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Poisson process with fuzzy rates

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Abstract

In a stochastic homogeneous Poisson process, interarrival times are independent and identically distributed (iid) exponential random variables whose parameter is called the rate of the process. By using fuzzy variables to describe the parameter, a Poisson process whose rates are fuzzy variables is established. Based on the random fuzzy theory, relationship between the renewal number and fuzzy rates is discussed. As an application, a random fuzzy compound Poisson process is investigated.

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References

  • Buckley J. J. (2005) Fuzzy probabilities and statistics. Springer, Heidelberg

    Google Scholar 

  • Dubois D., Prade H. (1980) Fuzzy sets and systems: Theory and applications. Academic Press, New York

    MATH  Google Scholar 

  • Hwang C. M. (2000) A theorem of renewal process for random fuzzy variables and its application. Fuzzy Sets and Systems 116: 237–244

    Article  MATH  MathSciNet  Google Scholar 

  • Kwakernaak H. (1978) Fuzzy random variables-I. Information Sciences 15: 1–29

    Article  MATH  MathSciNet  Google Scholar 

  • Kwakernaak H. (1979) Fuzzy random variables-II. Information Sciences 17: 253–278

    Article  MATH  MathSciNet  Google Scholar 

  • Li, S., Tang, W., & Zhao, R. (2005). Some results on fuzzy delayed renewal process. In: Proceedings of the 2005 international conference on machine learning and cybernetics, pp. 2624–2627.

  • Li S., Tang W., Zhao R. (2007) Fuzzy random delayed renewal process and fuzzy random equilibrium renewal process. Journal of Intelligent and Fuzzy Systems 18(2): 149–156

    MATH  Google Scholar 

  • Li, S., Tang, W., & Zhang, C. (2006). Poisson process with fuzzy time-dependent intensity. Lecture notes in computer sciences, pp. 786–797.

  • Li X., Liu B. (2006) New independence definition of fuzzy random variable and random fuzzy variable. World Journal of Modelling and Simulation 5(2): 338–342

    Google Scholar 

  • Liu B. (2002) Theory and practice of uncertain programming. Physica-Verlag, Heidelberg

    MATH  Google Scholar 

  • Liu Y., Liu B. (2003) Fuzzy random variables: A scalar expected value operator. Fuzzy Optimization and Decision Making 2: 143–160

    Article  MathSciNet  Google Scholar 

  • Liu B. (2006) A survey of credibility theory. Fuzzy Optimization and Decision Making 5: 387–408

    Article  MathSciNet  Google Scholar 

  • Liu B. (2004) Uncertainty theory. Springer, Berlin

    MATH  Google Scholar 

  • Liu B., Liu Y. K. (2002) Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems 10: 445–450

    Article  Google Scholar 

  • Liu Y. K., Liu B. (2003) Expected value operator of random fuzzy variable and random fuzzy expected value models. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11: 195–215

    Article  MATH  MathSciNet  Google Scholar 

  • Mcfadden T. A. (1965) The entropy of a point process. SIAM J. Appl. Math 13: 988–994

    Article  MATH  MathSciNet  Google Scholar 

  • Popova E., Wu H. C. (1999) Renewal reward processes with fuzzy rewards and their applications to T-age replacement policies. European Journal of Operational Research 117: 606–617

    Article  MATH  Google Scholar 

  • Ross S. M. (1996) Stochastic processes. John Wiley & Sons Inc., New York

    MATH  Google Scholar 

  • Stoyan D. (1983) Comparison methods for queues and other stochastic models. Wiley-Interscience, New York

    MATH  Google Scholar 

  • Zadeh L. A. (1965) Fuzzy sets. Information and Control 8: 338–353

    Article  MATH  MathSciNet  Google Scholar 

  • Zadeh L. A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3–28

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao R., Liu B. (2003) Renewal process with fuzzy interarrival times and rewards. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11: 573–586

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao R., Tang W., Yun H. (2006) Random fuzzy renewal process. European Journal of Operational Research 169: 189–201

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Science, Tianjin University of Commerce, Tianjin, 300134, China

    Shunqin Li

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  1. Shunqin Li
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Correspondence to Shunqin Li.

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Li, S. Poisson process with fuzzy rates. Fuzzy Optim Decis Making 9, 289–305 (2010). https://doi.org/10.1007/s10700-010-9082-1

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  • DOI: https://doi.org/10.1007/s10700-010-9082-1

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