Asymptotic property of a reparable multi-state device
Authors:
Houbao Xu, Jingyuan Yu and Guangtian Zhu
Journal:
Quart. Appl. Math. 63 (2005), 779-789
MSC (1991):
Primary 93D20, 90B25; Secondary 34D05
DOI:
https://doi.org/10.1090/S0033-569X-05-00986-0
Published electronically:
October 18, 2005
MathSciNet review:
2187931
Full-text PDF Free Access
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Abstract: This paper is devoted to studying the existence, uniqueness and asymptotic stability of a multi-state device’s time-dependent solution. $C_0$ semigroup theory is used to prove the existence of a unique non-negative solution of the device. Moveover, by analyzing the spectrum of the system operator generated by the device, this paper proves that 0 is the unique spectral point on the imaginary axis and the other spectra lie in the left half plane. As a result, the asymptotic behavior of a multi-state device is obtained.
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8 N. Satoh, M. Sasaki, T. Yuge and S. Yanagi, Reliability of 3-state device systems with simultaneous failure. IEEE Trans. Reliab., 42 (1993), 470-477.
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1 E. F. Moore and C. E. Shannon, Reliable circuits using less reliable relays. J. Franklin Institute, 9 (1956), 191-208.
2 C. L. Proctor and B. Singh, Reliability of three-state device network. Proc. Ann. Reliab. and Maintainability Symp. (1975), 311-316.
3 C. L. Proctor and B. Singh, The graphical evaluation of three-state device networks. Micro. Reliab., 15 (1975), 203-214.
4 C. L. Proctor and B. Singh, A reparable 3-state device. IEEE Trans. Reliab., 25 (1976), 210-211.
5 C. L. Proctor, Multistate dependent system. Proc. Ann. Reliab. and Maintainability Symp. (1977), 401-403.
6 E. A. Elsayed and A. Zebib, A reparable Multistate device. IEEE Trans.Reliab., 28 (1979), 81-82.
7 W. K. Chung, A reparable multistate device with arbitrarily distributed repair times. Micro. Reliab., 21 (1981), 255-256.
8 N. Satoh, M. Sasaki, T. Yuge and S. Yanagi, Reliability of 3-state device systems with simultaneous failure. IEEE Trans. Reliab., 42 (1993), 470-477.
9 G. Levitin, Reliability of multi-state systems with two failure-modes. IEEE Trans. Reliab., 52 (2003), 340-348.
10 W. Arendt, Resolvent positive operators. Proc. London Math. Soc., 54 (1987), 321-349.
11 A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983.
12 C. Jinhua and Ch. Kan, Introduction to Reliable Mathmatics, Science Press, Beijing, 1986.
13 Geni Gupur, Xue-zhi Li and Guang-tian Zhu. Functional Analysis Method in Queueing Theory. Research Information Ltd., United Kingdom, 2001.
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Additional Information
Houbao Xu
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
Address at time of publication:
Room 709, Institute 710, No. 16 FuCheng Road, Beijing 100037, China
Email:
xuhoubao@yahoo.com.cn
Jingyuan Yu
Affiliation:
Beijing Institute of Information and Control, Beijing 100037, China
Email:
yujy@nasic.spacechina.com
Guangtian Zhu
Affiliation:
Academy of Mathematic and System Sciences, C.A.S., Beijing 100080, China
Keywords:
Multi-state device,
asymptotic stability,
$C_0$-semigroup
Received by editor(s):
April 17, 2005
Received by editor(s) in revised form:
May 17, 2005
Published electronically:
October 18, 2005
Article copyright:
© Copyright 2005
Brown University
The copyright for this article reverts to public domain 28 years after publication.
