Anatoly Lisnianski

1. Anatoly Lisnianski EXTENDED RELIABILITY BLOCK DIAGRAM METHOD

2. Multi-state System (MSS)Basic Concepts • MSS is able to perform its task with partial performance • “all or nothing” type of failure criterion cannot be formulated

3. 1 D C E 3 2 G1(t) {0,1.5} G3(t) {0, 1.8, 4} 1 3 A 2 G2(t) {0,2} Oil Transportation system

4. Generic MSS model Performance stochastic processes for each system elementj: System structure functionthat produces the stochastic process corresponding to the output performance of the entire MSS

5. State-space diagram for the flow transmission MSS 1 1.5, 2, 4 3.5 4 2 0, 2, 4 1.5, 2, 1.8 2 1.8 3 1.5, 0, 4 1.5 5 6 8 0, 2, 1.8 1.5, 2, 0 0, 0, 4 1.8 0 0 7 1.5, 0, 1.8 1.5 10 9 0, 0, 1.8 0, 2, 0 0 0 11 1.5, 0, 0 0 12 0, 0, 0 0

6. Straightforward Reliability Assessmentfor MSS • Stage 1. State-space diagram building or model construction for MSS Difficult non-formalized process that may cause numerous mistakes even for relatively small MSS • Stage 2. Solving models with hundreds of states Can challenge the computer resources available

7. RBD Method: multi-state interpretation • each block of the reliability block diagram represents one multi-state element of the system • each block's j behavior is defined by the corresponding performance stochastic process • logical order of the blocks in the diagram is defined by the system structure function

8. Combined Universal Generating Function (UGF) and Random Processes Method • 1-st stage: a model of stochastic process should be built for every multi-state element. Based on this model a state probabilities for every MSS's element can be obtained. • 2-nd stage: an output performance distribution for the entire MSS at each time instant t should be defined using UGF technique

9. lk,k-1 mk-1,k lk-1,k-2 mk-2,k-1 ... ... ... k k-1 2 1 l3,2 m2,3 l2,1 m1,2 Multi-state Element Markov Model

10. Differential Equations forPerformance Distribution … = …

11. ENTIRE MULTI-STATE SYSTEM RELIABILITY EVALUATION • based on determined states probabilities for all elements, UGF for each individual element should be defined • by using composition operators over UGF of individual elements and their combinations in the entire MSS structure, one can obtain the resulting UGF for the entire MSS

12. … … Individual UGF Element j Individual UGF for element j

13. UGF for Entire MSS UGF for MSS with n elements and the arbitrary structure function  is defined by using composition operator:

14. Example: MSS consists of two elements G1(t) G2(t) 1 2 G(t)=min{G1(t),G2(t)}

15. UGF for Entire MSS

16. Polynomials “Multiplication”

17. Numerical Example 1 3 G(t)=min{G1(t)+G2(t), G3(t)} 2 Entire MSS

18. Element 3 Element 1 Element 2 g12=1.5 g33=4.0 g22=2.0 g32=1.8 g21=0 2 3 1 1 1 2 2 g11=0 g31=0 State-space diagrams of the system elements.

19. Differential Equations • For element 1: • For element 2:

20. For element 3:

21. Individual UGF

22. UGF for Entire MSS

23. UGF for parallel connected elements 1 and 2

24. UGF for elements connected in series

25. Resulting Performance Distribution for the Entire MSS

26. Probabilities of different performance levels p5(t) p2(t) p3 (t) p4(t) p1(t)

27. CONCLUSIONS • The presented method extends classical reliability block diagram method to repairable multi-state system. • The procedure is well formalized and based on natural decomposition of entire multi-state system. • Instead of building the complex model for the entire multi-state system, one should built n separate relatively simple models for system elements. • Instead of solving one high-order system of differential (for Markov process) or integral (for semi-Markov process) equations one has to solve n low-order systems for each system element.