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Findings about the two-state BMMPP for modeling point processes in reliability and queueing systems

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  • Yera Mora, Yoel Gustavo
  • Lillo Rodríguez, Rosa Elvira
  • Ramírez-Cobo, Pepa

Abstract

The Batch Markov Modulated Poisson Process (BMMPP) is a subclass of the versatile Batch Markovian Arrival process (BMAP) which have been widely used for the modeling of dependent and correlated simultaneous events (as arrivals, failures or risk events, real-time multimedia communications). Both theoretical and applied aspects are examined in this paper. On one hand, the identifiability of the stationary BMMPP2(K) is proven, where K is the maximum batch size. This is a powerful result when inferential tasks related to real data sets are carried out. On the other hand, some findings concerning the correlation and autocorrelation structures are provided.

Suggested Citation

  • Yera Mora, Yoel Gustavo & Lillo Rodríguez, Rosa Elvira & Ramírez-Cobo, Pepa, 2017. "Findings about the two-state BMMPP for modeling point processes in reliability and queueing systems," DES - Working Papers. Statistics and Econometrics. WS 24622, Universidad Carlos III de Madrid. Departamento de Estadística.
    • Handle: RePEc:cte:wsrepe:24622
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    References listed on IDEAS

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    1. Ozekici, S. & Soyer, R., 2003. "Reliability of software with an operational profile," European Journal of Operational Research, Elsevier, vol. 149(2), pages 459-474, September.
    2. K. Sikdar & S. K. Samanta, 2016. "Analysis of a finite buffer variable batch service queue with batch Markovian arrival process and server’s vacation," OPSEARCH, Springer;Operational Research Society of India, vol. 53(3), pages 553-583, September.
    3. 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.
    4. Liu, Baoliang & Cui, Lirong & Wen, Yanqing & Shen, Jingyuan, 2015. "A cold standby repairable system with working vacations and vacation interruption following Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 142(C), pages 1-8.
    5. Paul Fearnhead & Chris Sherlock, 2006. "An exact Gibbs sampler for the Markov‐modulated Poisson process," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(5), pages 767-784, November.
    6. Montoro-Cazorla, Delia & Pérez-Ocón, Rafael & del Carmen Segovia, Maria, 2009. "Replacement policy in a system under shocks following a Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 94(2), pages 497-502.
    7. A. D. Banik & M. L. Chaudhry, 2017. "Efficient Computational Analysis of Stationary Probabilities for the Queueing System BMAP / G /1/ N With or Without Vacation(s)," INFORMS Journal on Computing, INFORMS, vol. 29(1), pages 140-151, February.
    8. Landon, Joshua & Özekici, Süleyman & Soyer, Refik, 2013. "A Markov modulated Poisson model for software reliability," European Journal of Operational Research, Elsevier, vol. 229(2), pages 404-410.
    9. 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.
    10. Rodríguez, Joanna & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2016. "Dependence patterns for modeling simultaneous events," Reliability Engineering and System Safety, Elsevier, vol. 154(C), pages 19-30.
    11. Tetsuya Takine, 2016. "Analysis and computation of the stationary distribution in a special class of Markov chains of level-dependent M/G/1-type and its application to BMAP/M/ $$\infty $$ ∞ and BMAP/M/c+M queues," Queueing Systems: Theory and Applications, Springer, vol. 84(1), pages 49-77, October.
    12. S. Özekici & R. Soyer, 2006. "Semi-Markov modulated Poisson process: probabilistic and statistical analysis," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 125-144, August.
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