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Chap 9 reliability
- 1. 1 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Structure Function Reliability of Systems of Independent Components Bounds on the Reliability Function System Life as a Function of Component Lives Expected System Lifetime Reliability Theory9
- 2. 2 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Structure Functions • Consider a system consisting n components, each component is either functioning or has failed. = otherwise0 functionscomponentif1 i xi• Indicator variable: ( )nxxx ,...,,x 21=• State vector: • Suppose that: whether the system is functioning or nor depends on x. There exists a function Φ(x) such that: ( ) = x x x isvectorstatewhenfailssystemif0 isvectorstatewhenfunctionssystemif1 φ ( ) SystemtheofFunctionStructureThe:xφ
- 3. 3 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: Serial Structure • System functions only if all components function. Therefore: ( ) ( ) ∏= == n i in xxxx 1 21 ,...,,minxφ
- 4. 4 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: Parallel Structure • System functions only if at least one component functions. Hence: ( ) ( ) ( )∏= −−== n i in xxxx 1 21 11,...,,maxxφ
- 5. 5 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: k-out-of-n Structure • System functions if at least k of the n components function. ( ) < ≥ = ∑ ∑ = = kxif kxif n i i n i i 1 1 0 1 xφ The 2-out-of-3 system
- 6. 6 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: four-component Structure ( ) ( )[ ] ( )( )[ ] [ ]434321 43114321 111,max xxxxxx xxxxxxxx −+= −−−==xφ
- 7. 7 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Monotone system • Assumption: replacing a failed component by a functioning one will not lead to a deterioration of the system, i.e., Φ(x) is an increasing function of x ( ) ( ) ( )yx φφ ≤⇒=≤ niyx ii ,...,2,1 • We consider only systems with the above property as monotone systems
- 8. 8 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Minimal Path Set • x is called a path vector if Φ(x) = 1. Furthermore, if Φ(y)=0 for all y < x then x is said to be a minimal path vector. < =≤ ⇔< isomeforxy nixy Note ii ii ,...,2,1 : xy • If x is a minimal path vector then the set A{i, xi = 1} is called a minimal path set. • ⇒ A minimal path set is a set of components which ensures the functioning of the system.
- 9. 9 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example ( ) ( )[ ] ( )[ ] [ ][ ]5435432121 54321 ,max,max xxxxxxxxxx xxxxx −+−+= =xφ There are four minimal path sets: {1,3,4},{1,5},{2,3,4},{2,5}
- 10. 10 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Minimal Path Set (cont.) • Indicator function αj(x) of the minimal path set Aj{j = 1, 2…,s} is defined as: ( ) ( ) ∏∈ ∈ == j j Ai ii Ai j xxThen minxα • The system functions if all components of at least one minimal path set function. Hence, ( ) = otherwise0 functionofcomponentsallif1 j j A xα ( ) ( ) ( ) ( ) ( ) ∏∈ ==⇒ = = = jAi i j j j j j xα jα jα maxmax allfor0if0 somefor1if1 xx x x x φ φ Note that αj(x) is a serial structure function of the components of the jth minimal path set ⇒ Any system can be considered as a parallel arrangement of serial systems
- 11. 11 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example ( ) ( ) ( )( )( )( )5251432431 5251432431 11111 ,,,max xxxxxxxxxx xxxxxxxxxx −−−−−= =xφ The minimal path sets are: A1={1,3,4},A2={1,5}, A3={2,3,4}, A4={2,5} This is exactly the same function as before, isn’t it?
- 12. 12 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: The bridge system ( ) ( ) ( )( )( )( )5241531432 5243253141 11111 ,,,max xxxxxxxxxx xxxxxxxxxx −−−−−= =xφ • minimal path sets : A1={1,4}, A2={1,3,5}, A3={2,5}, A4={2,3,4} • The structure function can be expressed as:
- 13. 13 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Minimal Cut Set • x is called a cut vector if Φ(x) = 0. Furthermore, if Φ(y) = 1 for all y > x then x is said to be a minimal cut vector. • If x is a minimal cut vector then the set C{i: xi = 0} is called a minimal cut set. • ⇒ A minimal cut set is a minimal set of components whose failure ensure the failure of the system.
- 14. 14 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Minimal Cut Set (cont.) • Indicator function βj(x) of the minimal cut set Cj{j = 1, 2…,k} is defined as: ( ) ( ) ( )∏∈ ∈ −−== j j Ci ii Ci j xxThen 11maxxβ • The system is not functioning only if all components of at least one minimal cut set are not functioning. Hence, ( ) = otherwise0 functionsofcomponentsoneleastatif1 j j C xβ ( ) ( ) ( ) ( ) ( ) ( )∏∏ = ∈ = ==⇒ = = = k j i Ci k j j j j x j j j 11 max allfor1if1 somefor0if0 xx x x x βφ β β φ Note that βj(x) is a parallel structure function of the components of the jth minimal cut set ⇒ Any system can be considered as a serial arrangement of parallel systems
- 15. 15 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: The bridge system • minimal cut sets : C1={1,2}, C2={1,3,5}, C3={4,5}, C4={2,3,4} • Equivalent structure: ( ) ( ) ( ) ( ) ( ) ( )( )[ ] ( )( )( )[ ] ( )( )( )[ ] ( )( )[ ]54432 53121 5443253121 1111111 1111111 ,max,,max,,max,max xxxxx xxxxx xxxxxxxxxx −−−−−−−× −−−−−−−= =xφ
- 16. 16 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Reliability of Systems of Independent Components • Suppose that the state of the ith component, Xi, is a random variable such that: { } { }011 =−=== iii XPXPp pi is the reliability of component i. • Reliability of the system: ( ){ } ( )nXXXPr ,...,,where1 21=== XXφ Because the random variables Xi; i = 1,2,…,n are independent, r can be expressed as a function of component reliabilities: { } ( )npppprr ,...,,where 21== p • Note: { } ( ){ } ( )[ ] ( ) variable.random)(Bernoulii10aisBecause 1 − === X XXp φ EPr φφ
- 17. 17 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: Serial System ( ) ( ){ } { } ∏= = === == n i i i p niXP Pr 1 ,...,2,1allfor1 1Xp φ
- 18. 18 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: Parallel System ( ) ( ){ } ( ){ } { } { } ( )∏= −−= ==−= === ==== n i i i i i i p niXP niXP XPPr 1 11 ,...,2,1allfor01 ,...,2,1somefor1 1max1Xp φ
- 19. 19 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: 2-out-of-3 System ( ) ( ){ } { } { } { } { } ( ) ( ) ( ) 321323121 321321321321 2 111 )1,1,0()1,0,1()0,1,1()1,1,1( 1 ppppppppp pppppppppppp PPPP Pr −++= −+−+−+= =+=+=+== == XXXX Xp φ
- 20. 20 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: k-out-of-n System • Suppose that pi = p for all i = 1,2,…,n ( ) ( ){ } ( ) in n ki i n i i pp i n kXP Pr − = = − = ≥= == ∑ ∑ 1 1 1 Xp φ
- 21. 21 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Reliability of Systems of Independent Components Proposition: If r{p}is the reliability of a system of independent components, then r{p}is an increasing function of p. { } ( )[ ] ( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ]XX XX Xp ,01,1 011 Proof iiii iiii EpEp XEpXEp Er φφ φφ φ −+= =−+== = ( ) ( ) ( ) ( ) { } ( ) ( )[ ] ( )[ ] ( ) ( )[ ] { } ipr E EEpr XXXX XXXXwhere i ii iiii niii niii allforinincreasingis 0,0,1:functionincreasinganisSince ,0,0,1 ,...,,0,,...,,0 ,...,,1,,...,,1 111 111 p XX XXXp X X ⇒ ≥− +−=⇒ = = +− +− φφφ φφφ φ φ
- 22. 22 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Reliability of Systems of Independent Components • Consider the following problem: A system of n different components is to be built from a stockpile containing exactly two of each type of component. The question is whether: 1. To build two separate systems with the probability of functioning is: { } { } ( )( ) ( )( )[ ]p'p rr P P −−−= −= 111 functionsystemsofneither1 functionsystemstwotheofoneleastAt
- 23. 23 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Reliability of Systems of Independent Components 2. Or to build a single system whose ith component functions if at least one of the number i components function. In this case the probability that the system will function is: { } { } ( )( ) ( )( )[ ]p'p rr P P −−−= −= 111 functionsystemsofneither1 functionsystemstwotheofoneleastAt
- 24. 24 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Reliability of Systems of Independent Components Theorem: For any reliability function r and vectors p, p’ ( )( )[ ] ( )[ ] ( )[ ]'111'111 pppp rrr −−−≥−−− “Replication at the component level is more effective than replication on the system level”
- 25. 25 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Reliability of Systems of Independent Components Proof: Let X1, X2,…, Xn ; X1’, X2’,…, Xn’ be mutually independent 0 - 1 random variables with { } { } ( ){ } ( )( ) ( )( )[ ] ( )( )[ ] ( )( ) ( ) ( ) ( )( )[ ] ( ) ( )( )[ ] ( ) ( )( ){ } ( ) ( ){ } ( )[ ] ( )[ ]'111 0,01 1,max ,maxHence, max:oftymonotoniciBy max 1111,max 1'1 '' ' pp X'X X'X X'Xp'1p11 X'XX'X, X'X,p'1p11 rr P P Er and Er ppXXP XPpXPp iiii iiii −−−= ==−= == ≥−−− ≥ =−−−⇒ −−−== ==== φφ φφ φφ φφφφ φ ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }nn nnnn yxyxyx yxyxyxyyyxxx Notation ,max,...,,max,,max,max ,...,,,...,,;,...,, : 2211 22112121 = =⇒== yx xyyx
- 26. 26 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example • Consider the case of two types of component with 2 121 == pp • Replication at the system level: • Replication at the component level: ( )[ ] ( )[ ] 16 7 2 1 2 1 1 2 1 2 1 11'111 = − −−=−−−= pp rrrs ( )( )[ ] 16 7 16 9 4 3 2 1 11, 2 1 11 222 >= = −− −−=−−−= rrrC p'1p11
- 27. 27 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Bounds on the Reliability Function Method of Inclusion and Exclusion •Formula for the probability of the union of the events E1, E2, . . . , En: ( ) ( ) ( ) ( ) ( )n n kji kji ji ji n i i n i i EEEPEEEPEEPEPEP ...1... 21 1 11 + <<<== −+−+−= ∑∑∑ • Hence: ( ) ( ) ( ) ( ) ( ) ( ) ... ... 11 11 11 ≤ ≥ +−≤ −≥ ≤ ∑∑∑ ∑∑ ∑ <<<== <== == kji kji ji ji n i i n i i ji ji n i i n i i n i i n i i EEEPEEPEPEP EEPEPEP EPEP
- 28. 28 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Bounds on the Reliability Function Method of Inclusion and Exclusion (cont) •Let A1, A2, . . . ,As denote the minimal path sets of a given structure φ, and define the events E1, E2, . . . , Es by Ei = {all components in Ai function} •Since the system functions if and only if at least one of the events Ei occur, we have ( ) ( ) ( ) ∏∏∏ ∪∪∈∪∈∈ === kjijii AAAl lkji AAl lji Al li pEEEPpEEPpEP ;; • where: ( ) ( ) ( ) ( ) ... 1 11 ≤ −≥ ≤ = ∑∑ ∑ <= = ji ji s i i s i i s i EEPEP EPEPr p
- 29. 29 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: The bridge system ippi allfor= • minimal path sets : A1={1,4}, A2={1,3,5}, A3={2,5}, A4={2,3,4} • Because exactly five of the six unions of Ai and Aj contain four components (the exception being A2 ∪ A4, which contains all five components), we have ( ) ( ) ( ) ( ) 3 42 2 31 ; pEPEPpEPEP ==== ( ) ( ) ( ) ( ) ( ) ( ) 5 42 4 4332413121 ; pEEP pEEPEEPEEPEEPEEP = ===== • Hence, the first two inclusion–exclusion bounds yield ( ) ( ) ( )325432 252 pprpppp +≤≤−−+ p
- 30. 30 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Bounds on the Reliability Function Second Method for Obtaining Bounds on r(p): •Let A1, A2, . . . ,As denote the minimal path sets of a given structure φ, and define the events D1, D2, . . . , Ds by Di = {at least one component in Ai has failed} • since the system will have failed if and only if at least one component in each of the minimal path sets has failed we have: ( ) ( ) ( ) ( ) ( )12112121 .........1 −==− sss DDDDPDDPDPDDDPr p • We have: ( ) ( )iii DPDDDP ≥−11... • Hence, ( ) ( ) ( ) ( ) ∏ ∏∏ ∏ −−=−≤⇔ ≥− ∈i Aj j i i i i i pDPr DPr 111 1 p p
- 31. 31 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Bounds on the Reliability Function Second Method for Obtaining Bounds on r(p) (cont): •let C1, . . . ,Cr denote the minimal cut sets and define the events U1, . . . ,Ur by Ui = {at least one component in Ci is functioning} • since the system will function if and only if all of the events Ui occur, we have: ( ) ( ) ( ) ( ) ( ) ( ) ( )∏ ∏∏ −−=≥ = = ∈ − i Cj j i i rr r i pUP UUUUPUUPUP UUUPr 11 ...... ... 121121 21p • Finally, ( ) ( ) ∏ ∏∏ ∏ −−≤≤ −− ∈∈ i Aj j i Cj j ii prp 1111 p
- 32. 32 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE System Life as a Function of Component Lives • Letting F(t) denote the distribution of system lifetime ( ) ( ) { } { } ( ) ( )( )tPtPr tP tPtFtF n,..., at timegfunctioninissystem lifesystem1 1= = >=−= Where: ( ) { } { } ( )tF tiP tiPtP i i = >= = oflifetime at timegfunctioniniscomponent Hence, ( ) ( ) ( )( )tFtFrtF n,...,1=
- 33. 33 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example ( ) ( ) ( )∏∏ == =⇒= n i i n i i tFtFpr 11 p • Serial System: ( ) ( ) ( ) ( )∏∏ == −=⇒−−= n i i n i i tFtFpr 11 111p • Parallel System:
- 34. 34 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE The Failure Rate Function • the failure rate function λ(t) of continuous distribution G represents the probability intensity that a t-year-old item will fail. ( ) ( ) ( ) ( ) ( ) dt tdG tgwhere tG tg t ==λ • G is an increasing failure rate (IFR) distribution if λ(t) is an increasing function of t. • G is a decreasing failure rate (DFR) distribution if λ(t) is a decreasing function of t.
- 35. 35 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Example: The Weibull Distribution ( ) ( ) 0,1 ≥−= − tetG t α λ • A random variable is said to have the Weibull distribution if its distribution is given, for some λ > 0, α > 0, by • The failure rate function for a Weibull distribution: ( ) ( ) ( ) ( ) ( ) 1 1 − − −− == α λ αλ λαλ λλα λ α α t e te t t t • the Weibull distribution is IFR when α ≥ 1, and DFR when 0 < α ≤1. • when α = 1, G(t) = 1 − e−λt is the exponential distribution.
- 36. 36 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE The Hazard Function ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) .ondistributitheofthe: , ln 1 1 0 00 Fctionhazard fundsstwhere etFHence tFds sF sf dss tF tf t t t tt ∫ ∫∫ =Λ = −= − =⇒ − = Λ− λ λ λ • A distribution F is said to have increasing failure on the average (IFRA) if: ( ) ( ) 0forinincreases0 ≥= Λ ∫ tt t dss t t t λ
- 37. 37 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Expected System Lifetime • We have: { } ( )( ) ( ) ( ) ( )( )tFtFtwheretrtP n,...,lifesystem 1==> FF • And: { } ( ) ( ) ( ) [ ]XEdyyyfdxdyyfdydxyfdxxXP y x ====> ∫∫∫∫∫∫ ∞∞∞ ∞∞ 00 000 • Thus, [ ] ( )( )∫ ∞ = 0 lifesystem dttrE F
- 38. 38 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Examples ( ) ( )( ) > ≤≤ − = = > ≤≤ = 10,0 100, 10 10 Therefore, 3,2,1 10,1 100,10 3 t t t tr i t tt tFi F • A Series System of Uniformly Distributed Components: Consider a series system of three independent components each of which functions for an amount of time (in hours) uniformly distributed over (0, 10). Hence, r(p) = p1p2p3 [ ] 2 5 10 10 10 lifesystem 1 0 3 10 0 3 == − = ∫ ∫ dyy dt t E
- 39. 39 Assoc. Prof. Ho Thanh Phong Probability Models International University – Dept. of ISE Examples ( ) ( ) > ≤≤ = −++= 1,1 10, 2 321323121 t tt tF pppppppppr i p • A Two-out-of-Three System: Consider a two-out-of-three system of independent components, in which each component’s lifetime is (in months) uniformly distributed over (0, 1). [ ] ( ) ( )[ ] [ ] 2 1 2 1 1 23 1213lifesystem 1 0 32 1 0 32 =−= −= −−−= ∫ ∫ dyyy dtttE