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

4. Generic MSS model Performance stochastic processes for each system element j : System structure function that produces the stochastic process corresponding to the output performance of the entire MSS

5. State-space diagram for the flow transmission MSS

6. Straightforward Reliability Assessment for 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. Multi-state Element Markov Model k k-1 2 1 k,k-1 k-1,k-2 3,2 2,1...  k-1,k  k-2,k-1  2,3  1,2...

10. Differential Equations for Performance 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 Individual UGF for element j … … 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 12 G(t)=min{G1(t),G2(t)} G1(t)G2(t)

15. UGF for Entire MSS

16. Polynomials “Multiplication”

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

18. State-space diagrams of the system elements. Element 1Element 2 Element 3 g 22 =2.0 g 21 =0 g 33 =4.0 g 32 =1.8 g 31 =0 1 2 1 2 1 3 2 g 12 =1.5 g 11 =0

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 p 1 (t) p 3 (t) p 4 (t) p 2 (t) p 5 (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.