5.2. Estimating the mean residual time before the next parameter abnormalities

Unfortunately, in the world the universal approach to adequate prognosis of the future parameter conditions on the basis of current data is not created yet. The uncertainty level is too high. Nevertheless, in practice for each concrete case, subjective expert estimations, regression analysis of collected data, and simulation are often used. And, probabilistic models applied in some cases contain many simplifications, and they frequently do not consider an infrastructure of complex systems, heterogeneity of threats, distinctions in technologies of the control, and recovery of integrity for various elements of these systems [2–3]. The same aspects and also rarity of many random events (with some exceptions) do an ineffective statistical estimation of residual time before the next parameter abnormalities. At the same time, scientifically proven prognosis of a residual time resources is necessary for acceptance of preventive measures on timely elimination of the abnormality reasons. The above-stated characterizes an actuality of this and similar researches for different industrial areas [11–17].

Traced conditions of parameters are data about a condition before and on the current moment of time, but always the future is more important for all. With the use of current data, responsible staff (mechanics, technologists, engineers, etc.) should know about admissible time for work performance to maintain system operation. Otherwise, because of ignorance of a residual time resource before abnormality, the necessary works are not carried out. That is, because of ignorance of this residual time, measures for prevention of negative events after parameter abnormalities (failures, accidents, damages, and/or the missed benefit because of equipment time out) are not undertaken. And, on the contrary, knowing residual time before abnormality, these events may be avoided, or the system may be maintained accordingly. For monitored critical system, the probabilistic method to estimate the mean residual time before the next parameter abnormalities for each element and whole system is proposed.

For avoiding the possible crossing of a border of "Abnormality," a prediction of residual time, which is available for preventive measures, according to gathered data about parameter condition fluctuations considering ranges, should be carried out. For prediction the following are proposed: (1) a choice of probabilistic models for construction (PDF of time before the next abnormality for one element ("black box")), (2) development of the algorithm of generation (PDF of time before the next abnormality for complex system), and 3) formalization of calculative methods of estimating the mean residual time before the next parameter abnormal-

Probabilistic Methods and Technologies of Risk Prediction and Rationale of Preventive Measures by Using…

http://dx.doi.org/10.5772/intechopen.75109

41

The method allows to estimate residual time before the next parameter abnormality (i.e., time

The method allows to estimate residual time before the next parameter abnormality Tresid(1) for a given admissible risk Radm.(Treq) to lose integrity. The estimated Tresid(1) is the solution t0 of equation:

Here, R(Toccur, t, Tbetw, Tdiag, Тerr., Treq.) is the risk to lose integrity; it is addition to 1 for probability P(Treq) of providing system integrity ("probability of success"), and for calculations formulas (1)–(7) are used (see SubSection 3.1 of this article). So, for exponential PDF, formula

ð7Þ

ities for monitored critical system.

Figure 13. Example of system decomposition.

(1) transforms into formula.

before the first next coming into "red" range) [14].

concerning of unknown parameter t, i.e., Tresid(1) = t0.

By principles of system engineering (e.g., according to ISO/IEC/IEEE 15288), the complex system is decomposed to compound subsystems and elements with formal definition of states (see Figure 13).

For every valuable subsystem (element), monitored parameters are chosen, and for each parameter, the ranges of possible values of conditions are established: "In working limits," "Out of working range, but inside of norm," and "Abnormality" (interpreted similarly light signals ("green," "yellow," "red") (see Figure 14). The condition "Abnormality" characterizes a threat to lose system integrity (on the logic level, this range "Abnormality" may be interpreted analytically as failure, fault, unacceptable risk or quality, etc.).

Probabilistic Methods and Technologies of Risk Prediction and Rationale of Preventive Measures by Using… http://dx.doi.org/10.5772/intechopen.75109 41

Figure 13. Example of system decomposition.

where expression in square brackets is a probability of successful operation of analyzed critical infrastructure. Depending on the made risk definition in special cases, it can be interpreted as probability of safe or reliable operation or probability of norm observance for critical parameters of the equipment or others in the conditions of associated potential threats. The case

Unfortunately, in the world the universal approach to adequate prognosis of the future parameter conditions on the basis of current data is not created yet. The uncertainty level is too high. Nevertheless, in practice for each concrete case, subjective expert estimations, regression analysis of collected data, and simulation are often used. And, probabilistic models applied in some cases contain many simplifications, and they frequently do not consider an infrastructure of complex systems, heterogeneity of threats, distinctions in technologies of the control, and recovery of integrity for various elements of these systems [2–3]. The same aspects and also rarity of many random events (with some exceptions) do an ineffective statistical estimation of residual time before the next parameter abnormalities. At the same time, scientifically proven prognosis of a residual time resources is necessary for acceptance of preventive measures on timely elimination of the abnormality reasons. The above-stated characterizes an actuality of

Traced conditions of parameters are data about a condition before and on the current moment of time, but always the future is more important for all. With the use of current data, responsible staff (mechanics, technologists, engineers, etc.) should know about admissible time for work performance to maintain system operation. Otherwise, because of ignorance of a residual time resource before abnormality, the necessary works are not carried out. That is, because of ignorance of this residual time, measures for prevention of negative events after parameter abnormalities (failures, accidents, damages, and/or the missed benefit because of equipment time out) are not undertaken. And, on the contrary, knowing residual time before abnormality, these events may be avoided, or the system may be maintained accordingly. For monitored critical system, the probabilistic method to estimate the mean residual time before the next

By principles of system engineering (e.g., according to ISO/IEC/IEEE 15288), the complex system is decomposed to compound subsystems and elements with formal definition of states

For every valuable subsystem (element), monitored parameters are chosen, and for each parameter, the ranges of possible values of conditions are established: "In working limits," "Out of working range, but inside of norm," and "Abnormality" (interpreted similarly light signals ("green," "yellow," "red") (see Figure 14). The condition "Abnormality" characterizes a threat to lose system integrity (on the logic level, this range "Abnormality" may be interpreted

λ<sup>∑</sup> = λRMS means full capture of critical infrastructure by RMS capabilities.

40 Probabilistic Modeling in System Engineering

this and similar researches for different industrial areas [11–17].

parameter abnormalities for each element and whole system is proposed.

analytically as failure, fault, unacceptable risk or quality, etc.).

(see Figure 13).

5.2. Estimating the mean residual time before the next parameter abnormalities

For avoiding the possible crossing of a border of "Abnormality," a prediction of residual time, which is available for preventive measures, according to gathered data about parameter condition fluctuations considering ranges, should be carried out. For prediction the following are proposed: (1) a choice of probabilistic models for construction (PDF of time before the next abnormality for one element ("black box")), (2) development of the algorithm of generation (PDF of time before the next abnormality for complex system), and 3) formalization of calculative methods of estimating the mean residual time before the next parameter abnormalities for monitored critical system.

The method allows to estimate residual time before the next parameter abnormality (i.e., time before the first next coming into "red" range) [14].

The method allows to estimate residual time before the next parameter abnormality Tresid(1) for a given admissible risk Radm.(Treq) to lose integrity. The estimated Tresid(1) is the solution t0 of equation:

$$\mathbf{R(T\_{occur}, t\_{\nu} T\_{betw}, T\_{diag}, T\_{err}, T\_{req}) = R\_{adm.}(T\_{req})} \tag{7}$$

concerning of unknown parameter t, i.e., Tresid(1) = t0.

Here, R(Toccur, t, Tbetw, Tdiag, Тerr., Treq.) is the risk to lose integrity; it is addition to 1 for probability P(Treq) of providing system integrity ("probability of success"), and for calculations formulas (1)–(7) are used (see SubSection 3.1 of this article). So, for exponential PDF, formula (1) transforms into formula.

Figure 14. Elementary ranges for parameter conditions.

This formula is used for Eq. (7).

Toccur is the mathematical expectation of PDF Ωoccur (τ); it is defined by parameter statistics of transition from "green" into "yellow" range (see Figure 3). The other parameters Tbetw and Tdiag in formula (7) are known. The main practical questions are as follows: what about Treq. and what about the given admissible risk Radm.(Treq)? For answering we can use the properties of function R(Toccur, t, Tbetw, Tdiag, Тerr., Treq.):

The method is implemented by RMS. At once after crossing "yellow" border from "green," the automatic prediction of the mean residual time before the next parameter abnormalities (from the first input at the "yellow" range to the first input in the "red" range) is displayed (see

Probabilistic Methods and Technologies of Risk Prediction and Rationale of Preventive Measures by Using…

http://dx.doi.org/10.5772/intechopen.75109

43

Adequate reaction of responsible staff in real time is transparent for all interested parties.

5.3. About some effects from adequate probabilistic methods and technology applications

Some effects from the proposed adequate probabilistic methods and technologies of RMS are

Example 5.3.1. According to statistics from multifunctional safety system (MFSS), a frequency of occurrence of the latent or obvious threats is equal to once a month, and an average time of development of threats (from occurrence of the first signs of a critical situation up to failure) is about 1 day. A work shift is equal to 8 hours. The system control is used once for work shift, and a mean duration of the system control is about 10 minutes (it is supposed that recovery of object integrity is expected also for 10 minutes). The workers (they may be mechanics, technologists, engineers, etc.) of medium-level and skilled workers are capable to revealing signs of a critical situation after their occurrence, and workers of the initial level of proficiency are incapable. Medium-level workers can commit errors on the average not more often once a month, and skilled workers are not more often once a year. How consideration of the qualification level influences on predicted risks to lose object safety for a year and for 10 years?

The results of modeling. For workers of the initial level of proficiency, risks to lose object safety are near 1 (losses of integrity are inevitable). For workers of medium-level of proficiency, risk to lose object safety for a year is about 0.007 and for 10 years is about 0.067, and for skilled

estimated on the level of predicting risks to lose object safety (integrity) by PDF [16].

Figure 15).

Figure 15. Example of residual time and comments.


It means that the such maximal x exists when t = x and Treq. = x and 0 < R(Toccur, x, Tbetw, Tdiag, Тerr., x) < 1. That is, the residual time before the next parameter abnormality (i.e., time before the first next coming into "red" range) is equal to the defined x with the confidence level of admissible risk R(Toccur, x, Tbetw, Tdiag, Тerr., x).

For example, if Toccur = 100, Tbetw = 8 hours, Tdiag = 1 hour, Тerr. = 0, and Radm. = 0.05, unknown x is defined from equation, considering (1), (7):

So, if Toccur = 100 days, for Radm. = 0.01 residual time x ≈ 2.96 weeks (considering decisions of recovery problems of integrity every 8 hours).

Probabilistic Methods and Technologies of Risk Prediction and Rationale of Preventive Measures by Using… http://dx.doi.org/10.5772/intechopen.75109 43

Figure 15. Example of residual time and comments.

This formula is used for Eq. (7).

42 Probabilistic Modeling in System Engineering

of function R(Toccur, t, Tbetw, Tdiag, Тerr., Treq.):

Figure 14. Elementary ranges for parameter conditions.

admissible risk R(Toccur, x, Tbetw, Tdiag, Тerr., x).

is defined from equation, considering (1), (7):

recovery problems of integrity every 8 hours).

bigger, to lose integrity is less.

Toccur is the mathematical expectation of PDF Ωoccur (τ); it is defined by parameter statistics of transition from "green" into "yellow" range (see Figure 3). The other parameters Tbetw and Tdiag in formula (7) are known. The main practical questions are as follows: what about Treq. and what about the given admissible risk Radm.(Treq)? For answering we can use the properties

• If parameter t increases from 0 to ∞ for the same another parameters, the function R(…, t, …) is monotonously decreasing from 1 to 0, i.e., if the mean activation time of occurred danger (threat: from the first input at the "yellow" range to the first input in the "red" range) is

• If parameter Treq increases from 0 to ∞ for the same other parameters, the function R(…,Treq)

It means that the such maximal x exists when t = x and Treq. = x and 0 < R(Toccur, x, Tbetw, Tdiag, Тerr., x) < 1. That is, the residual time before the next parameter abnormality (i.e., time before the first next coming into "red" range) is equal to the defined x with the confidence level of

For example, if Toccur = 100, Tbetw = 8 hours, Tdiag = 1 hour, Тerr. = 0, and Radm. = 0.05, unknown x

So, if Toccur = 100 days, for Radm. = 0.01 residual time x ≈ 2.96 weeks (considering decisions of

is monotonously increasing from 0 to 1, i.e., for large Treq risk approaches to 1.

The method is implemented by RMS. At once after crossing "yellow" border from "green," the automatic prediction of the mean residual time before the next parameter abnormalities (from the first input at the "yellow" range to the first input in the "red" range) is displayed (see Figure 15).

Adequate reaction of responsible staff in real time is transparent for all interested parties.
