Research On Nonhomogeneous Poisson Process In Random Fuzzy Environments

On the base of the random fuzzy theory，the homogeneous Poisson process hasbeen carried on the corresponding research in the literature [26].As extensions of arandom fuzzy homogeneous Poisson process, this article mainly discusses thedefinition and some properties of the nonhomogeneous Poisson process in randomfuzzy environments. Furthermore, the problems of the nonhomogeneous Poissonprocess in random fuzzy environments about random selection are researched. Themain work for this article is as follows:(1) This paper gives a definition of nonhomogeneous Poisson process in randomfuzzy environment, Relations between the renewal number and the cumulative fuzzyintensity function x dxare analyzed.We then discussed the relatedproperties of this process.(2) By using probability p and p（s）to make a random choice fornonhomogeneous Poisson process in random fuzzy environments, we made definitionsfor p random selection process and p（s）random selection process ofnonhomogeneous Poisson process in random fuzzy environments, and then discussedthe related properties of these two new counting process.(3) This article made some verification work: when the fuzzy intensity function(x) degenerate into a constant number, that is,conclusions of nonhomogeneous Poisson process tally with the conclusions of thehomogeneous Poisson process in random fuzzy environments. Under the sameconditions, all conclusions of p random selection process of nonhomogeneousprocess tally with the conclusions of the p random selection process of homogeneous Poisson process in random fuzzy environments.In addition,when the selectionprobability p（s）degenerate into a constant number p,that is,（t）0p（s）ds pt,allconclusions of the p（s）random selection process of nonhomogeneous Poissonprocess tally with the conclusions of the p random selection process ofnonhomogeneous Poisson process in random fuzzy environments. This suggests thatthese conclusions are the generalization of homogeneous Poisson process in randomfuzzy environments.To a certain extent, these conclusions verify the results of thestudy.