Block diagram representation of the complex system

The initial idea is to use the differential equations of a probabilistic transfer model with hazard rates varying with the age of the molecules, i.e., to enlarge the limiting hypothesis (9.2). The objective is to find nonexponential families of survival distributions that are mathematically tractable and yet sufficiently flexible to fit the observed data. In the simplest case, the differential equation (9.7) links hazard rates and survival distributions. Nevertheless, this relation was at the origin of an erroneous use of the hazard function. In fact, substituting in this relation the age a by the exogenous time t, we obtain 

Since the exterior time t and the age of the molecules a are the same for the one-compartment model, we can use the previous equation to write 

The closed-form solutions are more difficult to obtain than those previously obtained by means of the survival functions. Numerical integration or quadrature can be used to solve the differential equation or the integral. For instance  

This form is very similar to the model often used when the molecules move across fractal media, e.g., the dissolution rate using a time-dependent coefficient given by (5.12) to describe phenomena that take place under dimensional constraints or understirred conditions [16]. The previous differential equation has the solution given by (9.9). 

In a pioneer work, Marcus established the link between some usual time-varying forms of h ( ) and / (a) in a single compartment [300]. For instance in h(t) = (f +/ ), a = 1 leads to A Gam(A,/3) and 1 a 2 defines the standard extreme stable-law density with exponent a. In the case of a = 1.5, the obtained distribution is known as the retention-time distribution of a Wiener process with drift. 

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Figure 9.5 Block diagram representation of the complex system shown in Figure 9.4.

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