Combined Markov and UGF Methods for Multi-State Repairable Phased Mission Systems

Document Type : Original Research Article


Department of Statistics, Faculty of Science, Ege University, Izmir, Turkey


The reliability analysis of multi-state phased mission systems (MS-PMS) is a crucial area of study in systems engineering and reliability engineering. An MS-PMS consists of multiple phases where the system can exist in different operational states in each phase. The system transitions from one phase to the next based on the success or failure of the current phase. The reliability of an MS-PMS depends on the reliabilities of each phase and the transition probabilities between system states across phases. By thoroughly analyzing the reliability of each phase and accurately estimating the probabilities of state transitions, the overall system reliability can be determined. Several methods are used for MS-PMS reliability analysis, such as Markov models, Universal Generating Function (UGF) technique, Petri nets, fault trees, etc. This study evaluated the reliability analysis of an MS-PMS with a combination of Markov and UGF techniques. This method is defined as a combined technique in the literature. The Markov modeling approach represents the system as a set of states with transitions between states based on the failure and repair of components. In addition, the UGF technique converts the Markov model into a set of algebraic equations that can be solved to obtain reliability metrics such as system availability, mean time to failure, etc. In this research, a three-phased multi-state repairable system was discussed. Transition diagrams were created based on components for all phases, and the resulting differential equations were solved. Then, the UGF method was applied according to the system structure of the phases, and the reliability metrics of the system were obtained.


Main Subjects

  1. D. Esary and H. Ziehms, “Reliability of Phased Missions, Reliability and Fault Tree Analysis”, Naval Postgraduate School, USA, pp. 213-236, 1975.
  2. Yuchang, “New Insights Into the BDD-Based Reliability Analysis of Phased-Mission Systems”, IEEE Transactions on Reliability, No.4, pp. 667-678, 2009.
  3. Yuchang, L. Xing and S. Amari, “A Multiple-Valued Decision Diagram Based Method for Efficient Reliability Analysis of Non-Repairable Phased-Mission Systems”, IEEE Transactions on Reliability, No.1, pp. 320-330, 2014.
  4. Peng, Q. Zhai, L. Xing and J. Yang, “Reliability Analysis and Optimal Structure of Series-Parallel Phased-Mission Systems Subject to Fault-Level Coverage”, IEEE Transactions on Reliability, No.8, pp.736-746, 2016.
  5. Smotherman and K. Zemoudeh, “A Non-Homogeneous Markov Model for Phased-Mission Reliability Analysis”, IEEE Transactions on Reliability, Vol.38, No.5, pp. 585-590, 1989.
  6. Li, H. Huanga, Y. Li and E. Zio, “Reliability Assessment of Multi-State Phased Mission System with Non-Repairable Multi-State Components”, Applied Mathematical Modelling, pp. 181-189, 2018.
  7. Cheng, J. Yang and L. Li, “Reliability Assessment of Multi-state Phased Mission Systems with Common Bus Performance Sharing Considering Transmission Loss and Performance Storage”, Reliability Engineering and System Safety, Vol. 199, 2020,
  8. Wu and L. Cui, “Reliability of Repairable Multi-State Two-Phase Mission Systems with Finite Number of Phase Switches”, Applied Mathematical Modelling, pp. 1229-1241, 2020.
  9. Shrestha, L. Xing and Y. Dai, "Reliability analysis of multi-state phased-mission systems," 2009 Annual Reliability and Maintainability Symposium, Fort Worth, TX, USA, pp. 151-156, 2009, Doi: 10.1109/RAMS.2009.4914667.
  10. Lisnianski and G. Levitin, Multi-State System Reliability: Assessment, Optimization and Applications, World Scientific Publishing 372 Co Pte Ltd, 2003.
  11. Qin J., Niu Y. and Li Z., “A combined method for reliability analysis of multi state system of minor repairable components”, Eksploatacja i Niezawodnosc Maintenance and Reliability, Vol.18, pp. 80-88, 2016.
  12. Basile, P. Dehombreux and F. Riane, “Identification of reliability models for non repairable and repairable systems with small samples”, Proceedings of IMS2004 Conference on Advances in Maintenance and Modeling, pp.8, 2004.
  13. Marshall, F. Proschan, “Classes of Distrubutions Applicable in Replacement with Renewal Thory Applications”, 6.Berkeley Symposium, 1972.
  14. A. Buzacott, “Markov Approach to Finding Failure Times of Repairable Systems”, IEEE Transaction on Reliability, R-19, No. 4, pp. 128-134, 1970.
  15. A. Ascher, “Evaluation of Repairable System Reliability Using the "Bad-As-Old" Concept”, IEEE Transaction on Reliability, R-17, No. 2, pp.103-110, 1968.
  16. R. Burdick, J. B. Fussell, D. M.  Rasmuson and J. R. Wilson, “Phased mission analysis: A review of new developments and an application”,IEEE Transactions on Reliability, Vol.26 No.1, pp. 43-49,1977.
  17. Lisnianski, I. Frenkel, Y. Ding, “Multi-state System Reliability Analysis and Optimization for Engineers and Industrial Managers”, Springer Science & Business Media, London, UK, 2010
  18. B., Misra, Handbook of Performability Engineering, Springer Verlag, London, 1316p, 2008.
  19. Wu, R. Peng and L. Xing, “Recent advances on reliability of Phased Mission Systems”, In: Stochastic Models in Reliability, Network Security and System Safety, JHC80 2019, pp 19-43 Vol. 1102, Springer, Singapore, 2019.
  20. Xing and J. B. Dugan, “Analysis of Generalized Phased-Mission System Reliability, Performance, and Sensitivity”, IEEE Transactions on Reliability, Vol.51 No.2, pp. 199-211, 2002.
  21. J. Dunnett and J. D. Andrews, “A binary decision diagram method for phased mission analysis of non – repairable systems”, Proceedings – ImechE Risk and Reliability, Vol.220, No.1, 2006.
  22. Lisnianski, “Extended block diagram method for a multi-state system reliability assesment”, Reliability Engineering and System Safety, Vol.92, No.12, pp. 1601-1607, 2006.
  23. Chew, “Systems reliability modelling for phased missions with maintenance-free operating periods”, PhD Thesis, Lougborough University, pp.339, 2010.
  24. Ding and S. Han, “Multi-state reliability analysis of rotor system using Semi-Markov model and UGF”, Journal of Vibroengineering ,Vol.20, No.5, pp. 2060-2072, 2018.
  25. Bondavalli, I. Mura and M. Nelli, “Analytical Modelling and Evaluation of Phased-Mission Systems for Space Applications”, IEEE Transactions on Reliability, pp. 85-91, 1997.
  26. Mandelli, T. Aldemir and E. Zio, “An Event Tree/Fault Tree/Embedded Markov Model Approach for the PSAM-8 Benchmark Problem Concerning A Phased Mission Space Propulsion subsystem”, Proc. of the 8th Int. Conf. On Probabilistic Safety Assessment and Management, pp. 9, 2006.
  27. Li, H. Huang ,Y. Li and E. Zio, “Reliability assessment of multi-state phased mission system with non-repairable multi-state components”, Applied Mathematical Modelling, pp. 181-199, 2018.
  28. Li, H. Huang, Y. Li and X. Xiong, “A Markov regenerative process model for phased mission systems under internal degradation and external shocks”, Reliability Engineering and System Safety, Vol. 215, 2021,
  29. Meshkat, L. Xing, S. Donohue and Y. Ou , “An overview of the phase-modular fault tree approach to phased Imission system analysis”, Proceedings of the International Conference on Space Mission Challenges for Information Technology, pp. 393-398., 2003.
  30. Cheng, J. Yang and L. Li., ”Reliability Assessment of Multi-State Phased Mission Systems With Common Bus Performance Sharing Subjected to Epistemic Uncertainty”, IEEE Transactions on Reliability, Vol. 71, No. 3, pp. 1281-1293, 2022.