**2. Basic concepts of the** *NHP*

Let us take *N t*ð Þ as the number of events that occurs in the time interval 0, ð �*t* with *t*>0 and *N*ð Þ¼ 0 0. The probability of *n* events occurring in this time interval is denoted by

$$P\_n(t) = P[N(t) = n], \qquad \qquad n = 0, 1, 2, \dots \tag{1}$$

According to Dubourdieu [6], an *MPP N t* f g ð Þ : *t*≥0 is a *PCP* with rate Λ, where the non-negative random variable Λ is called a structure variable. The *MPP* has been studied by several authors [7–9].

When Λ is a continuous random variable with probability density function (*pdf*), *f*ð Þ*λ* , we can find probability by

$$\underbrace{\mathbb{E}[P[N(t) = n | \Lambda]]}\_{} = \underset{0}{\stackrel{\circ}{\phantom{\mathbb{E}}}} \, P[N(t) = n | \Lambda = \lambda] f(\lambda) d\lambda \tag{2}$$

$$P[N(t) = n] \qquad = \underset{0}{\stackrel{\circ}{\phantom{\mathbb{E}}}} e^{-\lambda t} \frac{\left(\lambda t\right)^{n}}{n!} f(\lambda) d\lambda.$$

For *n* ¼ 0 and *t* >0 we have

$$P\_0(t) = \bigcap\_{\lambda=0}^{\infty} e^{-\lambda t} f(\lambda) d\lambda,\tag{3}$$

The higher order derivatives of the last expression with respect to *t* are

$$P\_0^{(n)}(t) = \frac{d^n}{dt^n} P\_0(t) = (-1)^n \int\_0^\infty \lambda^n e^{-\lambda t} f(\lambda) \, d\lambda. \tag{4}$$

By substituting (4) into (2) we get

$$P\_n(t) = \frac{t^n}{n!} \left[ (-\mathbf{1})^n P\_0^{(n)}(t) \right], \qquad \qquad n \ge 1 \tag{5}$$

The expressions (3) and (5) characterize an *MPP* with a continuous structure variable Λ. According to Hofmann [1], for the construction of examples, a special structure function is generally assumed, and from this the *pmf* is calculated by (3), (5). In most cases, this leads to formally complicated expressions. In ref. [1], Hofmann presents a *CP* called Hofmann process as an option to model the event number process given by (2) and whose general expression for (3) is as follows:

$$P\_0(t) = \exp\left\{-\theta(t)\right\} \qquad\qquad\qquad\theta(t) = \int\_0^t \lambda(\tau; a)d\tau \tag{6}$$

where *<sup>P</sup>*0ð Þ*<sup>t</sup>* is a completely monotonic function<sup>1</sup> . And *λ τ*ð Þ ; *a* is a function of three parameters: *a*≥0, *q*>0 and *κ* ≥0, which is a function infinitely divisible and given by

$$\lambda(\tau; a) = \frac{q}{(\mathbf{1} + \kappa \tau)^a} \qquad \qquad \qquad \qquad \forall \tau > 0. \tag{7}$$

Although *λ τ*ð Þ ; *a* depends on three parameters, we use this notation given that the parameter *a* provides various *CCPs*. We denote the *NHP* by Hð Þ *a*, *q*, *κ* , if the *pmf* of *N t*ð Þ satisfies the expressions (5) and (6).

Using the expression (7), we get by integrating that

$$\theta(t) = \begin{cases} \ln\left[\left(\mathbf{1} + \kappa t\right)^{q/\kappa}\right] & \text{if} \quad a = \mathbf{1} \\ \frac{q}{\kappa(\mathbf{1} - a)} \left[\left(\mathbf{1} + \kappa t\right)^{1-a} - \mathbf{1}\right] & \text{if} \quad a \neq \mathbf{1} \end{cases} \tag{8}$$

By substituting (8) into (6)

$$P\_0(t) = \begin{cases} \left(\mathbf{1} + \kappa t\right)^{-\frac{d}{\kappa}} & \text{if} \quad a = 1\\ \exp\left\{-\frac{q}{\kappa \cdot \left(\mathbf{1} - a\right)} \left[\left(\mathbf{1} + \kappa t\right)^{1-a} - \mathbf{1}\right]\right\} & \text{if} \quad a \neq 1 \end{cases} \tag{9}$$

Remark 1.1: If in the expression (9) for *a* ¼ 1 we take the limit as *κ* ! 0, we have:

$$\lim\_{\kappa \to 0} \left( \mathbf{1} + \kappa t \right)^{-\frac{q}{\kappa}} = e^{-qt}, \tag{10}$$

and the last expression agrees with the adequate *P*0ð Þ*t* of a *PCP* with rate *qt*.
