Abstract
We consider a Poisson process that is modulated in such a way that the arrival rate at any time depends on the state of a semi-Markov process. This presents an interesting generalization of Poisson processes with important implications in real life applications. Our analysis concentrates on the transient as well as the long term behaviour of the arrival count and the arrival time processes. We discuss probabilistic as well as statistical issues related to various quantities of interest.
Similar content being viewed by others
References
Berman M (1981) Inhomogeneous and modulated gamma processes. Biometrica 68:143–152
Çınlar E (1975) Introduction to stochastic processes. Prentice-Hall, Englewood Cliffs
Çınlar E, Özekici S (1987) Reliability of complex devices in random environments. Prob Eng Inf Sci 1:97–115
Cox DR (1955) Some statistical models connected with series of events. J R Stat Soc Ser B 17:129–164
Erdem A, Özekici S (2002) Inventory models with random yield in a random environment. Int J Prod Econ 78:239–253
Fischer W, Meier-Hellstern K (1992) The Markov-modulated Poisson process cookbook. Perform Eval 18:149–171
Kingman JFC (1964) On doubly stochastic Poisson processes. Proc Camb Philos Soc 60:923–960
Moler C, van Loan C (1978) Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev 20:801–836
Neuts MF (1978) The M/M/1 queue with randomly varying arrival and service rates. Opsearch 15:139–157
Özekici S (1996) Complex systems in random environments. In: Özekici S (ed) Reliability and maintenance of complex systems NATO ASI Series Vol. F154. Springer, Berlin Heidelberg New York, pp 137–157
Özekici S (1997) Markov modulated Bernoulli process. Math Methods of Operations Research 45:311–324
Özekici S, Soyer R (2003) Bayesian analysis of Markov modulated Bernoulli processes. Math Methods Oper Res 57:125–140
Özekici S, Soyer R (2003) Reliability of software with an operational profile. Eur J Oper Res 149:459–474
Prabhu NU, Zhu Z (1989) Markov-modulated queueing systems. Queueing Sys 5:215–246
Song JS, Zipkin P (1993) Inventory control in fluctuating demand environment. Oper Res 41:351–370
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Özekici, S., Soyer, R. Semi-Markov modulated Poisson process: probabilistic and statistical analysis. Math Meth Oper Res 64, 125–144 (2006). https://doi.org/10.1007/s00186-006-0067-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-006-0067-3