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By Hoang Pham

Security and probability Modeling provides the newest theories and techniques of safeguard and hazard with an emphasis on security and chance in modeling. It covers purposes in numerous components together with transportations and safety danger tests, in addition to purposes relating to present themes in safeguard and hazard. protection and threat Modeling is a necessary source for realizing the newest advancements in either qualitative and quantitative tools of defense and threat research and their functions in working environments.

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Reliability Modelling of Complex Systems 29 We will analyse the reliability of the subsystem S3 only. This subsystem consists of kn = 3 identical piping lines, each composed of ln = 360 steel pipe segments. In each of these lines there are pipe segments with Weibull reliability function RðtÞ ¼ exp½À0:0000000008t4 Š for t ! 0: We suppose that the system is good if at least two of its piping lines are not failed. Thus, according to Definitions 14–15, it may be considered as a homogeneous and regular series-‘‘2 out of 3’’ system, and according to Proposition 6, assuming an ¼ 1 1=a ðbln Þ ¼ 1 ð0:000000288Þ1=4 ; bn ¼ 0; and using (3), its reliability function is given by ð2Þ R3;360 ðtÞ   X 1    à 3 t ¼ exp Ài Á 0:000000288t4 an i i¼0  Ã3Ài for t !

N; and exponential reliability functions RðtÞ ¼ exp½ÀktŠ for t ! 0; k [ 0: Its lifetime and its reliability function, respectively, are given by T ð0Þ ¼ min fTi1 g; 1 i n Rn ðtÞ ¼ ½Rðtފn ¼ exp½ÀkntŠ; t ! 0: In order to improve the reliability of this series system the following exemplary methods can be used: – replacing the system components by the improved components with reduced failure rates by a factor q, 0 \ q \ 1, – a warm duplication (a single reservation) of system components, – a cold duplication of system components, – a mixed duplication of system components, – a hot system duplication, – a cold system duplication.

E 2 jl n 2 kn 6 Y 61 À 4 E k n j2 ... ;m Þ E kn j1 E 2 jm ... ;lknkn ðtÞ ¼ ... ... Fig. þrli mi " 1À li Y j¼1 Rij ðtÞ #ri " li Y j¼1 #1Àri Rij ðtÞ 3 7 7 for t 2 ðÀ1;1Þ; 5 " i ¼ li À mi ; i ¼ 1; 2; . ; kn : where m The scheme of a regular ‘‘m out of ln’’-series system is given in Fig. 15. Reliability Modelling of Complex Systems 31 Definition 17 The two-state ‘‘mi out of li’’-series system is called homogeneous if its component lifetimes Tij have an identical distribution function FðtÞ ¼ PðTij t 2 ðÀ1; 1Þ; tÞ; i ¼ 1; 2; .

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