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A Note on the Dependence Structure of the Two-State Markovian Arrival Process

  • Pepa Ramírez-Cobo (a1) and Emilio Carrizosa (a1)

Abstract

The Markovian arrival process generalizes the Poisson process by allowing for dependent and nonexponential interarrival times. We study the autocorrelation function of the two-state Markovian arrival process. Our findings show that the correlation structure of such a process has a very specific pattern, namely, it always converges geometrically to zero. Moreover, the signs of the autocorrelation coefficients are either constant or alternating.

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Copyright

Corresponding author

Postal address: Institute of Mathematics, University of Seville, Avda. Reina Mercedes s/n, 41012 Seville, Spain. Email address: jrcobo@us.es
∗∗ Postal address: Department of Statistics and Operations Research, University of Seville, Avda. Reina Mercedes s/n, 41012 Seville, Spain. Email address: ecarrizosa@us.es

References

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[1] Chakravarthy, S. R. (2001). The batch Markovian arrival process: a review and future work. In Advances in Probability and Stochastic Processes, eds Krishnamoorthy, A., Raju, N. and Ramaswami, V., Notable Publications, NJ, pp. 2149.
[2] Kang, S. H., Han Kim, Y., Sung, D. K. and Choi, B. D. (2002). An application of Markovian arrival process to modeling superposed ATM cell streams. IEEE Trans. Commun. 50, 633642.
[3] Lucantoni, D. M. (1993). The BMAP/G/1 queue: a tutorial. In Performance Evaluation of Computer and Communication Systems (Lecture Notes Comput. Sci. 729), eds Donatiello, L. and Nelson, R. D., Springer, New York, pp. 330358.
[4] Montoro-Cazorla, D., Pérez-Ocón, R. and Segovia, M. (2009). Replacement policy in a system under shocks following a Markovian arrival process. Reliab. Eng. System Safety 94, 497502.
[5] Neuts, M. F. (1979). A versatile Markovian point process. J. Appl. Prob. 16, 764779.
[6] Ramaswami, V. (1980). The N/G/1 queue and its detailed analysis. Adv. Appl. Prob. 12, 222261.
[7] Ramírez-Cobo, P., Lillo, R. and Wiper, M. (2010). Bayesian inference for the two-state Markovian arrival process. Working paper 10-43, Statistics and Econometrics Series 26, Universidad Carlos III de Madrid.
[8] Ramírez-Cobo, P., Lillo, R. E. and Wiper, M. P. (2010). Nonidentifiability of the two-state Markovian arrival process. J. Appl. Prob. 47, 630649.
[9] Rydén, T. (1996). An EM algorithm for estimation in Markov-modulated Poisson processes. Comput. Statist. Data Anal. 21, 431447.
[10] Scott, S. L. (1999). Bayesian analysis of the two state Markov modulated Poisson process. J. Comput. Graphical Statist. 8, 662670.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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