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- 1. Efficient Combinatorial Models for Reliability Analysis of Complex Dynamic Systems (基 C l D i S t (基 于组合模型的复杂动态系统可 靠性分析) Dr. Liudong Xing (邢留冬博士) ©2011 ASQ & Presentation Xing Presented live on Nov 09th, 2011http://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_liability Calendar/Webinars ‐_Chinese/Webinars_‐_Chinese.html
- 2. ASQ Reliability Division ASQ Reliability Division Chinese Webinar Series Chinese Webinar Series One of the monthly webinars One of the monthly webinars on topics of interest to reliability engineers. To view recorded webinar (available to ASQ Reliability Division members only) visit asq.org/reliability ) / To sign up for the free and available to anyone live webinars visit reliabilitycalendar.org and select English Webinars to find links to register for upcoming eventshttp://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_liability Calendar/Webinars ‐_Chinese/Webinars_‐_Chinese.html
- 3. Efficient Combinatorial Models forReliability Analysis of Complex Dynamic Systems(基于组合模型的复杂动态系统可靠性分析) Presented by Dr. Liudong Xing (邢留冬) E-mail: lxing@umassd.edu E mail: lxing@umassd edu Electrical and Computer Engineering Dept. University of Massachusetts Dartmouth, MA, USA www.massachusetts.edu www massachusetts edu ASQ Reliability Division Webinar Series
- 4. http://technicalhut.blogspot.com/2011/09/robo-earth-database-to-network-robots.htmlhttp://www.0592en.com/class/trans/2011/1031/1406.html, http://www.metrolic.com/us-nuclear-power-plant-funds-remain-unused-168212/http://www.geekwithlaptop.com/cloud-computing-takes-us-into-the-future-of-technology-chrome-os-leads-the-way, http://spie.org/x14634.xml?ArticleID=x14634 2http://www.infrastructurist.com/2009/05/19/a-vibrant-us-train-industry-would-employ-more-people-than-car-makers-do-now/ -- Image Sources
- 5. MotivationComputing and engineering systems areevolving t l i toward enabling much l d bli h largercollaboration & handling more complicatedmissions.The iTh increasing complexity and scale im l si m l xit d s l implythat reliability problems will not only continueto be a challenge but also require moreefficient models and solutions 3
- 6. @ This Talk --Reliability Analysis of Complex Dynamic SystemsEvaluation Methods p Complex Behavior Analytical methods Multiple states (多状态) o Combinatorial methods Multiple phases (多阶段) (fault trees, decision Sequence dependence (顺序相依) diagrams) Dynamic sparing (动态备用) o State space-based Imperfect coverage (不完全覆盖) methods (Markov models) Common-cause failures (共因故障) (共 障) Simulation methods Functional dependence (功能相依) Measurement-based Competing failures (竞争失效) p g Acknowledgment: US National Science Foundation (NSF) No. 0614652 & 0832594 & 1112947 4
- 7. AgendaOverview of complex b h iO i f l behaviorReliability and sensitivity analysis of multi- multistate systems 5
- 8. Multi-State (多状态)System & components: more than two levels ofpperformance (or states) varying from perfect poperation to complete failureBehaviors modeled: shared loads, performance ,pdegradation, imperfect coverage, multiple failuremodes, etc.Applications: power systems, transmission networks,communication networks circuits etc networks, circuits,Challenge:o dependence among multiple states 6
- 9. Multi-Phase (多阶段)A system supporting a mission characterized bymultiple, consecutive, and non-overlapping phases ofoperationSystem components subject to different stresses,environmental conditions, and reliability requirements i i t l diti s d li bilit i ts indifferent phasesApplications: aerospace (aircraft, rockets, spacecraft),nuclear power, airborne weapon systems, etcChallenge:o dynamics in system configuration, failure criteria, and y y g , , component failure behavioro s-dependencies across phases for a given component 7
- 10. Sequence Dependence (顺序相依)The order that fault events occur is important to thesystem reliabilityChallenge: sequence-dependent system f l h ll d d failurecriteria Failure F il Primary: P Switch: Sw Standby: P S Sw P S • Sw P: system fails Modeled using priority AND •P Sw: system OK y g gate in fault tree analysis y 8
- 11. Dynamic Sparing (动态备用) λPOne module is on-line & mponentsoperational, and one or λS Hot commore modules serve as t τ1 τ2standby units. λPWhen the on-line module components λS Coldexperiences a fault andthe fault is detected, it is c t τ1 τ2removed and replaced with λPa standby unit. omponents αS λS WarmChallenge: time/order-dependent failure co t τ1 τ2behavior 9
- 12. I Imperfect F lt C f t Fault Coverage (不完全覆盖)Imperfect detection, location or recovery of a detection location,component fault may cause an extensive damage to theentire system, despite presentence of redundancies. system redundanciesExtent of an uncovered fault damage can exhibitmultiple levels in hierarchical systems: if anundetected error escapes from one level, it may becovered at a higher level level.Challenge: multiple failure modes 10
- 13. Common Cause Failures (共因故障) Common-CauseSimultaneous failure of multiple components due to acommon causeChallenge: multiple dependent component failures External Cause Common Cause Global Effect on a Failure Internal Cause y y System/Subsystem (Propagated Failure) Selective Effect on System Components 11
- 14. Functional Dependence (功能相依)Occurrence of some event (trigger) causes othercomponents ( p n n components) to become mp n n (dependent mp n n ) minaccessible or unusableCascading f ilC di failures: multiple f il lti l failures i iti t d by th initiated b thetrigger of one component in the system resulting in achain reaction ord i effect ( h i i domino ff (common i power ingrids) FDEP FDEP A B C ...... 12
- 15. Competing Failures (竞争失效) Occur in systems subject to both functional dependence (FDEP) and propagated failures (PF) d d d d f il PF has different consequences due to competition in the time domain between trigger failure and failure propagated from dependent components components.Trigger f gg failure PF of dependent components: f f p mp failure isolationPF of dependent components Trigger failure: system fails 13
- 16. AgendaOverview of complex behaviorReliability and sensitivity analysis of multi- l l l lstate systems (MSS) y ( )o Basic conceptso MSS analysis methods l h do Examples E mp 14
- 17. MSS R li bilit ReliabilityMSS reliability at level d :o probability that the system performance level is greater than or equal to d. MRd = P (ϕ ( x) ≥ d )o φ( ) system structure function (x): f 15
- 18. MSS S Sensitivity M iti it MeasuresQuantify importance of components, and helpprioritize reliability improvement activitiesComposite importance measures (CIM): evaluatecontribution of a m f multi-state component as a whole to mpMSS reliabilityo Example: Birnbaum or average of the Sum of Absolute Deviation (SAD) ∑ ωi j =1 P(ϕ ( x) < d | x i = bij ) − P(ϕ ( x) < d ) MI SAD = ωi −1 i 16
- 19. MSS A l i M th d (1) Analysis MethodsSimulation-based methodso computationally expensive and time-consuming p y p go approximate resultso a complete new simulation must be performed when parameter values changeStateSt t space-based methods (M k models) b d th d (Markov d l )o more sever state explosion problem than analyzing binary systemsMulti-state minimal path/cut vectors (MMPV/MMCV) po doubly exponential complexity 17
- 20. MSS Analysis Methods (2) Decision diagrams (决策图)-based methodso Multi-state binary decision diagrams (MBDD)o Logarithmically-encoded binary decision diagrams Logarithmically encoded (LBDD)o Multi-state multi-valued decision diagrams (MMDD) 18
- 21. An Illustrative Example A Ill t ti E lEach board has 4 states B1 P1 M1o Bii,4 (both P & M are functional) 4 Buso Bi,3 (M is functional, P is down) B2o Bii,2 (P is functional, M is down) 2 P2 M2o Bi,1 (both P & M are down)The system has 3 stateso S3 (at least one P & both M are functional) f i l)o S2 (at least one P & exactly one M are functional)o S1 (no P or M is functional) 19
- 22. MBDD 4 Boolean variables to encode 4 board states o (B1,1, B1,2, B1,3, B1,4) for board B1 o (B2,1, B2,2, B2,3, B2,4) for board B2 , , ,3 , Board State B1,1 Board State B1,2 Board State B1,3 Board State B1,4 o numerous variables; o special operations to handle state dependencies in model generation and evaluationX. Zang, D.Wang, H. Sun, and K. S. Trivedi, “A BDD-based algorithm for analysis of multistate systemswith multistate components,” IEEE Trans. Computers, vol. 52, no. 12, pp. 1608–1618, Dec. 2003 20
- 23. LBDD 2 auxiliary Boolean variables to encode 4 board states y o (v1, v2) for board B1 o (w1, w2) for board B1 v1 v2 B1 states 1,3 0 0 B1,1 v1 0 1 B1,2 0 1 v2 0 1 1 0 B1,3 1 1 1 1 B1,4 o binary logic; no dependence among fewer auxiliary variables o state encoding and decoding are neededA. Shrestha and L. Xing, “A Logarithmic Binary Decision Diagrams-Based Method for Multistate SystemsAnalysis,” IEEE Trans. Reliability, Vol. 57, No. 4, pp. 595-606, Dec. 2008. 21
- 24. MMDD 1 multi-valued variable per multi-state component multi valued multi state o (B1) for board B1 o (B2) for board B2 B1 B1 B1 B1 1 4 1 4 1 4 1 4 2 3 2 3 2 3 2 3 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Board State B1,1 Board State B1,2 Board State B1,3 Board State B1,4 o no dependence among multi-valued variables o straightforward model generation and evaluation L. Xing and Y. Dai, “A New Decision Diagram Based Method for Efficient Analysis on Multi-State Systems,”IEEE Trans. Dependable and Secure Computing, vol. 6, no. 3, pp. 161-174, Jul.-Sep. 2009.S. V. Amari, L. Xing, A. Shrestha, J. Akers, and K. S. Trivedi, “Performability Analysis of Multi-StateComputing Systems Using Multi-Valued Decision Diagrams,” IEEE Trans. on Computers, vol. 59, no. 10, pp.1419-1433, 2010. 22
- 25. MFTExample Computer System MBDD, LBDD, MMDD 1 4 0 3 1 0 0 1 MBDD LBDD MMDD 23
- 26. Performance Comparison Microelectronics Center of North Carolina (MCNC) BBenchmarks h k o model size o # recursive calls o top-down recursive evaluation ti t d i l ti time o bottom-up evaluation timeA. Shrestha, L. Xing, and Y. Dai, “Decision Diagram-Based Methods, and Complexity Analysis for MultistateSystems,” IEEE Trans. Reliability, vol. 59, no. 1, pp. 145-161, Mar. 2010. 24
- 27. Name Inpu Outp Product t u Terms tMCNC Benchmarks N B h k 5xp1 9sym 7 9 10 1 75 87 alu2 10 8 68 alu4 14 8 1028Originally designed for b12 15 9 431Boolean switching functions bw 5 28 87 clip 9 5 167 con1 7 2 9Adapted to form MSS with p inc 7 9 34multistate components mdiv7 8 10 256 misex1 8 7 32 y po Each binary output ≡ a misex2 25 18 29 system state misex3c 14 14 305o A group of binary inputs ≡ g p y p postal 8 1 25 rd53 d 3 5 3 32 multistate component rd73 7 3 141o E.g., 4 binary inputs form 16- rd84 8 4 256 state components sao2 10 4 58 sn74181 14 8 1132 squar5 5 8 32 xor5 5 1 16 Z5xp1 7 10 128 25 Z9sym 9 1 420
- 28. Model Size M d l SiWMBDD > WLBDD > WMMDD 26
- 29. 10 100 1000 10000 100000 1000000 xor5 rd53 squar5 con1 misex1 postal rd73 inc bw rd84 Top-down 5xp1 Z9sym Z5xp1 9sym clip mdiv7 RMBDD > RMMDD > RLBDD MBDD sao2 misex2 alu2 LBDD b12 # of Top down Recursive Calls misex3c sn74181 alu4 MMDD27
- 30. Top-down R T d Recursive E l ti Time si Evaluation Ti TMBDD > TLBDD > TMMDD(in ms, time for decoding states is included for LBDD) MBDD LBDD MMDD 1000 100 10 1 0.1 0.01 v7 2 1 bw sq 3 73 84 1 r5 u2 u4 m n1 Z9 1 m o2 r5 c al p m x1 m x2 ym sn 3c xp 5 18 in p b1 cli st sy xo ua di co sa al al rd rd rd 5x ise ise x 9s Z5 po 74 m ise 28
- 31. 0 01 0.01 0.1 1 10 100 xo r5 rd 5 po 3 st al co n1 rd 7 sq 3 ua Z9 r 5 B tt sy m m ise x1 rd 84 5x p1 9s ym in c sa o Z5 2 xp 1 bw m ise x2 al TMBDD > TLBDD > TMMDD u2 MBDD b1 2 E l ti Ti cli p m Bottom-up Evaluation Time m div7 LBDD ise x sn 3c 74 18 1 al u4 MMDD29
- 32. S mm SummaryLBDD is a tradeoff that transforms multi-state domain into an equivalent auxiliary q ybinary domain, but offers reduced systemmodel size than MBDD MBDD.In general, MMDD is more efficient thanMBDD and LBDD. 30
- 33. AT Transmission N t smissi Network k 2 The system must 1 3 supply a demand >= 3 l d d s 5 7 t units from s to t. 4 6 Component Transmission Capacity State Probability 1 0 1 2 3 4 0.10 0.05 0.15 0.35 0.35 2 0 1 2 - - 0.10 0.05 0.85 - - 3 0 1 2 - - 0.10 0 10 0.05 0.85 0 05 0 85 - - 4 0 1 2 3 - 0.20 0.10 0.45 0.25 - 5 0 1 2 - - 0.10 0.05 0.85 - - 6 0 1 2 - - 0 10 0.10 0 05 0 85 0.05 0.85 - - 7 0 1 2 3 4 0.15 0.15 0.05 0.45 0.20A. Shrestha, L. Xing, D. W. Coit, “An Efficient Multi-State Multi-Valued Decision Diagram-Based Approachfor Multi-State System Sensitivity Analysis,” IEEE Trans. Reliability, vol. 59, no. 3, pp. 581-592, Sept. 2010. 31
- 34. x1 MMDD-based 3, 4 1 2 0 x2 x2 x2 0 1, 2 0 1 2 0 1 2 Analysis Results x3 x3 x3 x3 x3 0 1, 2 0 0 1, 2 2 1 0 2 1 0 1 2 1, 2 x4 x4 x4 3 3 2, 3 1, 2, 3 s 7 t 00, 1, 2 0, 1 5 x5 x5 2 x5 2 0 4 1 0 0 1 1, 2 6 x6 x6 1, 2 MRd =3 = P (ϕ ( x) ≥ 3) 0 2 0, 1 x7 0, 1, 2 3, 4 = 0.54541524 0 1 Rank j Birnbaum MAD MMAW MMFV 1 7 0.5559039984 0.3817906706 1.4199334287 0.4199334287 2 1 0.1750166234 0.1036475213 1.1140024163 0.1140024163 3 4 0.1011335000 0.0723422213 1.0795695635 0.0795695635 4 2 0.0622255969 0.0219048863 1.0240932917 0.0240932917 5 3 0.0622255969 0 0622255969 0.0219048863 0 0219048863 1.0240932917 1 0240932917 0.0240932917 0 0240932917 6 5 0.0169283156 0.0060528488 1.0066575580 0.0066575580 7 6 0.0169283156 0.0060528488 1.0066575580 0.0066575580 32
- 35. ConclusionDy mDynamic and dependent behavior has been recognized p gas a significant contribution to problems in complex reliability.system reliabilityo Multiple states (多状态), Multiple phases (多阶段), Sequence dependence (顺序相依), Dynamic sparing (动态备用), 序相依 动态备 Imperfect coverage (不完全覆盖), Common-cause failures (共 因故障), F 因故障) Functional dependence (功能相依) C ti ld d (功能相依), Competing ti failures (竞争失效), etc...Decision diagrams (决策图) are state-of-the-artcombinatorial models for efficient reliability analysis f ff y yof complex systems. 33
- 36. References: Multi State (多状态) Multi-Stateo G. Levitin, L. Podofillini, and E. Zio, “Generalised importance measures for multi-state elements based on , , , pperformance level restrictions,” Reliability Engineering & System Safety, vol. 82, no. 3, pp. 287–298, 2003.o A. Lisnianski and G. Levitin, Multi-State System Reliability: Assessment, Optimization, and Applications, vol. 6:Series of Quality, Reliability, and Engineering Statistics, World Scientific, 2003.o J E R i J. E. Ramirez-Marquez and D W C it “A Monte-Carlo simulation approach for approximating multi-state two- M d D. W. Coit, M t C l i l ti hf i ti lti t t tterminal reliability,” Reliability Engineering & System Safety, vol. 87, no. 2, pp. 253-264, Feb. 2005.o J. Huang and M. J. Zuo, “Dominant multi-state systems,” IEEE Trans. Reliability, vol. 53, no. 3, pp. 362–368, Sep.2004.o X. Zang, D.Wang, H. Sun, and K. S. Trivedi, “A BDD-based algorithm for analysis of multistate systems withmultistate components,” IEEE Trans. Computers, vol. 52, no. 12, pp. 1608–1618, Dec. 2003.o W. C. Yeh, “A fast algorithm for searching all multi-state minimal cuts,” IEEE Trans. Reliability, vol. 57, no. 4, pp.581–588, Dec 2008581 588 Dec. 2008.o L. Xing and Y. Dai, “A New Decision Diagram Based Method for Efficient Analysis on Multi-State Systems,”IEEE Trans. Dependable and Secure Computing, vol. 6, no. 3, pp. 161-174, Jul.-Sep. 2009.o S. V. Amari, L. Xing, A. Shrestha, J. Akers, and K. S. Trivedi, “Performability Analysis of Multi-State Computing g y y p gSystems Using Multi-Valued Decision Diagrams,” IEEE Trans. on Computers, Vol. 59, No. 10, pp. 1419-1433,October 2010.o A. Shrestha and L. Xing, “A Logarithmic Binary Decision Diagrams-Based Method for Multistate SystemsAnalysis,Analysis ” IEEE Trans Reliability Vol 57 No 4 pp 595-606 December 2008. Trans. Reliability, Vol. 57, No. 4, pp. 595-606, 2008o A. Shrestha, L. Xing, and Y. Dai, “Decision Diagram-Based Methods, and Complexity Analysis for MultistateSystems,” IEEE Trans. Reliability, vol. 59, no. 1, pp. 145-161, Mar. 2010.o etc... 34
- 37. References: Multi-Phase (多阶段) Multi Phaseo J. D. Esary and H. Ziehms, “Reliability analysis of phased missions,” in Reliability and Fault Tree Analysis, R. E. Barlow,J. B. Fussell, and N. D SiJ B F ll d N D. Singpurwalla, Edi ll Editors., pp. 213–236, 1975 213 236o A. K. Somani, J. A. Ritcey, and S. H. L. Au, "Computationally Efficient Phased-Mission Reliability Analysis for Systemswith Variable Configurations," IEEE Trans. Reliability, Vol. 41, No. 4, pp. 504-511, 1992.o Y. Ma and K.S. Trivedi, "An algorithm for reliability analysis of phased mission systems, Reliability Engineering & An phased-mission systems,"System Safety, Vol. 66, pp. 157–170, 1999.o A. Bondavalli, S. Chiaradonna, F. D. Giandomenico, and I. Mura, “Dependability modeling and evaluation of multiple-phased systems using DEEM,” IEEE Trans. Reliability, Vol. 53, No. 4, pp. 509–522, Dec. 2004.o M. K. Smotherman and K. Zemoudeh, “A non-homogeneous Markov model for phased-mission reliability analysis,” IEEETrans. Reliability, Vol. 38, No. 5, pp. 585–590, Dec. 1989.o L. Xing and J. B. Dugan, “Analysis of Generalized Phased Mission System Reliability, Performance and Sensitivity,”IEEE Trans. Reliability, vol. 51, no. 2, pp 199-211, Jun. 2002. y, , , pp. ,o L. Xing and J. B. Dugan, “A Separable TDD-Based Analysis of Generalized Phased-Mission Reliability,” IEEE Trans.Reliability, vol. 53, no. 2, pp. 174-184, Jun. 2004.o L. Xing, “Reliability Evaluation of Phased-Mission Systems with Imperfect Fault Coverage and Common-Cause Failures,”IEEE T Trans. on R li bili vol. 56, no. 1, pp. 58-68, M 2007 Reliability, l 56 1 58 68 Mar. 2007.o A. Shrestha and L. Xing, “Improved Modular Reliability Analyses of Hybrid Phased Mission Systems,” Journal of Riskand Reliability, Vol. 222, No. 4, 2008, pp. 507-520o A. Shrestha, L. Xing, and Y.S. Dai, “Reliability Analysis of Multi State Phased Mission Systems with Unordered and Reliability Multi-State Phased-MissionOrdered States,” IEEE Trans. Systems, Man, and Cybernetics, Part A: Systems & Humans , Vol. 41, No. 4, pp. 625-636, 2011.o S. V. Amari and L. Xing, "Reliability Analysis of k-out-of-n Systems with Phased-Mission Requirements," InternationalJournal of Performability Engineering, Vol. 7, No. 6, pp. 595-600, Nov. 2011.o etc... 35
- 38. References: Sequence Dependence (顺序相依)o J. B. Dugan, S. J. Bavuso, and M. A. Boyd, “Dynamic fault-tree models for fault-tolerant computer systems,” IEEETrans. on Reliability, vol. 41, no. 3, pp. 363-377, S 1992 l bl l 41 3 363 3 Sep. 1992.o W. Long, T. Zhang, Y. Lu, and M. Oshima, “On the quantitative analysis of sequential failure logic using MonteCarlo method for different distributions,” Proc. of Probabilistic Safety Assessment & Management, pp. 391-396, 2002.o T Yuge and S. Yanagi “Quantitative analysis of a fault tree with priority AND gates,” Reliability Engineering & T. S Yanagi, Quantitative gatesSystem Safety, vol. 93, no. 11, pp. 1577-1583, Nov. 2008.o L. Xing, A. Shrestha, and Y. Dai, "Exact Combinatorial Reliability Analysis of Dynamic Systems with Sequence-Dependent Failures," Reliability Engineering & System Safety, Vol. 96, No. 10, pp. 1375-1385, October 2011.o etc... 36
- 39. References: Dynamic Sparing (动态备用)o J. B. Dugan, S. J. Bavuso, and M. A. Boyd, “Dynamic fault-tree models for fault-tolerant computer systems,”IEEE Trans. Reliability, vol. 41, no. 3, pp. 363-377, Sep. 1992.o J She and M. G P h “R li bili of a k J. Sh d M G. Pecht, “Reliability f k-out-of-n W f Warm-Standby S S db System,” IEEE T ” Trans. R l b l Reliability, vol. 41, l 41no. 1, pp. 72-75, Mar. 1992o D. Liu, C. Zhang, W. Xing, R. Li, and H. Li, “Quantification of Cut Sequence Set for Fault Tree Analysis,”HPCC2007, Lecture Notes in Computer Science, no. 4782, pp 755-765, Springer-Verlag, 2007. , p , , pp. , p g g,o L. Xing, O. Tannous, and J. B. Dugan, "Reliability Analysis of Non-Repairable Cold-Standby Systems UsingSequential Binary Decision Diagrams," IEEE Trans. Systems, Man, and Cybernetics, Part A: Systems andHumans, in Press, DOI: 10.1109/TSMCA.2011.2170415 O. Tannous, L Xing, R. Po O T L. Xi R Peng, M Xi and S.H, N "R d d M. Xie, d S H Ng, "Redundancy All ti f S i P ll l Warm- Allocation for Series-Parallel WStandby Systems," Proc. of the IEEE International Conference on Industrial Engineering and EngineeringManagement, Singapore, Dec. 2011o P. Boddu and L. Xing, "Optimal Design of Heterogeneous Series-Parallel Systems with Common-CauseFailures," International Journal of Performability Engineering, Special Issue on Performance and DependabilityModeling of Dynamic Systems, Vol. 7, No. 5, pp. 455-466, Sep. 2011.o O. Tannous, L. Xing, and J. B. Dugan, “Reliability Analysis of Warm Standby Systems using SequentialBDD, Proc.BDD ” Proc of the 57th Annual Reliability & Maintainability Symposium, Jan 2011. Symposium Jan. 2011o etc... 37
- 40. References: Imperfect Coverage (不完全覆盖)o S. V. Amari, J. B. Dugan, and R. B. Misra, “A separable method for incorporating imperfect coverage in combinatorialmodel,” IEEE Trans. on Reliability, vol. 48, no. 3, pp. 267–274, Sep. 1999.o S. V. Amari, J. B. Dugan, and R. B. Misra, “Optimal reliability of systems subject to imperfect fault-coverage,” IEEETrans. on Reliability, vol. 48, no. 3, pp. 275 284, Sep. 1999. 275–284,o G. Levitin and S. V. Amari, “Multi-state systems with static performance dependent fault coverage,” Journal of Risk andReliability, vol. 222, pp. 95-103, 2008.o G. Levitin and S. V. Amari, “Multi-state systems with multi-fault coverage,” Reliability Engineering & System Safety, vol.93, pp. 1730-1739, 2008.o S. A. Doyle, J. B. Dugan, and A. Patterson-Hine, “A Combinatorial Approach to Modeling Imperfect Coverage,” IEEETransactions on Reliability, pp. 87-94, March 1995.o J B Dugan “Fault Trees and Imperfect Coverage,” IEEE Transactions on Reliability vol 38, no. 2, pp. 177 - 185 June J. B. Dugan, Fault Coverage Reliability, vol. 38 no 2 pp 185,1989.o L. Xing and J. B. Dugan, “Dependability Analysis of Hierarchical Systems with Modular Imperfect Coverage,” Proc. of the19th International System Safety Conference, Huntsville, Alabama, Sep. 2001o L. Xing, “Reliability Evaluation of Phased-Mission Systems with Imperfect Fault Coverage and Common-Cause Failures,”IEEE Trans. on Reliability, vol. 56, no. 1, pp. 58-68, Mar. 2007.o L. Xing and A. Shrestha, “Reliability Evaluation of Distributed Computer Systems Subject to Imperfect Coverage andDependent Common Cause Failures , Journal of Computer Sciences, Special Issue on Reliability and Autonomic Management, Common-Cause Failures”,vol. 2, no. 6, pp. 473-479, 2006.o A. Shrestha, L. Xing, and S. V. Amari, “Reliability and Sensitivity Analysis of Imperfect Coverage Multi-State Systems,”Proc. of The 56th Annual Reliability & Maintainability Symposium, San Jose, CA, USA, 2010.o etc... 38
- 41. References: CR f Common-Cause Failures (共因故障) C F ilo J. K. Vaurio, "Common cause failure probabilities in standby safety system fault tree analysis with testing—schemeand timing dependencies," Reliability Engineering & System Safety, Vol. 79, No. 1, pp. 43-57, January 2003.o S. Mitra, N. R. Saxena, and E. J. McCluskey, “Common-Mode Failures in Redundant VLSI Systems: A Survey,”IEEE Trans on Reliability Vol 49 No 3 pp 285-295 September 2000. Trans. Reliability, Vol. 49, No.3, pp. 285-295. 2000o J. K. Vaurio, “An Implicit Method for Incorporating Common-Cause Failures in System Analysis,” IEEE Trans. onReliability, Vol. 47, No.2, pp. 173-180, 1998.o K. N. Fleming, A. Mosleh, and A. P. Kelly, “On the analysis of dependent failures in risk assessment and reliabilityevaluation ” Nuclear Safety, vol 24, pp. 637–657, 1983.evaluation,” Safety vol. 24 pp 637 657 1983o Z. Tang, H. Xu, and J. B. Dugan, "Reliability analysis of phased mission systems with common cause failures,"Proceedings of Annual Reliability and Maintainability Symposium, pp. 313- 318, January 2005.o K.N. Fleming, A. Mosleh, “Common-cause data analysis and implications in system modeling,” Proceeding ofInternational Topical Meeting on Probabilistic S f M h d & A li iI i lT i lM i P b bili i Safety Methods Applications, V l 1 pp. 3/1 3/12 February 1985. Vol. 1. 3/1-3/12, F b 1985o G. Levitin, L. Xing, H. Ben-Haim, and Y. Dai, "Multi-state Systems with Selective Propagated Failures andImperfect Individual and Group Protections," Reliability Engineering and System Safety, in Press.o G. Levitin and L. Xing, "Reliability and Performance of Multi state Systems with Propagated Failures Having Reliability Multi-stateSelective Effect," Reliability Engineering and System Safety, vol. 95, no. 6, pp. 655-661, June 2010.o L. Xing, P. Boddu, Y. Sun, and W. Wang, “Reliability Analysis of Static and Dynamic Fault-Tolerant Systemssubject to Probabilistic Common-Cause Failures,” Journal of Risk and Reliability, vol. 224, no. 1, pp.43-53, 2010 .o L. Xing, A. Shrestha, L. Meshkat, and W. Wang, “Incorporating Common-Cause Failures into the ModularHierarchical Systems Analysis,” IEEE Trans. on Reliability, vol. 58, no. 1, pp. 10-19, Mar. 2009o L. Xing and S. V. Amari, “Effective Component Importance Analysis for the Maintenance of Systems withCommon Cause Failures,” Intl. Jnl. of Reliability, Quality and Safety Engineering, vol. 14, no. 5, pp 459-478, 2007. , f y, Q y f y g g, , , pp. ,o etc... 39
- 42. References: Functional Dependence (功能相依) & Competing Failures (竞争失效)o J. B. Dugan, S. J. Bavuso, and M. A. Boyd, “Dynamic fault-tree models for fault-tolerant computer systems,” IEEETrans. on Reliability, vol. 41, no. 3, pp. 363-377, Sep. 1992.o W. Li and H. Pham, “An inspection-maintenance model for systems with multiple competing processes,” IEEE i d h i i i d lf ih li l iTransactions on Reliability, 54(2), pp. 318-327, 2005.o H. Pham and D. M. Malon, “Optimal design of systems with competing failure modes,” IEEE Transactions onReliability, 43(2), pp. 251 – 254, 1994.o C. Bunea and T. A. Mazzuchi, “Competing failure modes in accelerated life testing,” Journal of Statistical Planningand Inference, 136(5), pp. 1608-1620, 2006.o A. Xu and Y. Tang , “Objective Bayesian analysis of accelerated competing failure models under Type-I censoring,”Computational Statistics & Data Analysis, 55(10), pp. 2830-2839, 2011o L. Xing, J. B. Dugan, and B. A. Morrissette, “Efficient Reliability Analysis of Systems with Functional DependenceLoops,” Maintenance and Reliability, pp. 65-69, No. 3/2009, 2009.o L. Xing, B. A. Morrissette , and J. B. Dugan, “Efficient Analysis of Imperfect Coverage Systems with Functional EfficientDependence,” Proc. of the 56th Annual Reliability & Maintainability Symposium, San Jose, CA, USA, Jan. 2010.o L. Xing and G. Levitin, "Combinatorial Algorithm for Reliability Analysis of Multi-State Systems with PropagatedFailures and Failure Isolation Effect," IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems andHumans , Vol. 41, No. 6, pp. 1156-1165, November 2011.o L. Xing and G. Levitin, "Combinatorial Analysis of Systems with Competing Failures Subject to Failure Isolation andPropagation Effects," Reliability Engineering and System Safety, Vol. 95, No. 11, pp. 1210-1215, November 2010.o etc... 40
- 43. Thank You! h k ! 谢谢! 41

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