**4.** *NHP* **in terms of** *CCP***s**

In this section, we present various approaches for the *NHP* using *CCP*.

### **4.1** *NHP* **as a non-homogeneous pure birth process**

We use logarithmic differentiation to find the derivative of (5) and we get

$$\frac{P\_{n'}(t)}{P\_n(t)} = \frac{n}{t} + \frac{P\_0^{(n+1)}(t)}{P\_0^{(n)}(t)}$$

Then

$$P\_{n'}(t) = \frac{n}{t} P\_n(t) + \frac{P\_0^{(n+1)}(t)}{P\_0^{(n)}(t)} P\_n(t) \tag{19}$$

From (5), we obtain

$$\begin{aligned} \frac{d^n}{dt^n} P\_n(t) &= -\frac{(-\mathbf{1})^{n-1}}{(n-1)!} t^{n-1} P\_0^{(n)}(t) = \frac{(-\mathbf{1})^{n-1}}{(n-1)!} t^{n-1} P\_0^{(n)}(t) \left( -\frac{P\_0^{(n-1)}(t)}{P\_0^{(n-1)}(t)} \right) \\ &= -\frac{P\_0^{(n)}(t)}{P\_0^{(n-1)}(t)} P\_{n-1}(t) \end{aligned}$$

By substituting in (19), we have

$$P\_{n'}(t) = \left(-\frac{P\_0^{(n)}(t)}{P\_0^{(n-1)}(t)}\right)P\_{n-1}(t) - \left(-\frac{P\_0^{(n+1)}(t)}{P\_0^{(n)}(t)}\right)P\_n(t). \tag{20}$$

*Some Results on the Non-Homogeneous Hofmann Process DOI: http://dx.doi.org/10.5772/intechopen.106422*

We denote

$$\lambda\_n(t; a) = -\frac{P\_0^{(n+1)}(t)}{P\_0^{(n)}(t)} = -\frac{d}{dt}\ln\left[(-1)^n P\_0^{(n)}(t)\right].\tag{21}$$

In ref. [18], Lundberg shows that this corresponds to the transition intensities. Then from (20) and (21), we can derive the following system of Kolmogorov differential equations that must be satisfied by the *NHP*:

$$\begin{aligned} P\_0'(t) &= -\lambda \mathfrak{o}(t; a) P \mathfrak{o}(t) \\ P\_n'(t) &= \lambda\_{n-1}(t; a) P\_{n-1}(t) - \lambda\_n(t; a) P\_n(t) \qquad \text{for} \qquad n \ge 1. \end{aligned} \tag{22}$$

By notation, we denote *λ*0ð Þ¼ *t*; *a θ*<sup>0</sup> ðÞ¼ *<sup>t</sup> <sup>q</sup>* ð Þ <sup>1</sup>þ*κ<sup>t</sup> <sup>a</sup> :* With initial conditions

$$P\_0(\mathbf{0}) = \mathbf{1} \quad \text{and} \quad P\_n(\mathbf{0}) = \mathbf{0} \quad \forall n \ge \mathbf{1} \tag{23}$$

Using the method given in ref. [18], we find that the solution of (22) is given by

$$P\_n(t) = \int\_0^t \lambda\_{n-1}(\tau; a) P\_{n-1}(\tau) \exp\left\{-\int\_\tau^t \lambda\_{n-1}(\nu; a) d\nu\right\} d\tau \qquad \text{for} \qquad n \ge 1.$$

From the system of equations given in (22), we have that the *NHP* is a nonhomogeneous pure birth process (*NHPBP*), which agrees with the definition given by Seal in ref. [19]. So, if *N t*ð Þ satisfies (6), then *N t*ð Þ is an *NHPBP* with transition intensities given by (21).
