Notes, Leemis, Reliability, structure function, reliability function

Reference: Lawrence Leemis, Reliability-Probabilistic models and statistical methods, second edition, 2009.

The state of the component is defined by a vector x_i=0 or 1 for a failed and functioning component.
Structure function of a system is $\phi(x)$ = 0 when system failed, and 1 if the system is functioning.

A series system functions if and only if all of its components function. Its structure function is:
  \phi(x) = min{x_1, x_2, ..., x_n}  = product_of x_i

A parallel system functions if and only if one or more component functions. Its structure function is:
 \phi(x) = max{x_1, x_2, ..., x_n} = 1 - product_of (1 - x_i)

Although the minimal and maximal values are very intuitive,  the mathematical formula for the parallel system took me a few minutes to grasp. To solve the parallel \phi(x), we first look for the opposite state vector, i.e., (1 - x_i).  Product of (1 - x_i) indicate a failed parallel system. So, the function system is 1 - product_of (1 - x_i).

Combination of series and parallel sub-units can be used to study complex systems.

Is there a formal proof that combination of these two basic subunit will form any configuration? This could be a very difficult combinatorial problem, and I need to consult a combinatorial mathematician on this topic. In practice, any complex systems can at least be 'reduced' to combinations of these two basic sub-units under some assumptions. This would be a convenient argument to study complex gene networks using these two basic configurations. 

System reliability function $r$ seems to be the viability function in aging (section 2.3, page 31). No, this does not seem to be case. Leemis defines survivor function on page 54:   $S(t) = P[T>=t]$.
Nevertheless, on page 72, example 3.9 show that reliability function can be used to calculate the survivor function.

What component should be we focus on to improve a systems's reliability? For parallel system, the limiting factor is the most reliable one (Page 39).

Mixture of survivor function is discussed in Secion 3.5, page 77. This is directly related to my power-law network model.

Leemis09 discussed Gompertz distribution on page 112. Leemis09 stated that Gompertz hazard function is derived from the assumption that Mill's ratio, 1/ harzard function = resistance to death, and h(t) decreases over time at a rate proportional to itself:
   $d(1/h(t) / dt = k (1/h(t))$
Solution to this assumption leads to a exponential function.

One relationship map for continuous univariate lifetime distributions is provided in figure 4.9 on page 114.

To find the reliability of a system at any time t, the component survivor functions should be used as argument in the reliability function (page 70): 
  S_system = r( S1(t), S2(t), S3(t), ..., Sn(t)). 
Q:How about mortality rate (hazard function)? It can be found by definition:
   h(t) = - S' / S

Page 71 gave an example for a two component in parallel configuration, which is equivalent to  synthetic lethalilty.
Structure function $r = 1 - (1-p1)(1-p2)$
Survivor function $S = 1 - (1-S1)(1-S2)$