**5. Optimization**

By using the models and software tools above the problems of optimization for an element, subsystem and system can be solved through calculating probabilities of success or failure during a given period on the timeline. This approach considers the threats, conditions of counteractions and the given admissible risk established by the precedent principle. Thus, the final choice of integrated measures is allocated on a payoff to the customer in view of specificity of the created or maintained system.

For example, the next general formal statements of problems for optimization can be used [6]:

points Тreq. from 0 to ∞ are computed, it means a trajectory of the PDF, depending on the characteristics of threats, periodic control, monitoring and recovery. And the building of PDF is the real base to prediction metrics P and R for given time Тreq.. In analogy with reliability, it is important to know a mean time between neighboring losses of integrity like mean time between neighboring failures in reliability (MTBF), but in application to quality,

For complex systems, parallel or serial structure existing models with known PDF can be developed by usual methods of probability theory. Let's consider the elementary structure from two independent parallel or series elements. Let PDF of time between losses of the ith

(t) = Р (τ<sup>i</sup> ≤ t); then: **1.** Time between losses of integrity for the system combined from series-connected indepen-

(i.e., the system goes into a state of lost integrity when either the first or second element integrity is lost). For this case the PDF of time between losses of system integrity is defined

**2.** Time between losses of integrity for system combined from parallel-connected indepen-

and second elements (i.e., the system goes into a state of lost integrity when both first and second elements have lost integrity). For this case the PDF of time between losses of system

Applying recurrently expressions (1), (2), it is possible to build PDF of time between losses of

All these ideas for analytical modeling operation processes are supported by the software tools "Mathematical modeling of system life cycle processes"—"know how" (registered by Rospatent №2,004,610,858), "Complex for evaluating quality of production processes" (regis-

By using the models and software tools above the problems of optimization for an element, subsystem and system can be solved through calculating probabilities of success or failure during a given period on the timeline. This approach considers the threats, conditions of counteractions and the given admissible risk established by the precedent principle. Thus, the final choice of integrated measures is allocated on a payoff to the customer in view of specific-

, τ2) ≤ t) = Р(τ1 ≤ t)Р(τ<sup>2</sup> ≤ t) = В1

, τ<sup>2</sup>

) ≤ t) = 1 − Р(min (τ1

dent elements (hot reservation) is equal to a maximum from two times τ<sup>i</sup>

(t)

integrity for any complex system with parallel and/or series structures.

: failure of first or second elements

: failure of first

(t) (2)

) > t) = 1 − Р(τ1 > t)Р(τ<sup>2</sup> > t)

(t) В<sup>2</sup>

] (1)

(t), that is, В<sup>i</sup>

dent elements is equal to a minimum from two times τ<sup>i</sup>

, τ<sup>2</sup>

] [1 − В<sup>2</sup>

(t)

tered by Rospatent №2,010,614,145) and others [1–4].

ity of the created or maintained system.

safety, etc.

element of integrity be В<sup>i</sup>

66 Probabilistic Modeling in System Engineering

by expression:

В(t) = Р(min (τ1

= 1–[1 − В1

integrity is defined by the expression:

В(t) = Р(max(τ1

**5. Optimization**


The combination of these formal statements also can be used in the system's life cycle.

The approach for using the developed models, methods and software tools to analyze and optimize system processes is illustrated in **Figure 9**.

**Figure 9.** The approach to analyze and optimize system processes.
