On the coherent systems subject to Marshall-Olkin type shocks
Coherent systems and Marshall-Olkin run shock models are combined. Coherent systems consisting of n components receive some kind of shocks from n+1 different sources similar to Marshall-Olkin type. More precisely, when the component j receives k consecutive fatal shocks from the source j or k consecutive fatal shocks from the source n+1, it fails, j = 1, …,n. When the interarrival time of shocks has phase-type distribution, reliability, mean time to failure (MTTF) and mean residual life (MRL) function of the coherent systems are studied. Numerical examples and graphical representations are provided.
___
- Marshall, A. W. and Olkin, I., A multivariate exponential distribution, J. Amer. Stat. Assoc. 62 (1967) 30--44.
- Ozkut, M. and Bayramoglu,I., On Marshall--Olkin type distribution with effect of shock magnitude, J. Comput. Appl. Math. 271 (2014) 150--162.
- Bayramoglu, I. and M. Ozkut, The reliability of coherent systems subjected to Marshall--Olkin type shocks, IEEE Trans. Rel. 64 (2015) 435-443.
- Durante, F., Girard, S. and Mazo, G., Marshall--Olkin type copulas generated by a global shock, J. Comput. Appl. Math. 296 (2016) 638--648.
- Ozkut, M. and Eryilmaz, S., Reliability analysis under Marshall--Olkin run shock model, J. Comput. Appl. Math. 349 (2019) 52--59.
- Neuts, M.F. and Meier, K.S, On the use of phase-type distributions in reliability modeling of systems with two components, OR Spektrum 2 (1981) 227--234.
- He, Q.M., Fundamentals of matrix-analytic methods, New York: Springer (2014).
- Pérez-Ocón, R. and Segovia, M.C., Shock models under a markovian arrival process. Math Comput Model 50 (2009) 879--884.
- Segovia, M.C. and Labeau, P.E, Reliability of a multi-state system subject to shocks using phase-type distributions, Appl Math Model 37 (2013) 4883--4904.
- Zhao, X., Guo, X. and Wang, X., Reliability and maintenance policies for a two-stage shock model with self-healing mechanism, Reliab Eng Syst Saf 172 (2018) 185--194.
- Cui, L. and Wu, B., Extended phase-type models for multistate competing risk systems, Reliab Eng Syst Saf 181 (2019) 1--16.
- Tank, F. and Eryilmaz, S., The distributions of sum, nima and maxima of generalized geometric random variables, Statist. Papers 56 (2015) 1191--1203.
- Bayramoglu, I. and Ozkut, M., The reliability of coherent systems subjected to Marshall--Olkin type shocks, IEEE Trans. Reliab. 64 (1) (2015) 434--443.
- Bairamov, I. and Parsi, S., Order statistics from mixed exchangeable random variables, J. Computat. Appl.Math., 235 (2011) 4629--4638.
- F. J. Samaniego, On closure of the IFR class under formation of coherent systems, IEEE Trans. Rel., 34(1) (1985) 1508--1527 .
- Kochar, S., Mukerjee, H. and Samaniego, F. J., The "signature" of a coherent system and its application to comparisons among systems, Naval Res. Logistics 46 (1999) 507--523.
- Navarro, J. and Rychlik, T., Reliability and expectation bounds for coherent systems with exchangeable components, J. Multivariate Anal. 98 (2007) 102--113.
- Eryilmaz, S., The number of failed components in a coherent system with exchangeable components, IEEE Trans. Reliab. 61 (2012) 203--207.
- Nama, M.K. and Asadi, M., Stochastic properties of components in a used coherent system, Methodol. Comput. Appl. Probab. 16 (2014) 675--691.
- Navarro, J. and Hernandez, P.J., Mean residual life functions of finite mixtures, order statistics and coherent systems, Metrika 67 (2008) 277-298.
- Ucer, B. and Gurler, S., On the mean residual lifetime at sys-tem level in two-component parallel systems for the FGMdistribution, J Math Stat 41 (2012) 139--145.
- Bayramoglu, I. and Ozkut, M., Mean residual life and inactivity time of a coherent system subjected to Marshall--Olkin type shocks, J. Comput. Appl. Math 298 (2016) 190--200.