Efficient Combinatorial Models for Reliability Analysis of 

C l D i S t (基Complex Dynamic Systems (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 XingPresented live on Nov 09th, 2011

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Efficient Combinatorial Models for Reliability Analysis of Complex Dynamic Reliability Analysis of Complex Dynamic

Systems (基于组合模型的复杂动态系统可靠性分析)(基于组合模型的复杂动态系统可靠性分析)

Presented byDr. Liudong Xing (邢留冬)E mail: lxing@umassd eduE-mail: lxing@umassd.edu

Electrical and Computer Engineering Dept.University of MassachusettsUniversity of Massachusetts

Dartmouth, MA, USAwww massachusetts eduwww.massachusetts.edu

ASQ Reliability Division Webinar Series

2http://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=x14634http://www.infrastructurist.com/2009/05/19/a-vibrant-us-train-industry-would-employ-more-people-than-car-makers-do-now/ -- Image Sources

MotivationComputing and engineering systems are

l i t d bli h l evolving toward enabling much larger collaboration & handling more complicated missions.

Th i si m l xit d s l im l The increasing complexity and scale imply that reliability problems will not only continue to be a challenge but also require more efficient models and solutions

3

@ This Talk --@ This Talk Reliability Analysis of Complex Dynamic Systems

Evaluation Methods Complex BehaviorAnalytical methods o Combinatorial methods

pMultiple states (多状态)Multiple phases (多阶段)o Combinatorial methods

(fault trees, decision diagrams)

Multiple phases (多阶段)Sequence dependence (顺序相依)Dynamic sparing (动态备用)

o State space-based methods (Markov models)

Dynamic sparing (动态备用)Imperfect coverage (不完全覆盖)Common-cause failures (共因故障)

Simulation methodsMeasurement-based

(共 障)Functional dependence (功能相依)Competing failures (竞争失效)p g

Acknowledgment: US National Science Foundation (NSF) No. 0614652 & 0832594 & 1112947 4

AgendaAgenda

O i f l b h iOverview of complex behaviorReliability and sensitivity analysis of multi-Reliability and sensitivity analysis of multistate systems

5

Multi-State (多状态)System & components: more than two levels of performance (or states) varying from perfect p poperation to complete failureBehaviors modeled: shared loads, performance , pdegradation, imperfect coverage, multiple failure modes, etc.modes, etc.Applications: power systems, transmission networks, communication networks circuits etccommunication networks, circuits, etcChallenge: o dependence among multiple stateso dependence among multiple states

6

Multi-Phase (多阶段)Multi-Phase (多阶段)A system supporting a mission characterized by A system supporting a mission characterized by multiple, consecutive, and non-overlapping phases of operationoperationSystem components subject to different stresses,

i t l diti s d li bilit i ts i environmental conditions, and reliability requirements in different phasesApplications: aerospace (aircraft, rockets, spacecraft), nuclear power, airborne weapon systems, etc Challenge:o dynamics in system configuration, failure criteria, and y y g , ,

component failure behavioro s-dependencies across phases for a given component

7

Sequence Dependence (顺序相依)Sequence Dependence (顺序相依)

The order that fault events occur is important to the The order that fault events occur is important to the system reliability

h ll d d f l Challenge: sequence-dependent system failure criteria

F ilPrimary:

P

Failure

Switch:Sw

Standby:S P S Sw P

• Sw P: system fails• P Sw: system OK

Modeled using priority AND gate in fault tree analysis

8

y g y

Dynamic Sparing (动态备用)Dynamic Sparing (动态备用)

λOne module is on-line & operational, and one or

mpo

nent

s Pλ

Sλ Hot

more modules serve as standby units.

Pλ

com

t1τ 2τ

When the on-line module experiences a fault and

com

pone

nts P

Sλ Cold

the fault is detected, it is removed and replaced with

c

t1τ 2τ

λa standby unit.Challenge: time/order-

ompo

nent

s Pλ

SλSα Warm

dependent failure behavior

co

t1τ 2τ

9

I f t F lt C (不完全覆盖) Imperfect Fault Coverage (不完全覆盖) Imperfect detection location or recovery of a Imperfect detection, location, or recovery of a component fault may cause an extensive damage to the entire system despite presentence of redundancies entire system, despite presentence of redundancies.

Extent of an uncovered fault damage can exhibit multiple levels in hierarchical systems: if an undetected error escapes from one level, it may be covered at a higher level covered at a higher level. Challenge: multiple failure modes

10

Common Cause Failures (共因故障)Common-Cause Failures (共因故障)Simultaneous failure of multiple components due to a Simultaneous failure of multiple components due to a common cause

Challenge: multiple dependent component failures

External Cause

Common Cause Failure

Internal Cause

Global Effect on a System/SubsystemInternal Cause

(PropagatedFailure)

y y

Selective Effect onSelective Effect on System Components

11

Functional Dependence (功能相依)

Occurrence of some event (trigger) causes other components (dependent components) to become mp n n ( p n n mp n n ) minaccessible or unusable

C di f il lti l f il i iti t d b th Cascading failures: multiple failures initiated by the trigger of one component in the system resulting in a h i i d i ff ( i chain reaction ordomino effect (common in power

grids)

FDEPFDEP

BA C ......

12

Competing Failures (竞争失效)

Occur in systems subject to both functional d d (FDEP) d d f il (PF)dependence (FDEP) and propagated failures (PF)

PF has different consequences due to competition in PF has different consequences due to competition in the time domain between trigger failure and failure propagated from dependent componentspropagated from dependent components.

Trigger failure PF of dependent components: failure isolationgg f f p mp fPF of dependent components Trigger failure: system fails

13

Agenda

Overview of complex behaviorl l l lReliability and sensitivity analysis of multi-

state systems (MSS)y ( )o Basic concepts

l h do MSS analysis methodso ExamplesE mp

14

MSS R li bilitMSS Reliability

MSS reliability at level d : o probability that the system performance level is o probability that the system performance level is

greater than or equal to d.

( ) f

))(( dPMRd ≥= xϕ

o φ(x): system structure function

15

MSS S iti it MMSS Sensitivity Measures

Quantify importance of components, and help prioritize reliability improvement activities Composite importance measures (CIM): evaluate contribution of a multi-state component as a whole to f m mpMSS reliabilityo Example: Birnbaum or average of the Sum of Absolute o Example: Birnbaum or average of the Sum of Absolute

Deviation (SAD)

))(()|)((1

<−=<=∑ =j ijiSAD

i

i dxPbxdxPMI

ϕϕω

1−ii ω

16

MSS A l i M th d (1)MSS Analysis Methods (1)

Simulation-based methodso computationally expensive and time-consumingp y p go approximate resultso a complete new simulation must be performed when o a complete new simulation must be performed when

parameter values change

St t b d th d (M k d l )State space-based methods (Markov models)o more sever state explosion problem than analyzing binary

systems

Multi-state minimal path/cut vectors (MMPV/MMCV)po doubly exponential complexity

17

MSS Analysis Methods (2)

Decision diagrams (决策图)-based methodso Multi-state binary decision diagrams (MBDD)o Logarithmically-encoded binary decision diagrams o Logarithmically encoded binary decision diagrams

(LBDD) o Multi-state multi-valued decision diagrams (MMDD)o Multi-state multi-valued decision diagrams (MMDD)

18

A Ill t ti E lAn Illustrative Example

Each board has 4 stateso Bi 4 (both P & M are functional)

P1 M1

B1

o Bi,4 (both P & M are functional) o Bi,3 (M is functional, P is down) o Bi 2 (P is functional, M is down) P M

B2

Bus

o Bi,2 (P is functional, M is down) o Bi,1 (both P & M are down)

P2 M2

The system has 3 stateso S3 (at least one P & both M are

f i l)functional)o S2 (at least one P & exactly one

M are functional)M are functional)o S1 (no P or M is functional)

19

MBDD

4 Boolean variables to encode 4 board stateso (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 o special operations to handle state dependencies in model

generation and evaluation

20X. Zang, D.Wang, H. Sun, and K. S. Trivedi, “A BDD-based algorithm for analysis of multistate systems with multistate components,” IEEE Trans. Computers, vol. 52, no. 12, pp. 1608–1618, Dec. 2003

LBDD2 auxiliary Boolean variables to encode 4 board states

LBDDy

o (v1, v2) for board B1

o (w1, w2) for board B1( 1, 2) 1

v1 v2 B1 states 1,31 2 1

0 0 B1,1

0 1 B1,20

1

v1

v

1,3

1

0

1 0 B1,3

1 1 B1,4

1

1 v2

1

0

o binary logic; no dependence among fewer auxiliary variableso state encoding and decoding are neededo state encoding and decoding are needed

21

A. Shrestha and L. Xing, “A Logarithmic Binary Decision Diagrams-Based Method for Multistate Systems Analysis,” IEEE Trans. Reliability, Vol. 57, No. 4, pp. 595-606, Dec. 2008.

MMDDMMDD1 multi-valued variable per multi-state component1 multi valued variable per multi state componento (B1) for board B1

o (B2) for board B2o (B2) for board B2

B1 B1 B1 B1

2 4

0 0 03

1

1

2 4

1 0 03

1

0

2 4

0 1 03

1

0

2 4

0 0 13

1

00 0 01 1 0 00 0 1 00 0 0 10

Board State B1,1 Board State B1,2 Board State B1,3 Board State B1,4

o no dependence among multi-valued variableso straightforward model generation and evaluationo 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.

22

S. V. Amari, L. Xing, A. Shrestha, J. Akers, and K. S. Trivedi, “Performability Analysis of Multi-State Computing Systems Using Multi-Valued Decision Diagrams,” IEEE Trans. on Computers, vol. 59, no. 10, pp. 1419-1433, 2010.

Example Computer System MFT

Example Computer System MBDD, LBDD, MMDD

3 4

1

0

10

10

MBDD LBDD MMDD

23

Performance Comparison Microelectronics Center of North Carolina (MCNC) B h k(MCNC) Benchmarkso model sizeo model sizeo # recursive calls

t d i l ti tio top-down recursive evaluation timeo bottom-up evaluation time

24A. Shrestha, L. Xing, and Y. Dai, “Decision Diagram-Based Methods, and Complexity Analysis for Multistate Systems,” IEEE Trans. Reliability, vol. 59, no. 1, pp. 145-161, Mar. 2010.

N B h kName Inpu

tOutp

ut

ProductTerms

MCNC Benchmarks 5xp1 7 10 75

9sym 9 1 87

alu2 10 8 68

Originally designed for Boolean switching functions

alu4 14 8 1028

b12 15 9 431

bw 5 28 87Boolean switching functions

Adapted to form MSS with

bw 5 28 87

clip 9 5 167

con1 7 2 9

inc 7 9 34pmultistate componentso Each binary output ≡ a

inc 7 9 34

mdiv7 8 10 256

misex1 8 7 32

misex2 25 18 29y p

system state o A group of binary inputs ≡

misex2 25 18 29

misex3c 14 14 305

postal 8 1 25

d 3 3 32g p y p

multistate componento E.g., 4 binary inputs form 16-

rd53 5 3 32

rd73 7 3 141

rd84 8 4 256

state components sao2 10 4 58

sn74181 14 8 1132

squar5 5 8 32

25

xor5 5 1 16

Z5xp1 7 10 128

Z9sym 9 1 420

M d l SiModel SizeW > W > WWMBDD > WLBDD > WMMDD

26

# of Top down Recursive Calls# of Top-down Recursive CallsR > R > RRMBDD > RMMDD > RLBDD

1000000

MBDD LBDD MMDD

10000

100000

1000

10

100

10

xor5

rd53

squa

r5

con1

mis

ex1

post

al

rd73 inc

bw

rd84

5xp1

Z9sy

m

Z5xp

1

9sym cl

ip

mdi

v7

sao2

mis

ex2

alu2 b12

mis

ex3c

sn74

181

alu4

27

T d R si E l ti Ti Top-down Recursive Evaluation Time TMBDD > TLBDD > TMMDD TMBDD > TLBDD > TMMDD

(in ms, time for decoding states is included for LBDD)

1000

MBDD LBDD MMDD

10

100

1

10

0.1

0.01

xor5

rd53sq

uar5

con1

misex1

posta

lrd73 inc bw rd845x

p1Z9s

ymZ5x

p19s

ym clipmdiv

7alu

2b1

2sa

o2mise

x2mise

x3c

sn74

181 alu4

28

B tt E l ti TiBottom-up Evaluation Time

T T TTMBDD > TLBDD > TMMDD

100

MBDD LBDD MMDD

10

1

0 01

0.1

0.01

xor5

rd53po

stal

con1 rd73

squa

r5Z9s

ymmise

x1 rd845x

p19s

ym inc sao2

Z5xp1 bw

misex2 alu

2b1

2clip

mdiv7

misex3

csn

74181 alu

4

29

S mmSummary

LBDD is a tradeoff that transforms multi-state domain into an equivalent auxiliary q ybinary domain, but offers reduced system model size than MBDD model size than MBDD.

In general, MMDD is more efficient than MBDD and LBDD.MBDD and LBDD.

30

A T smissi N t kA Transmission Network

The system must l d d 3

1

2

37s tsupply a demand >= 3

units from s to t.5

6

4

7s t

6

Component Transmission Capacity State ProbabilityComponent Transmission Capacity State Probability1 0 1 2 3 4 0.10 0.05 0.15 0.35 0.352 0 1 2 - - 0.10 0.05 0.85 - -3 0 1 2 0 10 0 05 0 853 0 1 2 - - 0.10 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 05 0 856 0 1 2 - - 0.10 0.05 0.85 - -7 0 1 2 3 4 0.15 0.15 0.05 0.45 0.20

31A. Shrestha, L. Xing, D. W. Coit, “An Efficient Multi-State Multi-Valued Decision Diagram-Based Approach for Multi-State System Sensitivity Analysis,” IEEE Trans. Reliability, vol. 59, no. 3, pp. 581-592, Sept. 2010.

MMDD-based 1

0

23, 4

x1

Analysis Results1 201, 20 0 1 2

x2x2 x2

x3x3x3 x3 x31, 202

10

1, 200

1

2

0

1, 2

x3x3x3 x3 x3

x4x4x4

1

2

31, 2, 3

00, 1

2, 3

0 21

0, 1, 2

3

2

x4x4x4

x5 x5 x5

3

54

7s t

)3)((3 ≥= xϕPMRd

0 1, 2

1, 20

1

2

0 1

x6x66

4

54541524.0 )3)((3

=≥= xϕPMRd

10

0, 1, 2 3, 4

0, 1 x7

Rank j Birnbaum MAD MMAW MMFV1 7 0.5559039984 0.3817906706 1.4199334287 0.41993342872 1 0.1750166234 0.1036475213 1.1140024163 0.11400241631 0.1750166234 0.1036475213 1.1140024163 0.11400241633 4 0.1011335000 0.0723422213 1.0795695635 0.07956956354 2 0.0622255969 0.0219048863 1.0240932917 0.02409329175 3 0 0622255969 0 0219048863 1 0240932917 0 0240932917

32

5 3 0.0622255969 0.0219048863 1.0240932917 0.02409329176 5 0.0169283156 0.0060528488 1.0066575580 0.00665755807 6 0.0169283156 0.0060528488 1.0066575580 0.0066575580

ConclusionConclusionDynamic and dependent behavior has been recognized Dy m p gas a significant contribution to problems in complex system reliabilitysystem reliability.o Multiple states (多状态), Multiple phases (多阶段), Sequence

序相依 动态备dependence (顺序相依), Dynamic sparing (动态备用), Imperfect coverage (不完全覆盖), Common-cause failures (共因故障) F ti l d d (功能相依) C ti 因故障), Functional dependence (功能相依), Competing failures (竞争失效), etc...

Decision diagrams (决策图) are state-of-the-art combinatorial models for efficient reliability analysis f ff y yof complex systems.

33

References: Multi State (多状态)References: Multi-State (多状态)o 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.

J E R i M d D W C it “A M t C l i l ti h f i ti lti t t to J. E. Ramirez-Marquez and D. W. Coit, “A Monte-Carlo simulation approach for approximating multi-state two-terminal 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 with multistate 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 Systems Analysis ” IEEE Trans Reliability Vol 57 No 4 pp 595-606 December 2008Analysis, IEEE Trans. Reliability, Vol. 57, No. 4, pp. 595-606, December 2008.o A. Shrestha, L. Xing, and Y. Dai, “Decision Diagram-Based Methods, and Complexity Analysis for Multistate Systems,” IEEE Trans. Reliability, vol. 59, no. 1, pp. 145-161, Mar. 2010. o etc...

34

References: Multi-Phase (多阶段)References: Multi Phase (多阶段)o J. D. Esary and H. Ziehms, “Reliability analysis of phased missions,” in Reliability and Fault Tree Analysis, R. E. Barlow, J B F ll d N D Si ll Edi 213 236 1975J. B. Fussell, and N. D. Singpurwalla, Editors., pp. 213–236, 1975o A. K. Somani, J. A. Ritcey, and S. H. L. Au, "Computationally Efficient Phased-Mission Reliability Analysis for Systems with 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 &o Y. Ma and K.S. Trivedi, An algorithm for reliability analysis of phased mission systems, Reliability Engineering & 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,” IEEE Trans. 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 R li bili l 56 1 58 68 M 2007IEEE Trans. on Reliability, vol. 56, no. 1, pp. 58-68, Mar. 2007.o A. Shrestha and L. Xing, “Improved Modular Reliability Analyses of Hybrid Phased Mission Systems,” Journal of Risk and Reliability, Vol. 222, No. 4, 2008, pp. 507-520 o A. Shrestha, L. Xing, and Y.S. Dai, “Reliability Analysis of Multi-State Phased-Mission Systems with Unordered ando A. Shrestha, L. Xing, and Y.S. Dai, Reliability Analysis of Multi State Phased Mission Systems with Unordered and Ordered 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," International Journal of Performability Engineering, Vol. 7, No. 6, pp. 595-600, Nov. 2011. o etc...

35

References: Sequence Dependence 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,” IEEE l b l l 41 3 363 3 S 1992Trans. on Reliability, vol. 41, no. 3, pp. 363-377, Sep. 1992.

o W. Long, T. Zhang, Y. Lu, and M. Oshima, “On the quantitative analysis of sequential failure logic using Monte Carlo 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 &o T. Yuge and S. Yanagi, Quantitative analysis of a fault tree with priority AND gates, Reliability Engineering & System 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

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.

J Sh d M G P h “R li bili f k f W S db S ” IEEE T R l b l l 41o J. She and M. G. Pecht, “Reliability of a k-out-of-n Warm-Standby System,” IEEE Trans. Reliability, vol. 41, no. 1, pp. 72-75, Mar. 1992 o 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 Using Sequential Binary Decision Diagrams," IEEE Trans. Systems, Man, and Cybernetics, Part A: Systems and Humans, in Press, DOI: 10.1109/TSMCA.2011.2170415

O T L Xi R P M Xi d S H N "R d d All ti f S i P ll l Wo O. Tannous, L. Xing, R. Peng, M. Xie, and S.H, Ng, "Redundancy Allocation for Series-Parallel Warm-Standby Systems," Proc. of the IEEE International Conference on Industrial Engineering and Engineering Management, Singapore, Dec. 2011 o P. Boddu and L. Xing, "Optimal Design of Heterogeneous Series-Parallel Systems with Common-Cause Failures," International Journal of Performability Engineering, Special Issue on Performance and Dependability Modeling 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 Sequential BDD ” Proc of the 57th Annual Reliability & Maintainability Symposium Jan 2011BDD, Proc. of the 57th Annual Reliability & Maintainability Symposium, Jan. 2011.o etc...

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References: Imperfect Coverage (不完全覆盖) 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 combinatorial model,” 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,” IEEE Trans. on Reliability, vol. 48, no. 3, pp. 275–284, Sep. 1999.Trans. on Reliability, vol. 48, no. 3, pp. 275 284, Sep. 1999.o G. Levitin and S. V. Amari, “Multi-state systems with static performance dependent fault coverage,” Journal of Risk and Reliability, 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,” IEEE Transactions 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 Juneo J. B. Dugan, Fault Trees and Imperfect Coverage, IEEE Transactions on Reliability, vol. 38, no. 2, pp. 177 185, June 1989.o L. Xing and J. B. Dugan, “Dependability Analysis of Hierarchical Systems with Modular Imperfect Coverage,” Proc. of the 19th International System Safety Conference, Huntsville, Alabama, Sep. 2001 o 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 and Dependent Common-Cause Failures”, Journal of Computer Sciences, Special Issue on Reliability and Autonomic Management,Dependent Common Cause Failures , Journal of Computer Sciences, Special Issue on Reliability and Autonomic Management, 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...

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R f C C F il (共因故障)References: Common-Cause Failures (共因故障)

o J. K. Vaurio, "Common cause failure probabilities in standby safety system fault tree analysis with testing—scheme and 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 2000IEEE Trans. on Reliability, Vol. 49, No.3, pp. 285-295. September 2000. o J. K. Vaurio, “An Implicit Method for Incorporating Common-Cause Failures in System Analysis,” IEEE Trans. on Reliability, 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 reliability evaluation ” Nuclear Safety vol 24 pp 637 657 1983evaluation,” Nuclear Safety, vol. 24, pp. 637–657, 1983.o 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 of I i l T i l M i P b bili i S f M h d & A li i V l 1 3/1 3/12 F b 1985International Topical Meeting on Probabilistic Safety Methods & Applications, Vol. 1. pp. 3/1-3/12, February 1985.o G. Levitin, L. Xing, H. Ben-Haim, and Y. Dai, "Multi-state Systems with Selective Propagated Failures and Imperfect 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 Havingo G. Levitin and L. Xing, Reliability and Performance of Multi state Systems with Propagated Failures Having Selective 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 Systems subject 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 Modular Hierarchical 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 with Common 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...

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References: Functional Dependence (功能相依)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,” IEEE Trans. on Reliability, vol. 41, no. 3, pp. 363-377, Sep. 1992.

i d h i i i d l f i h l i l io W. Li and H. Pham, “An inspection-maintenance model for systems with multiple competing processes,” IEEE Transactions 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 on Reliability, 43(2), pp. 251 – 254, 1994.Reliability, 43(2), pp. 251 254, 1994.o C. Bunea and T. A. Mazzuchi, “Competing failure modes in accelerated life testing,” Journal of Statistical Planning and 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 Dependence Loops,” 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 Functionalo L. Xing, B. A. Morrissette , and J. B. Dugan, Efficient Analysis of Imperfect Coverage Systems with Functional Dependence,” 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 Propagated Failures and Failure Isolation Effect," IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans , 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 and Propagation Effects," Reliability Engineering and System Safety, Vol. 95, No. 11, pp. 1210-1215, November 2010.o etc...

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h k !Thank You!谢谢!谢谢!

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