## Partially Damaged Redundant System

In the preceding section, we examined a reliability model for a system consisting of m series-connected blocks with numbers of elements distributed according to the Poisson law. In this section, we consider a more general case in which the probability of an element being initially functional can take any possible value: 0 < q < 1 (see Gavrilov and Gavrilova, 1991; 2001 for more detail).

In the general case, the distribution of blocks in the organism according to the number of initially functional elements is described by the binomial rather than Poisson distribution.

If an organism can be presented as a system constructed of m series-connected blocks with binomially distributed elements, its failure rate is given by the following formula:

It is proposed to call a parameter x0 the initial virtual age of the system, IVAS (Gavrilov and Gavrilova, 1991; 2001). Indeed, this parameter has the dimension of time, and corresponds to the age by which an initially ideal system would have accumulated as many defects as a real n-1

system already has at the initial moment in time (at x = 0). In particular, when q = 1, i.e., when all the elements are functional at the beginning, the initial virtual age of the system is zero and the failure rate grows as a power function of age (the Weibull law), as described in the causes of failure rate increase with age section. However, when the system is not initially ideal (q < 1), we obtain the binomial law of mortality (see basic failure models).

In the case when x0 > 0, there is always an initial period of time, such that x ^ x0 and the following approximation to the binomial law is valid:

Hence, for any value of q < 1 there always exists a period of time x when the number of newly formed defects is much less than the original number, and the failure rate grows exponentially with age.

So, if the system is not initially ideal (q < 1), the failure rate in the initial period of time grows exponentially with age according to the Gompertz law. A numerical example provided in Figure 5.2 (see the reliability approach to system's failure in aging section) shows that increase in the initial system's damage load (initial virtual age) converts the observed mortality trajectory from the Weibull to the Gompertz one.

The model discussed here not only provides an explanation for the exponential increase in the failure rate with age, but it also explains the compensation law of mortality (see Gavrilov and Gavrilova, 1991; 2001).

The compensation law of mortality is observed whenever differences in mortality are caused by differences in initial redundancy (the number of elements in a block, n), while the other parameters, including the ''true aging rate'' (rate of elements' loss are similar in populations of a given species (presumably because of homeostasis—stable body temperature, glucose concentration, etc.). For lower organisms with poor homeostasis there may be deviations from this law. Our analysis of data published by Pletcher et al. (2000) revealed that in Drosophila this law holds true for male-female comparisons (keeping temperature the same), but not for experiments conducted at different temperatures, presumably because temperature may influence the rate of element loss.

The failure rate of the blocks asymptotically approaches an upper limit which is independent of the number of initially functional elements and is equal to Therefore the failure rate of a system consisting of m blocks in series tends asymptotically with increased age to an upper limit m^,, independently of the values of n and q.

Thus the reliability model described here provides an explanation for a general pattern of aging and mortality in biological species: the exponential growth of failure rate in the initial period, with the subsequent mortality deceleration and leveling-off, as well as the compensation law of mortality.

This model might also be called the model of series-connected blocks with varying degrees of redundancy or distributed redundancy. The basic conclusion of the model might be reformulated as follows: if vital components of a system differ in their degree of redundancy, the mortality rate initially grows exponentially with age (according to the Gompertz law) with subsequent levelingoff in later life. 