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On Dependent Multi-State Semi-Coherent Systems Based on Multi-State Joint Signature

Abstract

In this paper, by considering two multi-state semi-coherent systems sharing some components, we define the multi-state joint signature. Then, some properties of this multi-state joint signature are discussed in detail, and some preservation results are established in terms of stochastic ordering. A south-east shift order for comparing the multi-state joint signature matrices is then discussed. Finally, some numerical examples are presented for illustrating all the theoretical results established here.

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References

  1. Arcones MA, Kvam PH, Samaniego FJ (2002) Nonparametric estimation of a distribution subject to a stochastic precedence constraint. J Am Stat Assoc 97:170–182

    MathSciNet  MATH  Article  Google Scholar 

  2. Balakrishnan N, Volterman W (2014) On the signatures of ordered system lifetimes. J Appl Probab 51:82–91

    MathSciNet  MATH  Article  Google Scholar 

  3. Balakrishnan N, Volterman W (2016) Exact nonparametric inference for component and system lifetime distributions based on joint signatures. IEEE Trans Rel 65:179–186

    Article  Google Scholar 

  4. Balakrishnan N, Ng HKT, Navarro J (2011a) Exact nonparametric inference for component lifetime distribution based on lifetime data from systems with known signatures. J Nonparametr Stat 23:741–752

    MathSciNet  MATH  Article  Google Scholar 

  5. Balakrishnan N, Ng NKT, Navarro J (2011b) Linear inference for type-II censored lifetime data of reliability systems with known signatures. IEEE Trans Rel 60:426–440

    Article  Google Scholar 

  6. Bieniek M, Burkschat M, Rychlik T (2020) Comparisons of the expectations of system and component lifetimes in the failure dependent proportional hazard model. Methodol Comput Appl Probab 22:173–189

    MathSciNet  MATH  Article  Google Scholar 

  7. Boland PJ (2001) Signatures of indirect majority systems. J Appl Probab 38:597–603

    MathSciNet  MATH  Article  Google Scholar 

  8. Da Costa BV (2013) A multistate monotone system signature. Stat Probab Lett 83:2583–2591

    MathSciNet  MATH  Article  Google Scholar 

  9. Da GF, Hu TZ (2013) On bivariate signatures for systems with independent modules. In: Li H, Li X (eds) Stochastic orders in reliability and risk. Springer, New York

    Google Scholar 

  10. Da GF, Chan PS, Xu MC (2018a) On the signature of complex system: a decomposed approach. Eur J Oper Res 265:1115–1123

    MathSciNet  MATH  Article  Google Scholar 

  11. Da GF, Xu MC, Chan PS (2018b) An efficient algorithm for computing the signatures of systems with exchangeable components and applications. IISE Trans 50:584–595

    Article  Google Scholar 

  12. Eryilmaz S, Tuncel A (2016) Generalizing the survival signature to unrepairable homogeneous multi-state systems. Nav Res Logist 63:593–599

    MathSciNet  MATH  Article  Google Scholar 

  13. Gertsbakh I, Shpungin Y (2012a) Multidimensional spectra of multistate systems with binary components. In: Lisnianski A, Frenkel I (eds) Recent advances in system reliability. Springer, London

    MATH  Google Scholar 

  14. Gertsbakh I, Shpungin Y (2012b) Network reliability and resilience. Springer, London

    MATH  Google Scholar 

  15. Gertsbakh I, Shpungin Y, Spizzichino F (2011) Signatures of coherent systems built with separate modules. J Appl Probab 48:843–855

    MathSciNet  MATH  Article  Google Scholar 

  16. Gertsbakh I, Shpungin Y, Spizzichino F (2012) Two-dimensional signatures. J Appl Probab 49:416–429

    MathSciNet  MATH  Article  Google Scholar 

  17. Kochar S, Mukerjee H, Samaniego FJ (1999) The “signature” of a coherent system and its application to comparisons among systems. Nav Res Logist 46:507–523

    MathSciNet  MATH  Article  Google Scholar 

  18. Koutras MV, Triantafyllou IS, Eryilmaz S (2016) Stochastic comparisons between lifetimes of reliability systems with exchangeable components. Methodol Comput Appl Probab 18:1081–1095

    MathSciNet  MATH  Article  Google Scholar 

  19. Levitin G, Gertsbakh I, Shpungin Y (2011) Evaluating the damage associated with intentional network disintegration. Reliab Eng Syst Saf 96:433–439

    Article  Google Scholar 

  20. Lisnianski A, Frenkel I, Ding Y (2010) Multi-state system reliability analysis and optimization for engineers and industrial managers. Springer, London

    MATH  Book  Google Scholar 

  21. Marichal JL, Mathonet P (2013) Computing system signatures through reliability functions. Stat Probab Lett 83:710–717

    MathSciNet  MATH  Article  Google Scholar 

  22. Marichal JL, Mathonet P, Navarro J, Paroissin C (2017) Joint signature of two or more systems with applications to multistate systems made up of two-state components. Eur J Oper Res 263:559–570

    MathSciNet  MATH  Article  Google Scholar 

  23. Mohammadi L (2017a) The joint signature of parallel systems for different permutations of failure times. Comput Stat 32:1727–1746

    MathSciNet  MATH  Article  Google Scholar 

  24. Mohammadi L (2017b) The joint signature of coherent systems. Nav Res Logist 64:566–579

    MathSciNet  MATH  Article  Google Scholar 

  25. Natvig B (2010) Multistate systems: reliability theory with applications. Wiley, Hoboken

    MATH  Google Scholar 

  26. Navarro J, Rychlik T (2007) Reliability and expectation bounds for coherent systems with exchangeable components. J Multivar Anal 98:102–113

    MathSciNet  MATH  Article  Google Scholar 

  27. Navarro J, Spizzichino FJ (2020) Aggregation and signature based comparisons of multi-state systems via decompositions of fuzzy measures. Fuzzy Sets Syst 396:115–137

    MathSciNet  MATH  Article  Google Scholar 

  28. Navarro J, Ruiz JM, Sandoval CJ (2007) Properties of coherent systems with dependent components. Commun Stat Theory Methods 36:175–191

    MathSciNet  MATH  Article  Google Scholar 

  29. Navarro J, Samaniego FJ, Balakrishnan N, Bhattacharya D (2008) On the application and extension of system signatures in engineering reliability. Nav Res Logist 55:313–327

    MathSciNet  MATH  Article  Google Scholar 

  30. Navarro J, Samaniego FJ, Balakrishnan N (2010a) The joint signature of coherent systems with shared components. J Appl Probab 47:235–253

    MathSciNet  MATH  Article  Google Scholar 

  31. Navarro J, Spizzichino F, Balakrishnan N (2010b) Applications of average and projected systems to the study of coherent systems. J Multivar Anal 101:1471–1482

    MathSciNet  MATH  Article  Google Scholar 

  32. Navarro J, Samaniego FJ, Balakrishnan N (2013) Mixture representations for the joint distribution of lifetimes of two coherent systems with shared components. Adv Appl Probab 45:1011–1027

    MathSciNet  MATH  Article  Google Scholar 

  33. Reed S (2017) An efficient algorithm for exact computation of system and survival signatures using binary decision diagrams. Reliab Eng Syst Saf 165:257–267

    Article  Google Scholar 

  34. Samaniego FJ (1985) On closure of the IFR class under formation of coherent systems. IEEE Trans Rel 34:69–72

    MATH  Article  Google Scholar 

  35. Samaniego FJ (2007) System signatures and their applications in engineering reliability. Springer, New York

    MATH  Book  Google Scholar 

  36. Yang YD, Ng HKT, Balakrishnan N (2016) A stochastic expectation-maximization algorithm for the analysis of system lifetime data with known signature. Comput Stat 31:609–641

    MathSciNet  MATH  Article  Google Scholar 

  37. Yang YD, Ng HKT, Balakrishnan N (2019) Expectation-maximization algorithm for system-based lifetime data with unknown system structure. AStA Adv Stat Anal 103:69–98

    MathSciNet  MATH  Article  Google Scholar 

  38. Yi H, Cui LR (2018) A new computation method for signature: Markov process method. Nav Res Logist 65:410–426

    MathSciNet  MATH  Article  Google Scholar 

  39. Yi H, Balakrishnan N, Cui LR (2020) On the multi-state signatures of ordered system lifetimes. Adv Appl Probab 52:291–318

    MathSciNet  MATH  Article  Google Scholar 

  40. Yi H, Balakrishnan N, Cui LR (2021a) Comparisons of multi-state systems with binary components of different sizes. Methodol Comput Appl Probab. (to appear). https://doi.org/10.1007/s11009-020-09805-x

  41. Yi H, Cui LR, Balakrishnan N (2021b) Computation of survival signatures for multi-state consecutive-k systems. Reliab Eng Syst Saf 208:107429

    Article  Google Scholar 

  42. Zarezadeh S, Mohammadi L, Balakrishnan N (2018) On the joint signature of several coherent systems with some shared components. Eur J Oper Res 264:1092–1100

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

This work has been supported by the National Natural Science Foundation of China (No.72001016 and No.71631001). Our thanks also go to the anonymous reviewers and the Editor-in-Chief for their useful comments and suggestions on the earlier version of this manuscript which led to this improved version.

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Correspondence to He Yi.

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Yi, H., Balakrishnan, N. & Cui, L. On Dependent Multi-State Semi-Coherent Systems Based on Multi-State Joint Signature. Methodol Comput Appl Probab (2021). https://doi.org/10.1007/s11009-021-09877-3

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Keywords

  • Multi-state
  • Joint signature
  • Stochastic ordering
  • Preservation property
  • South-east shift order

Mathematics Subject Classification (2010)

  • Primary 60E15
  • Secondary 62H10