**Abstract**

The classical counting processes (Poisson and negative binomial) are the most traditional discrete counting processes (*DCPs*); however, these are based on a set of rigid assumptions. We consider a non-homogeneous counting process (which we name non-homogeneous Hofmann process – *NHP*) that can generate the classical counting processes (*CCPs*) as special cases, and also allows modeling counting processes for event history data, which usually exhibit under- or over-dispersion. We present some results of this process that will allow us to use it in other areas and establish both the probability mass function (*pmf*) and the cumulative distribution function (*cdf*) using transition intensities. This counting process (*CP*) will allow other researchers to work on modelling the *CP*, where data dispersion exists in an efficient and more flexible way.

**Keywords:** mixed Poisson Process, Hofmann process, variance-to-mean ratio, transition intensity
