Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T05:50:40.063Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparing lifetimes of coherent systems with dependent components operating in random environments

Part of: Operations research and management science Special processes Distribution theory - Probability

Published online by Cambridge University Press:  01 October 2019

Nil Kamal Hazra  and
Maxim Finkelstein
Nil Kamal Hazra*
Affiliation:
Indian Institute of Technology Jodhpur
Maxim Finkelstein*
Affiliation:
University of the Free State
*
*Postal address: Department of Mathematics, Indian Institute of Technology Jodhpur, Karwar 342037, India.
*Postal address: Department of Mathematics, Indian Institute of Technology Jodhpur, Karwar 342037, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the impact of a random environment on lifetimes of coherent systems with dependent components. There are two combined sources of this dependence. One results from the dependence of the components of the coherent system operating in a deterministic environment and the other is due to dependence of components of the system sharing the same random environment. We provide different sets of sufficient conditions for the corresponding stochastic comparisons and consider various scenarios, namely, (i) two different (as a specific case, identical) coherent systems operate in the same random environment; (ii) two coherent systems operate in two different random environments; (iii) one of the coherent systems operates in a random environment and the other in a deterministic environment. Some examples are given to illustrate the proposed reasoning.

Keywords

Coherent systemdistortion/domination functiondependent componentk-out-of-n systemrandom environmentstochastic order

MSC classification

Primary: 90B25: Reliability, availability, maintenance, inspection
Secondary: 60E15: Inequalities; stochastic orderings 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 937 - 957
Copyright
© Applied Probability Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amini-Seresht, E., Zhang, Y. and Balakrishnan, N. (2018). Stochastic comparisons of coherent systems under different random environments. J. Appl. Prob. 55, 459–472.CrossRefGoogle Scholar
Bada, F. G., Sangüesa, C. and Cha, J. H. (2014). Stochastic comparison of multivariate conditionally dependent mixtures. J. Multivar. Anal. 129, 82–94.CrossRefGoogle Scholar
Balakrishnan, N., Barmalzan, G. and Haidari, A. (2016). Multivariate stochastic comparisons of multivariate mixture models and their applications. J. Multivar. Anal. 145, 37–43.CrossRefGoogle Scholar
Balakrishnan, N. and Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Probab. Engrg. Inform. Sci. 27, 403–443.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing . Holt, Rinehart and Winston, New York.Google Scholar
Belzunce, F., Franco, M., Ruiz, J. M. and Ruiz, M. C. (2001). On partial orderings between coherent systems with different structures. Probab. Engrg. Inform. Sci. 15, 273–293.CrossRefGoogle Scholar
Belzunce, F., Mercader, J. A., Ruiz, J. M. and Spizzichino, F. (2009). Stochastic comparisons of multivariate mixture models. J. Multivar. Anal. 100, 1657–1669.CrossRefGoogle Scholar
Cha, J. H. and Finkelstein, M. (2018). Point Processes for Reliability Analysis. Shocks and Repairable Systems. Springer, London.CrossRefGoogle Scholar
Dewan, I. and Khaledi, B. E. (2014). On stochastic comparisons of residual lifetime at random time. Statist. Probab. Lett. 88, 73–79.CrossRefGoogle Scholar
Esary, J. D. and Proschan, F. (1963). Reliability between system failure rate and component failure rates. Technometrics 5, 183–189.CrossRefGoogle Scholar
Finkelstein, M. (1999). Wearing-out of components in a variable environment. Reliab. Engrg. System Safety 66, 235–242.CrossRefGoogle Scholar
Finkelstein, M. (2008). Failure Rate Modeling for Reliability and Risk. Springer, London.Google Scholar
Hazra, N. K., Kuiti, M. R., Finkelstein, M. and Nanda, A. K. (2017). On stochastic comparisons of maximum order statistics from the location-scale family of distributions. J. Multivar. Anal. 160, 31–41.CrossRefGoogle Scholar
Hazra, N. K. and Nanda, A. K. (2016). Stochastic comparisons between used systems and systems made by used components. IEEE Trans. Rel . 65, 751–762.CrossRefGoogle Scholar
Karlin, S. (1968). Total Positivity. Stanford University Press, CA.Google Scholar
Kenzin, M. and Frostig, E. (2009). M-out-of-n inspected systems subject to shocks in random environment. Reliab. Engrg. System Safety 94, 1322–1330.CrossRefGoogle Scholar
Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999). The ‘signature’ of a coherent system and its application to comparisons among systems. Naval Res. Logist . 46, 507–523.3.0.CO;2-D>CrossRefGoogle Scholar
Lindqvist, B. H., Samaniego, F. J. and Huseby, A. B. (2016). On the equivalence of systems of different sizes, with applications to system comparisons. Adv. Appl. Prob. 48, 332–348.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
Misra, A. K. and Misra, N. (2012). Stochastic properties of conditionally independent mixture models. J. Statist. Plann. Inference 142, 1599–1607.CrossRefGoogle Scholar
Mulero, J., Pellery, F. and Rodríguez-Griñolo, R. (2010). Negative aging and stochastic comparisons of residual lifetimes in multivariate frailty models. J. Statist. Plann. Inference 140, 1594–1600.CrossRefGoogle Scholar
Nakagawa, T. (1979). Further results of replacement problem of a parallel system in random environment. J. Appl. Prob. 16, 923–926.CrossRefGoogle Scholar
Nanda, A. K., Jain, K. and Singh, H. (1998). Preservation of some partial orderings under the formation of coherent systems. Statist. Probab. Lett. 39, 123–131.CrossRefGoogle Scholar
Navarro, J., Águila, Y. D., Sordo, M. A. and Suárez-Liorens, A. (2013). Stochastic ordering properties for systems with dependent identical distributed components. Appl. Stoch. Models Bus. Ind. 29, 264–278.CrossRefGoogle Scholar
Navarro, J., Águila, Y. D., Sordo, M. A. and Suárez-Liorens, A. (2014). Preservation of reliability classes under the formation of coherent systems. Appl. Stoch. Models Bus. Ind. 30, 444–454.CrossRefGoogle Scholar
Navarro, J., Águila, Y. D., Sordo, M. A. and Suárez-Liorens, A. (2016). Preservation of stochastic orders under the formation of generalized distorted distributions: applications to coherent systems. Methodol. Comput. Appl. Probab. 18, 529–545.CrossRefGoogle Scholar
Navarro, J., Pellerey, F. and Di Crescenzo, A. (2015). Orderings of coherent systems with randomized dependent components. Eur. J. Operat. Res. 240, 127–139.CrossRefGoogle Scholar
Navarro, J. and Rubio, R. (2010). Comparisons of coherent systems using stochastic precedence. TEST 19, 469–486.CrossRefGoogle Scholar
Nelsen, R. B. (1999). An Introduction to Copulas. Springer, New York.CrossRefGoogle Scholar
Persona, A., Sgarbossa, F. and Pham, H. (2016). Systemability: a new reliability function for different environments. In Quality and Reliability Management and Its Applications, pp. 145–193. Springer, London.CrossRefGoogle Scholar
Petakos, K. and Tsapelas, T. (1997). Reliability analysis for systems in a random environment. J. Appl. Prob. 34, 1021–1031.CrossRefGoogle Scholar
Pledger, P. and Proschan, F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In Optimizing Methods in Statistics, ed. Rustagi, J. S., pp. 89–113. Academic Press, New York.Google Scholar
Proschan, F. and Sethuraman, J. (1976). Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J. Multivar. Anal. 6, 608–616.CrossRefGoogle Scholar
Rde, L. (1976). Reliability systems in random environment. J. Appl. Prob. 13, 407–410.CrossRefGoogle Scholar
Samaniego, F. J. (2007). System Signature and Their Applications in Engineering Reliability. Springer, New York.CrossRefGoogle Scholar
Samaniego, F. J. and Navarro, J. (2016). On comparing coherent systems with heterogeneous components. Adv. Appl. Prob. 48, 88–111.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar