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A Markovian canonical form of second-order matrix-exponential processes

Author

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  • Bodrog, L.
  • Heindl, A.
  • Horváth, G.
  • Telek, M.

Abstract

Besides the fact that - by definition - matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that - in two dimensions - the respective sets, MAP(2) and MEP(2), are indeed identical with respect to the stationary behavior. Furthermore, this equivalence extends to acyclic MAPs, i.e., AMAP(2), so that AMAP(2)[reverse not equivalent]MAP(2)[reverse not equivalent]MEP(2). For higher orders, these equivalences do not hold. The second-order equivalence is established via a novel canonical form for the (correlated) processes. An explicit moment/correlation-matching procedure to construct the canonical form from the first three moments of the interarrival time distribution and the lag-1 correlation coefficient shows how these compact processes may conveniently serve as input models for arrival/service processes in applications.

Suggested Citation

  • Bodrog, L. & Heindl, A. & Horváth, G. & Telek, M., 2008. "A Markovian canonical form of second-order matrix-exponential processes," European Journal of Operational Research, Elsevier, vol. 190(2), pages 459-477, October.
    • Handle: RePEc:eee:ejores:v:190:y:2008:i:2:p:459-477
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    References listed on IDEAS

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    1. Asmussen, Søren & Bladt, Mogens, 1999. "Point processes with finite-dimensional conditional probabilities," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 127-142, July.
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    Cited by:

    1. Ramírez-Cobo, Pepa & Carrizosa, Emilio & Lillo, Rosa E., 2021. "Analysis of an aggregate loss model in a Markov renewal regime," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    2. Rodríguez, Joanna & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2015. "Failure modeling of an electrical N-component framework by the non-stationary Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 134(C), pages 126-133.
    3. Sunkyo Kim, 2016. "Minimal LST representations of MAP(n)s: Moment fittings and queueing approximations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 63(7), pages 549-561, October.
    4. Rodríguez César, Joanna Virginia & Ramírez-Cobo, Pepa, 2012. "On the identifiability of the two-state BMAP," DES - Working Papers. Statistics and Econometrics. WS ws120201, Universidad Carlos III de Madrid. Departamento de Estadística.
    5. Casale, Giuliano & Sansottera, Andrea & Cremonesi, Paolo, 2016. "Compact Markov-modulated models for multiclass trace fitting," European Journal of Operational Research, Elsevier, vol. 255(3), pages 822-833.
    6. Pepa Ramírez-Cobo & Rosa Lillo & Michael Wiper, 2014. "Identifiability of the MAP 2 /G/1 queueing system," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 274-289, April.
    7. Joanna Rodríguez & Rosa E. Lillo & Pepa Ramírez-Cobo, 2016. "Nonidentifiability of the Two-State BMAP," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 81-106, March.
    8. Pepa Ramírez-Cobo & Rosa E. Lillo, 2012. "New Results About Weakly Equivalent MAP 2 and MAP 3 Processes," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 421-444, September.
    9. Yera, Yoel G. & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2019. "Fitting procedure for the two-state Batch Markov modulated Poisson process," European Journal of Operational Research, Elsevier, vol. 279(1), pages 79-92.
    10. Yera, Yoel G. & Lillo, Rosa E. & Nielsen, Bo F. & Ramírez-Cobo, Pepa & Ruggeri, Fabrizio, 2021. "A bivariate two-state Markov modulated Poisson process for failure modeling," Reliability Engineering and System Safety, Elsevier, vol. 208(C).

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