Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T22:04:00.440Z Has data issue: false hasContentIssue false
  • English
  • Français

A functional equation related to a repairable system subjected to priority rules

Published online by Cambridge University Press:  18 March 2016

E. J. VANDERPERRE  and
S. S. MAKHANOV
E. J. VANDERPERRE
Affiliation:
Reliability Engineering Research Unit, Ruzettelaan 183, Bus 158, 8370 Blankenberge, Belgium email: [email protected]
S. S. MAKHANOV*
Affiliation:
School of Information and Computer Technology, Sirindhorn International Institute of Technology, Thammasat University, Tiwanont Road, T. Bangkadi, A. Muang, Pathum Thani 12000, Thailand email: [email protected]
*
†Corresponding author
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse the survival time of a general duplex system sustained by an auxiliary cold standby unit and subjected to priority rules. The duplex system is attended by two general repairmen Rp and Rh . Repairman Rp has priority in repairing failed units with regard to repairman Rh provided that both repairmen are jointly idle. Otherwise, the priority is overruled. The auxiliary unit has its own repair facility. The duplex system has overall, break-in priority (often called pre-emptive priority) in operation and in standby with regard to the auxiliary unit. The analysis of the survival time is based on advanced complex function theory (sectionally holomorphic functions). The main problem is to convert a functional equation into a (parameter dependent) Sokhotski–Plemelj problem.

Keywords

duplex systempriority rulesurvival functionstopping timesecurity intervalsectionally holomorphic function
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 1 , February 2017 , pp. 123 - 140
Copyright
Copyright © Cambridge University Press 2016 

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] Apostol, T. M. (1978) Mathematical Analysis, Addison-Wesley P.C., London.Google Scholar
[2] Birolini, A. (2007) Reliability Engineering. Theory and Practice, Springer, Berlin.Google Scholar
[3] Brémaud, P. (1991) Point Processes and Queues, Springer Series in Statistics, Springer-Verlag, Berlin.Google Scholar
[4] Cox, D. R. (1955) A use of complex probabilities in the theory of stochastic processes. Math. Proc. Cambridge Phil. Soc. 51, 313–319. DOI:10.1017/80305004100030231.Google Scholar
[5] Doob, J. L. (1994) Measure Theory, Springer, Berlin.Google Scholar
[6] Epstein, B. & Weissmann, I. (2008) Mathematical Models for System Reliability, Chapman and Hall, CRC Press, Baton Rouge.Google Scholar
[7] Gakhov, L. D. (1996) Boundary Value Problems, Pergamon Press, Oxford.Google Scholar
[8] Gnedenko, B. V. & Ushakov, I. A. (1995) Probabilistic Reliability Engineering, Falk, J. (editor), John Wiley & Sons Inc., New York City.Google Scholar
[9] Kim, D. S., Lee, S. M., Jung, J.-H., Kim, T. H., Lee, S. & Park, J. S. (2012) Reliability and availability analysis for an on board computer in a satellite system using standby redundancy and rejuvenation. J. Mech. Sci. Technol. 26 (7), 2059–2063.Google Scholar
[10] Leung, K. N. F., Zhang, Y. L. & Lai, K. K. (2010) A bivariate optimal replacement policy for a cold standby repairable system with repair priority. Naval Res. Logist. 57, 149–158.CrossRefGoogle Scholar
[11] Leung, K. N. F., Zhang, Y. L. & Lai, K. K. (2011) Analysis for a two-dissimilar-component cold standby system with priority. Reliab. Eng. Syst. Saf. 96, 314–321.Google Scholar
[12] Lu, J.-K. (1993) Boundary Value Problems for Analytic Functions. Series in Pure Mathematics, World Scientific, Hong Kong.Google Scholar
[13] Ozaki, H., Kara, A. & Cheng, Z. (2012) User-perceived reliability of unrepairable shared protection systems with functionally identical units. Int. J. Syst. Sci. 45 (5), 869–883.CrossRefGoogle Scholar
[14] Pólya, G. & Szegő, G. (1978) Problems and Theorems in Analysis I, Springer, Berlin.Google Scholar
[15] Roos, B. W. (1996) Analytic Functions and Distributions in Physics and Engineering, John Wiley & Sons Inc., New York City.Google Scholar
[16] Ruiz-Castro, J. E., Pérez-Ocón, R. & Fernández-Villodre, G. (2008) Modelling a reliability system by discrete phase-type distributions. Reliab. Eng. Syst. Saf. 93, 1650–1657.Google Scholar
[17] Shi, D. H. & Liu, L. (1996) Availability analysis of a two-unit series system with a priority shut-off rule. Naval Res. Logist. 43, 1009–1024.3.0.CO;2-J>CrossRefGoogle Scholar
[18] Shingarewa, L. & Lizarraga-Celaya, C. (2009) Maple and Mathematica: A Problem Solving Approach for Mathematics, Springer, Berlin.CrossRefGoogle Scholar
[19] Ushakov, I. A. (2012) Stochastic Reliability Models, John Wiley & Sons Inc., New York City.Google Scholar
[20] Vanderperre, E. J. (1998) On the reliability of Gaver's parallel system sustained by a cold standby unit and attended by two repairmen. J. Oper. Res. Soc. Japan 41, 171–180.Google Scholar
[21] Vanderperre, E. J. (2000) Long-run availability of a two-unit standby system subjected to a priority rule. Bull. Belgian Math. Soc.-Simon Stevin 7, 355–364.Google Scholar
[22] Vanderperre, E. J. & Makhanov, S. S. (2014) On the availability of a warm standby system: A numerical approach. J. Spanish Soc. Stat. Oper. Res. 22, 644–657.Google Scholar
[23] Vanderperre, E. J. & Makhanov, S. S. (2014) Reliability analysis of a repairable duplex system. Int. J. Syst. Sci. 45, 1970–1977.Google Scholar
[24] Vanderperre, E. J. & Makhanov, S. S. (2015) Overall availability and risk analysis of a general robot–safety device system. Int. J. Syst. Sci. 46, 1889–1896.Google Scholar
[25] Wang, H. X. & Xu, G. Q. (2012) A cold system with two different components and a single vacation of the repairman. Appl. Math. Comput. 219, 2614–2657.Google Scholar
[26] Wu, Q. (2012) Reliability analysis of a cold standby system attacked by shocks. Appl. Math. Comput. 218, 11654–11673.Google Scholar