# A discrete MMAP for analysing the behaviour of a multi-state complex dynamic system subject to multiple events

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## Abstract

A complex multi-state system subject to different types of failures, repairable and/or non-repairable, external shocks and preventive maintenance is modelled by considering a discrete Markovian arrival process with marked arrivals (D-MMAP). The internal performance of the system is composed of several degradation states partitioned into minor and major damage states according to the risk of failure. Random external events can produce failures throughout the system. If an external shock occurs, there may be an aggravation of the internal degradation, cumulative external damage or extreme external failure. The internal performance and the cumulative external damage are observed by random inspection. If major degradation is observed, the unit goes to the repair facility for preventive maintenance. If a repairable failure occurs then the system goes to corrective repair with different time distributions depending on the failure state. Time distributions for corrective repair and preventive maintenance depend on the failure state. Rewards and costs depending on the state at which the device failed or was inspected are introduced. The system is modelled and several measures of interest are built into transient and stationary regimes. A preventive maintenance policy is shown to determine the effectiveness of preventive maintenance and the optimum state of internal and cumulative external damage at which preventive maintenance should be taken into account. A numerical example is presented, revealing the efficacy of the model. Correlations between the numbers of different events over time and in non-overlapping intervals are calculated. The results are expressed in algorithmic-matrix form and are implemented computationally with Matlab.

## Keywords

Reliability Complex multi-state systems Phase-type distribution Marked Markovian Arrival Process (MMAP) Correlations## Mathematics subject classification

60 J10 90B25 60 K10## Notes

### Acknowledgements

This paper is partially supported by the Junta de Andalucía, Spain, under the grant FQM-307 and by the Ministerio de Economía y Competitividad, España, under Grant MTM2017-88708-P and by the European Regional Development Fund (ERDF).

## References

- Buchholz P, Kriege J, Felko I (2014) Input modeling with phase-type distributions and Markov models. Theory and applications. Springer, HeidelbergGoogle Scholar
- Dewanji A, Segupta D, Chakraborty K (2011) A discrete time model for software reliability with application to a flight control software. Appl Stoch Model Bus Ind 27(6):723–731MathSciNetCrossRefzbMATHGoogle Scholar
- Eryilmaz S (2010) Mean residual and mean past lifetime of multi-state systems with identical components. IEEE Trans Reliab 59(4):644–649CrossRefGoogle Scholar
- He QM (2014) Fundamental of matrix analytic methods. Springer Science+Business Media, New YorkCrossRefGoogle Scholar
- Laggounea R, Chateauneuf A, Aissania D (2010) Preventive maintenance scheduling for a multi-component system with non-negligible replacement time. Int J Syst Sci 41(7):747–761MathSciNetCrossRefGoogle Scholar
- Li W, Pham H (2005) Reliability modeling of multi-state degraded systems with multi-competing failures and random shocks. IEEE Trans Reliab 54(2):297–303CrossRefGoogle Scholar
- Lisnianski A, Frenkel I, Ding Y (2010) Multi-state system reliability analysis and optimization for engineers and industrial managers. Springer-Verlag, LondonCrossRefzbMATHGoogle Scholar
- Liu Z, Ma X, Shenc L, Zhaob Y (2016) Degradation-shock-based reliability models for fault-tolerant systems. Qual Reliab Eng Int 32(3):949–955CrossRefGoogle Scholar
- Mahfoud H, El Barkany A, El Biyaali A (2016) Preventive maintenance optimization in healthcare domain: status of research and perspective. J Qual Reliab Eng 2016(Article ID 5314312):10Google Scholar
- Murchland J (1975) Fundamental concepts and relations for reliability analysis of multistate systems. In: Barlow RE, Fussell JB, Singpurwalla N (eds) Reliability and fault tree analysis: theoretical and applied aspects of system reliability. SIAM, Philadelphia, pp 581–618Google Scholar
- Neuts MF (1975) Probability distributions of phase type. In: Liber Amicorum Prof. Emeritus H. Florin. Department of Mathematics, University of Louvain, Belgium, pp 183–206Google Scholar
- Neuts MF (1979) A versatile Markovian point process. J Appl Probab 16:764–779MathSciNetCrossRefzbMATHGoogle Scholar
- Neuts MF (1981) Matrix geometric solutions in stochastic models. An algorithmic approach. John Hopkins, University Press, BaltimorezbMATHGoogle Scholar
- Osaki S, Asakura A (1970) A two-unit standby redundant system with repair and preventive maintenance. J Appl Probab 7:641–648MathSciNetCrossRefzbMATHGoogle Scholar
- Ruiz-Castro JE (2016a) Markov counting and reward processes for analyzing the performance of a complex system subject to random inspections. Reliab Eng Syst Saf 145:155–168CrossRefGoogle Scholar
- Ruiz-Castro JE (2016b) Complex multi-state systems modelled through Marked Markovian Arrival Processes. Eur J Oper Res 252(3):852–865MathSciNetCrossRefzbMATHGoogle Scholar
- Ruiz-Castro JE, Li Q-L (2011) Algorithm for a general discrete
*k*-out-of-*n*:*G*system subject to several types of failure with an indefinite number of repairpersons. Eur J Oper Res 211:97–111MathSciNetCrossRefzbMATHGoogle Scholar - Shatnawi O (2016) An integrated framework for developing discrete-time modelling in software reliability engineering. Qual Reliab Eng Int 32(8):2925–2943CrossRefGoogle Scholar
- Warrington L, Jones JA (2003) Representing complex systems within discrete event simulation for reliability assessment. In: Proceedings Annual Reliability and Maintainability Symposium, pp 487–492Google Scholar
- Yin H, Yang X, Peng R (2015) Generalized Accelerated Failure Time Frailty Model for Systems Subject to Imperfect Preventive Maintenance. Math Probl Eng 2015(Article ID 908742):8zbMATHGoogle Scholar