Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T09:25:06.845Z Has data issue: false hasContentIssue false
  • English
  • Français

RELIABILITY ANALYSIS OF A SIMPLE REPAIRABLE SYSTEM

Part of: Stability

Published online by Cambridge University Press:  05 September 2011

L. N. GUO ,
H. B. XU ,
C. GAO  and
G. T. ZHU
L. N. GUO*
Affiliation:
Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, PR China (email: [email protected])
H. B. XU
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China (email: [email protected])
C. GAO
Affiliation:
Department of Engineering, Beijing Institute of Information and Control, Beijing 100037, PR China (email: [email protected])
G. T. ZHU
Affiliation:
Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a new kind of simple repairable system consisting of a repairman with multiple delayed-vacation strategy. A common technique in reliability studies is to substitute the steady-state reliability indexes for instantaneous ones because the dynamic solution of the system is difficult or even impossible to obtain. However, this substitution is not always valid. Therefore, it is important to study the existence, uniqueness and expression for the system’s dynamic solution, and to discuss the system’s stability. The purpose of this paper is threefold: to study the uniqueness and existence of the dynamic solution, and its expression, using C0-semigroup theory; to discuss the exponential stability of the system by analysing the spectral distribution and quasi-compactness of the system operator; to derive some reliability indexes of the system from an eigenfunction point of view, which is different from the traditional Laplace transform technique, and present a profit analysis to determine the optimal vacation time in order to achieve the maximum system profit.

Keywords

dynamic solutionsteady-state solutionC0-semigroup theoryexponential stabilityreliability indexprofit analysis

MSC classification

Secondary: 93D20: Asymptotic stability
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 2 , October 2010 , pp. 203 - 217
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Barlow, R. and Hunter, L., “Optimum preventive maintenance policies”, Oper. Res. 8 (1960) 90–100, doi:10.1287/opre.8.1.90.CrossRefGoogle Scholar
[2]Brown, M. and Proschan, F., “Imperfect repair”, J. Appl. Probab. 20 (1983) 851–859, doi:10.2307/3213596.CrossRefGoogle Scholar
[3]Cao, J. H. and Cheng, K., Introduction to reliability mathematics (Higher Education Press, Beijing, 2006).Google Scholar
[4]Cao, J. H. and Wu, Y. H., “Reliability analysis of a multistate system with a replaceable repair facility”, Acta Math. Appl. Sin. 4 (1988) 113–121, doi:10.1007/BF02006059.CrossRefGoogle Scholar
[5]Clément, Ph., Heijmans, H. J. A. M., Angenent, S., van Duijn, C. J. and de Pagter, B., One-parameter semigroups, Volume 5 of CWI Monographs (North-Holland, Amsterdam, 1987).Google Scholar
[6]Gopalan, M. N. and Murulidhar, N. N., “Cost analysis of a one-unit repairable system subject to on-line preventive maintenance and/or repair”, Microelectronics Reliability 31 (1991) 223–228, doi:10.1016/0026-2714(91)90203-J.CrossRefGoogle Scholar
[7]Guo, T. D. and Cao, J. H., “Reliability analysis of a multistate one-unit repairable system operating under a changing environment”, Microelectronics Reliability 32 (1992) 439–443, doi:10.1016/0026-2714(92)90074-U.CrossRefGoogle Scholar
[8]Gupur, G., Li, X. Z. and Zhu, G. T., “Existence and uniqueness of nonnegative solution of M/GB/1 queueing model”, Comput. Math. Appl. 39 (2000) 199–209, doi:10.1016/S0898-1221(00)00076-6.CrossRefGoogle Scholar
[9]Hu, W. W., Xu, H. B., Yu, J. Y. and Zhu, G. T., “Exponential stability of a reparable multi-state device”, J. Syst. Sci. Complex. 20 (2007) 437–443, doi:10.1007/s11424-007-9039-9.CrossRefGoogle Scholar
[10]Hu, W. W., Xu, H. B. and Zhu, G. T., “Exponential stability of a parallel repairable system with warm standby”, Acta Anal. Funct. Appl. 9 (2007) 311–319.Google Scholar
[11]Jain, M., Rakhee, and Singh, M., “Bilevel control of degraded machining system with warm standbys, setup and vacation”, Appl. Math. Model. 28 (2004) 1015–1026, doi:10.1016/j.apm.2004.03.009.CrossRefGoogle Scholar
[12]Ke, J. C. and Wang, K. H., “Vacation policies for machine repair problem with two type spares”, Appl. Math. Model. 31 (2007) 880–894, doi:10.1016/j.apm.2006.02.009.CrossRefGoogle Scholar
[13]Kella, D., “The threshold policy in the M/G/1 queue with server vacations”, Naval Res. Logist. 36 (1989) 111–123, doi:10.1002/1520-6750(198902)36:1%3C111::AID-NAV3220360109%3F3.0.CO;2-3.3.0.CO;2-3>CrossRefGoogle Scholar
[14]Lam, Y., “Geometric processes and replacement problem”, Acta Math. Appl. Sin. 4 (1998) 366–377, doi:10.1007/BF02007241.Google Scholar
[15]Liu, R. B. and Tang, Y. H., “One-unit repairable system with multiple delay vacations”, Chinese J. Engrg. Math. 23 (2006) 721–724.Google Scholar
[16]Nagel (ed.), R., One-parameter semigroups of positive operators, Volume 1184 of Lecture Notes in Mathematics (Springer, Berlin, 1986).Google Scholar
[17]Pazy, A., Semigroups of linear operators and applications to partial differential equations (Springer, New York, NY, 1983).CrossRefGoogle Scholar
[18]Tang, Y. H., “Some new reliability problems and results for one-unit repairable system”, Microelectronics Reliability 36 (1996) 465–468, doi:10.1016/0026-2714(95)00137-9.CrossRefGoogle Scholar
[19]Taylor, A. E. and Lay, D. C., Introduction to functional analysis (Wiley, New York, NY, 1980).Google Scholar
[20]Utkin, L. V. and Gurov, S. V., “Steady-state reliability of repairable systems by combined probability and possibility assumptions”, Fuzzy Sets and Systems 97 (1998) 193–202, doi:10.1016/S0165-0114(96)00362-4.CrossRefGoogle Scholar
[21]Wang, G. J. and Zhang, Y. L., “A bivariate mixed policy for a simple repairable system based on preventive repair and failure repair”, Appl. Math. Model. 33 (2009) 3354–3359, doi:10.1016/j.apm.2008.11.008.CrossRefGoogle Scholar
[22]Zhang, Y. L., “A bivariate optimal replacement policy for a repairable system”, J. Appl. Probab. 31 (1994) 1123–1127, doi:10.2307/3215336.CrossRefGoogle Scholar
[23]Zhang, Z. G. and Love, C. E., “The threshold policy in the M/G/1 queue with an exceptional first vacation”, INFOR Inf. Syst. Oper. Res. 36 (1998) 193–204.Google Scholar