Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T07:38:05.927Z Has data issue: false hasContentIssue false
  • English
  • Français

ON RELATIVE AGING COMPARISONS OF COHERENT SYSTEMS WITH IDENTICALLY DISTRIBUTED COMPONENTS

Published online by Cambridge University Press:  12 February 2020

Nil Kamal Hazra Open the ORCID record for Nil Kamal Hazra [Opens in a new window]  and
Neeraj Misra
Nil Kamal Hazra
Affiliation:
Department of Mathematics, Indian Institute of Technology Jodhpur, Karwar 342037, Rajasthan, India E-mail: nilkamal@iitj.ac.in
Neeraj Misra
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India
Get access Rights & Permissions [Opens in a new window]

Abstract

The relative aging is an important notion which is useful to measure how a system ages relative to another one. Among the existing stochastic orders, there are two important orders describing the relative aging of two systems, namely, aging faster orders in the cumulative hazard and the cumulative reversed hazard rate functions. In this paper, we give some sufficient conditions under which one coherent system ages faster than another one with respect to the aforementioned stochastic orders. Further, we show that the proposed sufficient conditions are satisfied for k-out-of-n systems. Moreover, some numerical examples are given to illustrate the applications of proposed results.

Keywords

coherent systemdual distortion/domination functionk-out-of-n systemstochastic orders
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 3 , July 2021 , pp. 481 - 495
Copyright
© Cambridge University Press 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

1.Amini-Seresht, E., Zhang, Y., & Balakrishnan, N. (2018). Stochastic comparisons of coherent systems under different random environments. Journal of Applied Probability 55: 459–472.CrossRefGoogle Scholar
2.Balakrishnan, N. & Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: A review with an emphasis on some recent developments. Probability in the Engineering and Informational Sciences 27: 403–443.CrossRefGoogle Scholar
3.Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing. New York: Holt, Rinehart and Winston.Google Scholar
4.Bartoszewicz, J. (1985). Dispersive ordering and monotone failure rate distributions. Advances in Applied Probability 17: 472–474.CrossRefGoogle Scholar
5.Belzunce, F., Franco, M., Ruiz, J.M., & Ruiz, M.C. (2001). On partial orderings between coherent systems with different structures. Probability in the Engineering and Informational Sciences 15: 273–293.CrossRefGoogle Scholar
6.Belzunce, F., Martínez-Riquelme, C., & Mulero, J. (2016). An introduction to stochastic orders. New York: Academic Press.Google Scholar
7.Champlin, R., Mitsuyasu, R., Elashoff, R., & Gale, R.P. (1983). Recent advances in bone marrow transplantation. In Gale, R.P. (ed.), UCLA symposia on molecular and cellular biology, vol. 7. New York, pp. 141–158.Google Scholar
8.Deshpande, J.V. & Kochar, S.C. (1983). Dispersive ordering is the same as tail-ordering. Advances in Applied Probability 15: 686–687.CrossRefGoogle Scholar
9.Di Crescenzo, A. (2000). Some results on the proportional reversed hazards model. Statistics and Probability Letters 50: 313–321.CrossRefGoogle Scholar
10.Ding, W., Fang, R., & Zhao, P. (2017). Relative aging of coherent systems. Naval Research Logistics 64: 345–354.CrossRefGoogle Scholar
11.Ding, W. & Zhang, Y. (2018). Relative ageing of series and parallel systems: Effects of dependence and heterogeneity among components. Operations Research Letters 46: 219–224.CrossRefGoogle Scholar
12.Esary, J.D. & Proschan, F. (1963). Reliability between system failure rate and component failure rates. Technometrics 5: 183–189.CrossRefGoogle Scholar
13.Finkelstein, M. (2006). On relative ordering of mean residual lifetime functions. Statistics and Probability Letters 76: 939–944.CrossRefGoogle Scholar
14.Finkelstein, M. (2008). Failure rate modeling for reliability and risk. London: Springer.Google Scholar
15.Hazra, N.K. & Misra, N. (2019). On relative ageing of coherent systems with dependent identically distributed components. arXiv:1906.08488v1.Google Scholar
16.Hazra, N.K. & Nanda, A.K. (2015). A note on warm standby system. Statistics and Probability Letters 106: 30–38.CrossRefGoogle Scholar
17.Hazra, N.K. & Nanda, A.K. (2016). On some generalized orderings: In the spirit of relative ageing. Communications in Statistics – Theory and Methods 45: 6165–6181.CrossRefGoogle Scholar
18.Kalashnikov, V.V. & Rachev, S.T. (1986). Characterization of queueing models and their stability. In Prohorov, Yu.K. (eds.), Probability theory and mathematical statistics, vol. 2. Amsterdam: VNU Science Press, pp. 37–53.Google Scholar
19.Karlin, S. (1968). Total positivity. Stanford, CA: Stanford University Press.Google Scholar
20.Kayid, M., Izadkhah, S., & Zuo, M.J. (2017). Some results on the relative ordering of two frailty models. Statistical Papers 58: 287–301.Google Scholar
21.Kochar, S., Mukerjee, H., & Samaniego, F.J. (1999). The ‘signature’ of a coherent system and its application to comparisons among systems. Naval Research Logistics 46: 507–523.3.0.CO;2-D>CrossRefGoogle Scholar
22.Kochar, S.C. & Wiens, D.P. (1987). Partial orderings of life distributions with respect to their ageing properties. Naval Research Logistics 34: 823–829.3.0.CO;2-R>CrossRefGoogle Scholar
23.Lai, C. & Xie, M. (2006). Stochastic ageing and dependence for reliability. New York: Springer.Google Scholar
24.Li, C. & Li, X. (2016). Relative ageing of series and parallel systems with statistically independent and heterogeneous component lifetimes. IEEE Transactions on Reliability 65: 1014–1021.CrossRefGoogle Scholar
25.Misra, N. & Francis, J. (2015). Relative ageing of (n−k+1)-out-of-n systems. Statistics and Probability Letters 106: 272–280.CrossRefGoogle Scholar
26.Misra, N. & Francis, J. (2018). Relative aging of (n−k+1)-out-of-n systems based on cumulative hazard and cumulative reversed hazard functions. Naval Research Logistics 65: 566–575.CrossRefGoogle Scholar
27.Misra, N., Francis, J., & Naqvi, S. (2017). Some sufficient conditions for relative aging of life distributions. Probability in the Engineering and Informational Sciences 31: 83–99.CrossRefGoogle Scholar
28.Nanda, A.K., Hazra, N.K., Al-Mutairi, D.K., & Ghitany, M.E. (2017). On some generalized ageing orderings. Communications in Statistics – Theory and Methods 46: 5273–5291.CrossRefGoogle Scholar
29.Nanda, A.K., Jain, K., & Singh, H. (1998). Preservation of some partial orderings under the formation of coherent systems. Statistics and Probability Letters 39: 123–131.CrossRefGoogle Scholar
30.Navarro, J. (2018). Stochastic comparisons of coherent systems. Metrika 81: 465–482.CrossRefGoogle Scholar
30.Navarro, J., Águila, Y.D., Sordo, M.A., & Suárez-Llorens, A. (2013). Stochastic ordering properties for systems with dependent identical distributed components. Applied Stochastic Models in Business and Industry 29: 264–278.CrossRefGoogle Scholar
32.Navarro, J., Águila, Y.D., Sordo, M.A., & Suárez-Llorens, A. (2014). Preservation of reliability classes under the formation of coherent systems. Applied Stochastic Models in Business and Industry 30: 444–454.CrossRefGoogle Scholar
33.Navarro, J., Águila, Y.D., Sordo, M.A., & Suárez-Llorens, A. (2016). Preservation of stochastic orders under the formation of generalized distorted distributions: Applications to coherent systems. Methodology and Computing in Applied Probability 18: 529–545.CrossRefGoogle Scholar
34.Navarro, J., Pellerey, F., & Di Crescenzo, A. (2015). Orderings of coherent systems with randomized dependent components. European Journal of Operational Research 240: 127–139.CrossRefGoogle Scholar
35.Navarro, J. & Rubio, R. (2010). Comparisons of coherent systems using stochastic precedence. TEST 19: 469–486.CrossRefGoogle Scholar
36.Nelsen, R.B. (1999). An introduction to copulas. New York: Springer.CrossRefGoogle Scholar
37.Pledger, P. & Proschan, F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In J.S., Rustagi (ed.), Optimizing methods in statistics. New York: Academic Press, pp. 89–113.Google Scholar
38.Pocock, S.J., Gore, S.M., & Keer, G.R. (1982). Long-term survival analysis: The curability of breast cancer. Statistics in Medicine 1: 93–104.CrossRefGoogle ScholarPubMed
39.Proschan, F. & Sethuraman, J. (1976). Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. Journal of Multivariate Analysis 6: 608–616.CrossRefGoogle Scholar
40.Razaei, M., Gholizadeh, B., & Izadkhah, S. (2015). On relative reversed hazard rate order. Communications in Statistics – Theory and Methods 44: 300–308.CrossRefGoogle Scholar
41.Samaniego, F.J. & Navarro, J. (2016). On comparing coherent systems with heterogeneous components. Advances in Applied Probability 48: 88–111.CrossRefGoogle Scholar
42.Sengupta, D. & Deshpande, J.V. (1994). Some results on the relative ageing of two life distributions. Journal of Applied Probability 31: 991–1003.CrossRefGoogle Scholar
43.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar