Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-21T14:13:40.406Z Has data issue: false hasContentIssue false
  • English
  • Français

On relative ageing of coherent systems with dependent identically distributed components

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

Published online by Cambridge University Press:  29 April 2020

Nil Kamal Hazra Open the ORCID record for Nil Kamal Hazra [Opens in a new window]  and
Neeraj Misra
Nil Kamal Hazra*
Affiliation:
Indian Institute of Technology Jodhpur
Neeraj Misra*
Affiliation:
Indian Institute of Technology Kanpur
*
*Postal address: Department of Mathematics, Indian Institute of Technology Jodhpur, Karwar-342037, India. Email address: nilkamal.nilu@gmail.com
**Postal address: Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

Relative ageing describes how one system ages with respect to another. The ageing faster orders are used to compare the relative ageing of two systems. Here, we study ageing faster orders in the hazard and reversed hazard rates. We provide some sufficient conditions for one coherent system to dominate another with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

Keywords

Coherent systemdual distortion/domination functionk-out-of-n systemredundancyresidual lifetimestochastic orders

MSC classification

Primary: 90B25: Reliability, availability, maintenance, inspection
Secondary: 60E15: Inequalities; stochastic orderings 60K10: Applications (reliability, demand theory, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 1 , March 2020 , pp. 348 - 376
Copyright
© Applied Probability Trust 2020

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
Arriaza, A., Sordo, M. A. and Suárez-Liorens, A. (2017). Comparing residual lives and inactivity times by transform stochastic orders. IEEE Trans. Rel. 66, 366–372.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. Prob. Eng. Inf. 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
Bartoszewicz, J. (1985). Dispersive ordering and monotone failure rate distributions. Adv. Appl. Prob. 17, 472–474.CrossRefGoogle Scholar
Belzunce, F., Franco, M., Ruiz, J. M. and Ruiz, M. C. (2001). On partial orderings between coherent systems with different structures. Prob. Eng. Inf. Sci. 15, 273–293.CrossRefGoogle Scholar
Belzunce, F., Martínez-Riquelme, C. and Mulero, J. (2016). An Introduction to Stochastic Orders. Academic Press, New York.Google Scholar
Boland, P. J. and El-Neweihi, E. (1985). Component redundancy versus system redundancy in the hazard rate ordering. IEEE Trans. Rel. 44, 614–619.CrossRefGoogle Scholar
Champlin, R., Mitsuyasu, R., Elashoff, R. and Gale, R. P. (1983). In Recent Advances in Bone Marrow Transplantation (UCLA Symposia on Molecular and Cellular Biology), ed. Gale, R. P., pp. 141–158. Alan R. Liss, New York.Google Scholar
Cox, D. R. (1972). Regression models and life-tables. J. R. Statist. Soc. B [Statist. Methodology] 34, 187–220.Google Scholar
Da, G. and Ding, W. (2016). Component level versus system level k-out-of-n assembly systems. IEEE Trans. Rel. 65, 425–433.CrossRefGoogle Scholar
Deshpande, J. V. and Kochar, S. C. (1983). Dispersive ordering is the same as tail-ordering. Adv. Appl. Prob. 15, 686–687.CrossRefGoogle Scholar
Di Crescenzo, A. (2000). Some results on the proportional reversed hazards model. Statist. Prob. Lett. 50, 313–321.CrossRefGoogle Scholar
Ding, W. and Zhang, Y. (2018). Relative ageing of series and parallel systems: effects of dependence and heterogeneity among components. Operat. Res. Lett. 46, 219–224.CrossRefGoogle Scholar
Ding, W., Fang, R. and Zhao, P. (2017). Relative aging of coherent systems. Naval Res. Logistics 64, 345–354.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. (2006). On relative ordering of mean residual lifetime functions. Statist. Prob. Lett. 76, 939–944.CrossRefGoogle Scholar
Finkelstein, M. (2008). Failure Rate Modeling for Reliability and Risk. Springer, London.Google Scholar
Gupta, N. (2013). Stochastic comparisons of residual lifetimes and inactivity times of coherent systems. J. Appl. Prob. 50, 848–860.CrossRefGoogle Scholar
Gupta, N., Misra, N. and Kumar, S. (2015). Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components. Europ. J. Operat. Res. 240, 425–430.CrossRefGoogle Scholar
Hazra, N. K. and Nanda, A. K. (2014). Component redundancy versus system redundancy in different stochastic orderings. IEEE Trans. Rel. 63, 567–582.CrossRefGoogle Scholar
Hazra, N. K. and Nanda, A. K. (2015). A note on warm standby system. Statist. Prob. Lett. 106, 30–38.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
Hazra, N. K. and Nanda, A. K. (2016). On some generalized orderings: in the spirit of relative ageing. Commun. Statist. Theory Meth. 45, 6165–6181.CrossRefGoogle 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
Kalashnikov, V. V. and Rachev, S. T. (1986). Characterization of queueing models and their stability. In Probability Theory and Mathematical Statistics, eds Prohorov, Y. K.et al., pp. 37–53. VNU Science Press, Amsterdam.Google Scholar
Karlin, S. (1968). Total Positivity. Stanford University Press, Stanford, CA.Google Scholar
Kayid, M., Izadkhah, S. and Zuo, M. J. (2017). Some results on the relative ordering of two frailty models. Statist. Papers 58, 287–301.CrossRefGoogle Scholar
Kochar, S. C. and Wiens, D. P. (1987). Partial orderings of life distributions with respect to their ageing properties. Naval Res. Logistics 34, 823–829.3.0.CO;2-R>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. Logistics 46, 507–523.3.0.CO;2-D>CrossRefGoogle Scholar
Lai, C. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.Google Scholar
Li, C. and Li, X. (2016). Relative ageing of series and parallel systems with statistically independent and heterogeneous component lifetimes. IEEE Trans. Rel. 65, 1014–1021.CrossRefGoogle Scholar
Li, X. and Lu, X. (2003). Stochastic comparison on residual life and inactivity time of series and parallel systems. Prob. Eng. Inf. Sci. 17, 267–275.CrossRefGoogle Scholar
Mantel, N. and Stablein, D. M. (1988). The crossing hazard function problem. J. R. Statist. Soc. D 37 59–64.Google Scholar
Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
Misra, N. and Francis, J. (2015). Relative ageing of (n—k+1)-out-of-n systems. Statist. Prob. Lett. 106, 272–280.CrossRefGoogle Scholar
Misra, N. and Francis, J. (2018). Relative aging of (n—k+1)-out-of-n)-out-of-n systems based on cumulative hazard and cumulative reversed hazard functions. Naval Res. Logistics 65, 566–575.CrossRefGoogle Scholar
Misra, N., Dhariyal, I. D. and Gupta, N. (2009). Optimal allocation of active spares in series systems and comparison of component and system redundancies. J. Appl. Prob. 46 19–34.CrossRefGoogle Scholar
Misra, N., Francis, J. and Naqvi, S. (2017). Some sufficient conditions for relative aging of life distributions. Prob. Eng. Inf. Sci. 31, 83–99.CrossRefGoogle Scholar
Nanda, A. K. and Hazra, N. K. (2013). Some results on active redundancy at component level versus system level. Operat. Res. Lett. 41, 241–245.CrossRefGoogle Scholar
Nanda, A. K., Hazra, N. K., Al-Mutairi, D. K. and Ghitany, M. E. (2017). On some generalized ageing orderings. Commun. Statist. Theory Meth. 46, 5273–5291.CrossRefGoogle Scholar
Nanda, A. K., Jain, K. and Singh, H. (1998). Preservation of some partial orderings under the formation of coherent systems. Statist. Prob. Lett. 39, 123–131.CrossRefGoogle Scholar
Navarro, J. and Rubio, R. (2010). Comparisons of coherent systems using stochastic precedence. Test 19, 469–486.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 Business Industry 29, 264–278.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. Methodology Comput. Appl. Prob. 18, 529–545.CrossRefGoogle Scholar
Navarro, J., Fernández-Martinez, P., Fernández-Sánchez, J. and Arriaza, A. (2019). Relationships between importance measures and redundancy in systems with dependent components. Prob. Eng. Inf. Sci. doi:10.1017/S0269964819000159.CrossRefGoogle Scholar
Navarro, J., Pellerey, F. and Di Crescenzo, A. (2015). Orderings of coherent systems with randomized dependent components. Europ. J. Operat. Res. 240, 127–139.CrossRefGoogle Scholar
Nelsen, R. B. (1999). An Introduction to Copulas. Springer, New York.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
Pocock, S. J., Gore, S. M. and Keer, G. R. (1982). Long-term survival analysis: the curability of breast cancer. Statist. Med. 1, 93–104.CrossRefGoogle ScholarPubMed
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
Razaei, M., Gholizadeh, B. and Izadkhah, S. (2015). On relative reversed hazard rate order. Commun. Statist. Theory Meth. 44, 300–308.CrossRefGoogle Scholar
Samaniego, F. J. and Navarro, J. (2016). On comparing coherent systems with heterogeneous components. Adv. Appl. Prob. 48, 88–111.CrossRefGoogle Scholar
Sengupta, D. and Deshpande, J. V. (1994). Some results on the relative ageing of two life distributions. J. Appl. Prob. 31 991–1003.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Zhang, Y., Amini-Seresht, E. and Ding, W. (2017). Component and system active redundancies for coherent systems with dependent components. Appl. Stoch. Models Business Industry 33, 409–421.Google Scholar
Zhao, P., Zhang, Y. and Li, L. (2015). Redundancy allocation at component level versus system level. Europ. J. Operat. Res. 241, 402–411.CrossRefGoogle Scholar