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FURTHER RESULTS ON STOCHASTIC ORDERINGS AND AGING CLASSES IN SYSTEMS WITH AGE REPLACEMENT

Published online by Cambridge University Press:  05 February 2021

Josué Corujo Open the ORCID record for Josué Corujo [Opens in a new window]  and
José E. Valdés
Josué Corujo
Affiliation:
CEREMADE, Université Paris-Dauphine, Université PSL, CNRS, 75016 Paris, France Institut de Mathématiques de Toulouse, Institut National des Sciences Appliquées, 31077 Toulouse, France E-mail: [email protected]
José E. Valdés
Affiliation:
Facultad de Matématica y Computación, Universidad de La Habana, 10400 Habana, Cuba E-mail: [email protected]
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Abstract

Reliability properties associated with the classic models of systems with age replacement have been a usual topic of research. Most previous works have checked the aging properties of the lifetime of the working units using stochastic comparisons among the systems with age replacement at different times. However, from a practical point of view, it would also be interesting to deduce to which aging classes the lifetime of the system belongs, making use of the aging properties of the lifetime of its working units. The first part of this article deals with this problem. Further along, stochastic orderings are established between the systems with replacement at the same time using several stochastic comparisons among the lifetimes of their working units. In addition, the lifetimes of two systems with age replacement are compared as well. This is performed assuming stochastic orderings between the number of replacement until failure, and the lifetimes of their working units conditioned to be less or equal than the replacement time. Similar comparisons are accomplished considering two systems with age replacement where the replacements occur at a random time. Illustrative examples are presented throughout the paper.

Keywords

age replacementaging classesparallel systemsrandom time replacementstationary pointwise availabilitystochastic orders
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 3 , July 2022 , pp. 660 - 689
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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References

Angus, J.E., Meng-Lai, Y., & Trivedi, K. (2012). Optimal random age replacement for availability. International Journal of Reliability, Quality and Safety Engineering 19(05): 1250021.CrossRefGoogle Scholar
Anis, M.Z. & Basu, K. (2014). Tests for exponentiality against NBUE alternatives: a Monte Carlo comparison. Journal of Statistical Computation and Simulation 84(2): 231–247.CrossRefGoogle Scholar
Asha, G. & Unnikrishnan Nair, N. (2010). Reliability properties of mean time to failure in age replacement models. International Journal of Reliability, Quality and Safety Engineering 17(1): 15–26.CrossRefGoogle Scholar
Barlow, R.E. & Proschan, F. (1965). Mathematical theory of reliability. With contributions by Larry C. Hunter. The SIAM Series in Applied Mathematics. New York-London-Sydney: John Wiley & Sons, Inc.Google Scholar
Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing: probability models. Silver Spring, MD: To Begin With.Google Scholar
Belzunce, F., Ortega, E.-M., & Ruiz, J.M. (2005). A note on replacement policy comparisons from NBUC lifetime of the unit. Statistical Papers 46(4): 509–522.CrossRefGoogle Scholar
Belzunce, F., Martínez-Riquelme, C., & Mulero, J. (2016). An introduction to stochastic orders. Amsterdam: Elsevier/Academic Press.Google Scholar
Block, H.W., Langberg, N.A., & Savits, T.H. (1990). Maintenance comparisons: block policies. Journal of Applied Probability 27(3): 649–657.CrossRefGoogle Scholar
Bobotas, P. & Koutras, M.V. (2019). Distributions of the minimum and the maximum of a random number of random variables. Statistics & Probability Letters 146: 57–64.CrossRefGoogle Scholar
Eryilmaz, S. (2017). A note on optimization problems of a parallel system with a random number of units. International Journal of Reliability, Quality and Safety Engineering 24(05): 1750022.CrossRefGoogle Scholar
Ito, K., Zhao, X., & Nakagawa, T. (2017). Random number of units for K-out-of-n systems. Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems 45: 563–572.Google Scholar
Izadi, M., Sharafi, M., & Khaledi, B.-E. (2018). New nonparametric classes of distributions in terms of mean time to failure in age replacement. Journal of Applied Probability 55(4): 1238–1248.CrossRefGoogle Scholar
Jain, K. (2009). Stochastic comparison of repairable systems. American Journal of Mathematical and Management Sciences 29(3–4): 477–495.CrossRefGoogle Scholar
Karlin, S. (1968). Total positivity, vol. I. Stanford, CA: Stanford University Press.Google Scholar
Kayid, M., Ahmad, I.A., Izadkhah, S., & Abouammoh, A.M. (2013). Further results involving the mean time to failure order, and the decreasing mean time to failure class. IEEE Transactions on Reliability 62(3): 670–678.CrossRefGoogle Scholar
Kayid, M., Izadkhah, S., & Alshami, S. (2016). Laplace transform ordering of time to failure in age replacement models. Journal of the Korean Statistical Society 45(1): 101–113.CrossRefGoogle Scholar
Khan, R.A., Bhattacharyya, D., & Mitra, M. (2020). A change point estimation problem related to age replacement policies. Operations Research Letters 48(2): 105–108.CrossRefGoogle Scholar
Klefsjö, B. (1982). On aging properties and total time on test transforms. Scandinavian Journal of Statistics. Theory and Applications 9(1): 37–41.Google Scholar
Knopik, L. (2005). Some results on the ageing class. Control and Cybernetics 34(4): 1175–1180.Google Scholar
Knopik, L. (2006). Characterization of a class of lifetime distributions. Control and Cybernetics 35(2): 407–414.Google Scholar
Lai, C.-D. & Xie, M. (2006). Stochastic ageing and dependence for reliability. New York: Springer. With a foreword by Richard E. Barlow.Google Scholar
Li, X. & Xu, M. (2008). Reversed hazard rate order of equilibrium distributions and a related aging notion. Statistical Papers 49(4): 749–767.CrossRefGoogle Scholar
Marshall, A.W. & Olkin, I. (2007). Life distributions. Springer Series in Statistics. New York: Springer. Structure of nonparametric, semiparametric, and parametric families.Google Scholar
Marshall, A.W. & Proschan, F. (1972). Classes of distributions applicable in replacement with renewal theory implications. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, CA, 1970/1971), Vol. I: Theory of statistics, pp. 395–415.CrossRefGoogle Scholar
McCool, J.I. (2012). Using the Weibull distribution. Wiley Series in Probability and Statistics. Hoboken, NJ: John Wiley & Sons, Inc. Reliability, modeling, and inference.CrossRefGoogle Scholar
Mercier, S. & Castro, I.T. (2019). Stochastic comparisons of imperfect maintenance models for a gamma deteriorating system. European Journal of Operational Research 273(1): 237–248.CrossRefGoogle Scholar
Müller, A. & Stoyan, D. (2002). Comparison methods for stochastic models and risks. Wiley Series in Probability and Statistics. Chichester: John Wiley & Sons, Ltd.Google Scholar
Nair, N.U., Sankaran, P.G., & Balakrishnan, N. (2013). Quantile-based reliability analysis. Statistics for Industry and Technology. New York: Birkhäuser/Springer. With a foreword by J. R. M. Hosking.CrossRefGoogle Scholar
Nakagawa, T. (2005). Maintenance theory of reliability. London: Springer-Verlag.Google Scholar
Nakagawa, T. (2014). Random maintenance policies. London: Springer-Verlag.CrossRefGoogle Scholar
Nakagawa, T. & Zhao, X. (2012). Optimization problems of a parallel system with a random number of units. IEEE Transactions on Reliability 61(2): 543–548.CrossRefGoogle Scholar
Nanda, A.K. & Shaked, M. (2008). Partial ordering and aging properties of order statistics when the sample size is random: a brief review. Communications in Statistics. Theory and Methods 37(11–12): 1710–1720.CrossRefGoogle Scholar
Ohnishi, M. (2002). Stochastic orders in reliability theory. In S. Osaki (Ed.), Stochastic models in reliability and maintenance. Berlin: Springer, pp. 31–63.CrossRefGoogle Scholar
Park, M., Jung, K.M., Kim, J.J.-Y., & Park, D.H. (2019). Efficiency consideration of generalized age replacement policy. Applied Stochastic Models in Business and Industry 35(3): 671–680.CrossRefGoogle Scholar
Ross, S.M. (1996). Stochastic processes, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. New York: John Wiley & Sons, Inc.Google Scholar
Safaei, F., Ahmadi, J., & Balakrishnan, N. (2019). A repair and replacement policy for repairable systems based on probability and mean of profits. Reliability Engineering & System Safety 183: 143–152.CrossRefGoogle Scholar
Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. Springer Series in Statistics. New York: Springer.CrossRefGoogle Scholar
Shanthikumar, J.G. & Yao, D.D. (1991). Bivariate characterization of some stochastic order relations. Advances in Applied Probability 23(3): 642–659.CrossRefGoogle Scholar
Sreelakshmi, N., Kattumannil Sudheesh, K., & Asha, G. (2018). Quantile based tests for exponentiality against DMRQ and NBUE alternatives. Journal of the Korean Statistical Society 47(2): 185–200.Google Scholar
Sudheesh, K.K., Asha, G., & Jagathnath Krishna, K.M. (2019). On the mean time to failure of an age-replacement model in discrete time. Communications in Statistics. Theory and Methods, 1–17. https://doi.org/10.1080/03610926.2019.1672742.Google Scholar
Taylor, J.M. (1983). Comparisons of certain distribution functions. Mathematische Operationsforschung und Statistik Series Statistics 14(3): 397–408.Google Scholar
Weiss, G.H. (1957). On the theory of replacement of machinery with a random failure time. Naval Research Logistics Quarterly 3: 279–293, 1956.CrossRefGoogle Scholar
Zhao, X. & Nakagawa, T. (2012). Optimization problems of replacement first or last in reliability theory. European Journal of Operational Research 223(1): 141–149.CrossRefGoogle Scholar
Zhao, X., Al-Khalifa, K.N., Magid Hamouda, A., & Nakagawa, T. (2017). Age replacement models: a summary with new perspectives and methods. Reliability Engineering & System Safety 161: 95–105.CrossRefGoogle Scholar
Zhao, X., Mizutani, S., Chen, M., & Nakagawa, T. (2020). Preventive replacement policies for parallel systems with deviation costs between replacement and failure. Annals of Operations Research. https://doi.org/10.1007/s10479-020-03791-6.Google Scholar