1. Consecutively connected systems Radio relay system Pipeline

2. Consecutively connected systems Binary consecutive k-out-of-n system k n h Multi-state generalization SiSi i i+1 … i+h Element state distribution

3. j j+1 … j+h The most remote node connected with node j by element i The most remote node connected with node j by all of the elements located at this node Connectivity model

4. The most remote node connected with node 1 by all of the elements located at nodes 1,2,…,j j+1 j j Recursive algorithm

5. Retransmission delay model 11 22 33 11 22 33 T=1+3T=1+3 T=2T=2

6. Random vector G i (j) = {G i (j) (1),…,G i (j) (n+1)} G i (j) (h) is the random time of the signal arrival to node C h since it has arrived at C j. ii SiSi j j+1 … j+h 88 For multi-state element i located at node C j. State variables

7. j j+1 … j+h Retransmission time provided by group of elements located at node C j.

8. m m+1 … h f(G (m),G (m+1) )(h) =min{G (m) (h),G (m) (m+1) + G (m+1) (h)} G (m+1) (h) G (m) (m+1) G(m)(h)G(m)(h) Delays of a signal retransmitted by all of the MEs located at C 1, …, C m+1

9. 2p2-p222p2-p22 p2p2 p1p1 p1+p2p1+p2 I II C 1 C 2 C 3 e1e1 e2e2 e 1, e 2 C 1 C 2 C 3 R I = 2p 2  p 2 2 R II = p 2 +p 1 (p 1 +p 2 ) Optimal Element Allocation in a Linear Multi-state Consecutively Connected System Connectivity model

10. Optimal Element Allocation in a Linear Multi-state Consecutively Connected System Retransmission delay model C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 6 8 1 7 4 3 2 5 Receiver 6 7 8 2 3 4 1 5 A C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 8 3 6 2 5 7 1 4 Receiver C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 B C

11. Optimal Element Allocation in a in the Presence of CCF 2p2-p222p2-p22 p2p2 p1p1 p1+p2p1+p2 I II C 1 C 2 C 3 e1e1 e2e2 e 1, e 2 C 1 C 2 C 3 R I = s (2p 2  p 2 2 ) R II = s p 2 + s 2 p 1 (p 1 +p 2 )

12. Multi-state Acyclic Networks Linear Consecutively Connected System Acyclic network

13. Multi-state Acyclic Networks Single terminal Multiple terminals Tree structure Connectivity Model Random vector G i (j) = {G i (j) (1), …, G i (j) (n)} Set of nodes connected to C i

14. CaCa CbCb CcCc CdCd CeCe CfCf CaCa CbCb CcCc CdCd CeCe CfCf Several elements located at the same node Random vector G i (j) = {G i (j) (1), …, G i (j) (n)} Set of nodes connected to C i

15. Set of nodes connected to C 1 by MEs located at C 1, C 2, …, C h. CaCa CbCb CcCc CdCd h+1 CfCf h CaCa CbCb CfCf CaCa CbCb CcCc CdCd CfCf h CaCa CbCb CcCc CdCd CfCf for h = 1,…,n  2

16. G i (j) = {G i (j) (1), …, G i (j) (n)} Model with capacitated arcs  ser (X, *) =  ser (*, X) = * for any X  par (X, *) =  par (*, X) = X for any X Transmission time:  par (X, Y) = min(X, Y)  ser (X, Y) = X+Y Max flow path capacity:  par (X, Y) = max(X, Y)  ser (X, Y) =min( X,Y(

17.  par (G (i) (f),  ser  G (i) (i+1),G (i+1) (f)))=  ser  G (i) (i+1),G (i+1) (f)) G(i)(e)G(i)(e) G(i)(d)G(i)(d) G (i) (i+1) G (i+1) (e) G(i)(f)G(i)(f) CfCf CeCe CdCd C i+1 CiCi CiCi CfCf CeCe CdCd  par (G (i) (d),  ser  G (i) (i+1),G (i+1) (d)))=G (i) (d)  par (G (i) (e),  ser  G (i) (i+1),G (i+1) (e))) Transformation of two elements into an equivalent one

18. Optimal element allocation in multi-state acyclic networks A C1C1 C2C2 p 1{2{ p 1{3} p 1{2,3} C1C1 C2C2 p 1{2} p 1{3} p 1{2,3{ B p 2{3{ C3C3 C3C3 S A = s{2(p 1{3} +p 1{2,3} )  (p 1{3} +p 1{2,3} ) 2 } S B =s{p 1{3} +p 1{2,3} } +s 2 (1  p 1{3}  p 1{2,3}  p 1  )(1  p 1  )

19. Optimal network reliability enhancement