**1. Introduction**

In ref. [1], Hofmann introduced a new class of infinitely divisible mixed Poisson process (*MPP*), this broader class of *CP* allows obtaining other *CCP* by simply modifying or choosing its parameters, as well as Poisson, negative binomial, Poisson-Pascal among other distributions (see [2]). The family of distributions defined by Hofmann has been used in many types of applications of modelling and simulation studies that include topics such as accident models [3].

In this chapter, we analysed the event of number process f g *N t*ð Þ, *t*≥0 and used a broader *CP,* which is based on the Hofmann process. The appeal of this *CP* is that, analogous to the family of frequency distributions, it allows to generate several known *CP*. Through an *NHP,* we can generate the following as special cases: the Poisson counting process (*PCP*), the negative binomial counting process (*NBCP*) and the Poisson-Pascal process among other *CCPs*, and this allows us to obtain models for *CP* with under- or over-dispersion. The *NHP* was introduced by Hofmann [1] and has been used by other researchers [3–5]. Some properties of the *NHP* found by Walhin

[2] are presented in this chapter, and we used the transition intensities to describe additional properties of the *NHP*.

The objective of this chapter is to present a unified view of related results on the *NHP*. The chapter is organised as follows: in Section 2, we present the *NHP*; in Section 3, we present some statistical properties, such as *pmf* and probability generating function (pgf), and formulas for the mean and variance are derived; in Section 4, we present various approaches for the *NHP* using *CCP*; in Section 5, we present other properties for *NHP*; finally, conclusions are presented.
