fluid dynamics and solid mechanics

Page 40 of 407

  • STATIONARY MARKOVIAN ARRIVAL PROCESSES: RESULTS AND OPEN PROBLEMS

  • Part of
  • AZAM ASANJARANI, YONI NAZARATHY
  • Journal:
    The ANZIAM Journal / Volume 64 / Issue 1 / January 2022
    Published online by Cambridge University Press:
    22 February 2022, pp. 54-68
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  • We consider two classes of irreducible Markovian arrival processes specified by the matrices C and D: the Markov-modulated Poisson process (MMPP) and the Markov-switched Poisson process (MSPP). The former exhibits a diagonal matrix D while the latter exhibits a diagonal matrix C. For these two classes we consider the following four statements: (I) the counting process is overdispersed; (II) the hazard rate of the event-stationary interarrival time is nonincreasing; (III) the squared coefficient of variation of the event-stationary process is greater than or equal to one; (IV) there is a stochastic order showing that the time-stationary interarrival time dominates the event-stationary interarrival time. For general MSPPs and order two MMPPs, we show that (I)–(IV) hold. Then for general MMPPs, it is easy to establish (I), while (II) is shown to be false by a counter-example. For general simple point processes, (III) follows from (IV). For MMPPs, we conjecture that (IV) and thus (III) hold. We also carry out some numerical experiments that fail to disprove this conjecture. Importantly, modelling folklore has often treated MMPPs as “bursty”, and implicitly assumed that (III) holds. However, to the best of our knowledge, proving this is still an open problem.

  • 1 - Background

  • Peter G. Baines, University of Melbourne
  • Book:
    Topographic Effects in Stratified Flows
    Published online:
    04 April 2025
    Print publication:
    27 January 2022, pp 1-15
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Page 40 of 407