Multi-state system is defined as a system, where components and system are allowed to have an arbitrary number of performance levels.
Bibliography:
Last updated Feb. 10. 2005
(Compiled by A. Lisnianski, G. Levitin, E. Korczak)
Main
definitions, properties, reliability measures
[1.1]. G. Epstein, G. Trieder, D. C. Rine, The development of multivalued logic as related to computer science, Computer, vol 7, 1974 September, pp 20-32.
[1.2]. R. M. Barton, W. W. Damon, Reliability in
a Multi-State System, Proceedings Sixth Annual Southeastern Symposium on
Systems Theory,
[1.3]. J. Murchland, Fundamental concepts and
relations for reliability analysis of multi-state systems. In Reliability
and Fault Tree Analysis, ed. R. Barlow, J. Fussell,
[1.4]. R. E. Barlow, A. S. Wu, Coherent systems with multi-state components, Mathematics of Operations Research, vol 3, 1978, Nov, pp. 275-281.
[1.5]. E. El-Neweihi, F. Proschan, J. Seturaman, Multistate coherent systems, J. Applied Probability, vol 15, 1978 Dec, pp. 675-688.
[1.6]. S. Contini, S Fumagalli, S. Garriba, P. Mussio, F. Naldi, G. Volta, Multiple-valued logic in modeling of nuclear safety systems and automated search for fault states, paper XIII-3 in Proceed. Topical Meeting on Probabilistic Analysis of Nuclear Reactor Safety, vol. 3, Eds. E.N. Cramer, D. Okrent, American Nuclear Society, 1978.
[1.7]. S. M. Ross, Multivalued state component systems, Annals of Probability, vol 7, 1979, pp 379-383.
[1.8]. B. Natvig, Multi-state coherent systems. In Encyclopedia of Statistical Sciences, vol. 5, ed. N. Jonson and S. Kotz, Wiley&Sons, NY, 1984.
[1.9]. T. Aven, U. Jensen, Stochastic models in reliability, Springer-Verlag, NY., 1999
[1.10]. T. Aven, On performance measures for multistate monotone systems, Reliability Engineering and System Safety, vol 41, 1993, pp 259-266.
[1.11]. R. D. Brunelle, K. C.
Kapur, Issues in modeling
system reliability from customer's perspective, IEEE International Conference on Systems, Man, and Cybernetics,
Vol.5, pp.4693-4698, 1998.
[1.12]. K. Yu, I. Koren, Y. Guo, Generalized Multistate Monotone Coherent Systems, IEEE Transactions on Reliability, vol. 43, NO. 2, 1994 June, pp 242-250.
[1.13].
[1.14]. J. C. Hudson, K. C. Kapur, Reliability Theory for Multistate Systems with Multistate Components, Microelectronics and Reliability, vol. 22, No. 1, Jaunary 1982, pp. 1-7.
[1.15]. J. C. Hudson, K. C. Kapur, Reliability analysis of multistate systems with multistate components, Transactions of Institute of Industrial Engineers, vol 15, No. 2, 1983 Jun, pp 127-135.
[1.16]. T. Aven, Reliability Evaluation of Multistate Systems with Multistate Components, IEEE Transactions on Reliability, vol. R-34, NO. 5, 1985, December, pp 473-479.
[1.17]. J. Xue, On Multistate System Analysis, IEEE Transactions on Reliability, vol R-34, 1985 October, pp 329-337.
[1.18]. T. Aven, Availability evaluation of flow networks with varying throughput-demand and deferred repairs. IEEE Transactions on Reliability, vol. 38, 1990, pp 499-505.
[1.19]. R. Boedigheimer. Customer-driven
reliability models for multistate coherent systems. Ph.D Thesis ,
[1.20]. J. Xue, F. Xu, Modular decomposition of
Multistate systems, Reliability Theory and Applications: Proc. China-Japan
Reliability Symp. Edited
by Shunji Osaki and Jin Hua Cao. 1987 September, World Scientific
Publishing Co. pp.402-410.
[1.21]. J. Shao, K. C. Kapur, Multilevel Modular Decomposition for Multistate Systems, Proceedings of Annual Reliability and Maintainability Symposium, January 1989, pp. 102-107.
[1.22]. I. Ushakov, A universal generating function. Soviet Journal Computer Systems Science, 1987; 25: pp. 61-73, (in Russian).
[1.23]. I. Ushakov, Reliability analysis of multi-state systems by means of modified generating function. Journal Information Process. Cybernet. , 1988, vol. 24, no. 3, pp. 131-135, (in Russian).
[1.24]. Zhang Ming, Xie Hong Wei, Model characteristics of multistate monotone coherent systems. Mohu Xitong yu Shuxue ( Fuzzy Systems and Mathematics ), Vol.11, No.3, (1997), pp.45-49. (in Chinese)
[1.25]. R. Boedigheimer, K. Kapur, Involving the customer in the development and evaluation of Multistate reliability models', Proceedings of the 39-th Annual Reliability and Mainainability Symposium, Atlanta, Georgia, January, 1993.
[1.26]. R. Boedigheimer, K. Kapur, Customer-Driven Reliability Models for Multistate Coherent Systems, IEEE Transactions on Reliability, vol. 43, NO. 1, 1994 March, pp 46-50.
[1.27]. I. Ushakov, The method of generalized generating sequences, European Journal of Operational Research, 2000, 125 (2), pp. 316-323.
[1.28]. T. Aven, Availability analysis of monotone systems, In Reliability and Maintenance of Complex Systems, S. Ozekici (ed.), NATO ASI Series F, Springer, Berlin, pp. 206-223, 1996
[1.29]. J. Huang, M. Zuo, Multi-State k-out-of-n System Model and its Applications, Proceedings Annual RELIABILITY and MAINTAINABILITY Symposium, 2000 pp 264-268.
[1.30]. H. Block, T. Savits, A Decomposition of Multistate monotone system, J. Applied Probability, vol 19, 1982 June, pp. 391-402.
[1.31]. H. Block, T. Savits, Multistate systems, In Reliability in the Acquisitions Process (Eds., D.J. DePriest, R.L. Launer), Mercel Dekker, New York, 1983, pp. 47-53.
[1.32]. H. Block, T. Savits, Decomposition for multistate monotone systems, In Reliability Theory and Models: Stochastic Failure Models, Optimal Maintenance Policies, Life Testing and Structures, Vol. 10 (Eds.,M. Abdel-Hameed, E. Cinlar and J. Quinn), Academic Press Inc., Orlando1984,pp. 231-241.
[1.33]. M. Finkelstein, Probabilistic approach to some problems of system safety, Microelectron. Reliab. 34, 1994, pp. 1441-1457.
[1.34].
[1.35]. B. Gnedenko and
[1.36]. K. Reinshke, Systems consisting of
multi-state units. in Reliability of technical systems: Handbook, Ed. by
I. Ushakov, pp. 517-521, Radio i Sviaz,
[1.37]. J. Huang, M. Zuo, Y. Wu, Generalized multi-state k-out-of-n:G systems, IEEE Transactions on Reliability, vol 49, 2000, March, pp 105-111.
[1.38]. K. Andrzejczak, Disjunctive and
conjuctive representations of reliability structure of multistate coherent
systems. The Fourth International Conference RELCOMEX'87, Reliability and
Exploitation of Computer Systems . Ksi??
[1.39]. K. Andrzejczak, Some properties of multistate BW-systems. SERDICA. Bulgaricae Mathematicae Publicationes , Vol.13, No.4, (1987). pp.341-346.
[1.40]. K. Andrzejczak, Deterministic properties
of the multistate systems. Performance Evaluation, Reliability and
Exploitation of Computer Systems. Proceeding of the Fifth Anniversary
International Conference RELCOMEX'89 ,
[1.41]. K. Andrzejczak, Structure analysis of multistate coherent systems. Optimization, [Optimization. A Journal of Mathematical Programming and Operations Research] Vol.25, No.2-3, (1992). pp.301-316.
[1.42]. V.C. Bueno, A note on the component lifetime estimation of a multistate monotone system through the system lifetime. Advances in Applied Probability , Vol.20, (1988), pp.686-689.
[1.43]. E. El-Neweihi, F. Proschan, Multistate
reliability models: a survey. In: Multivariate Analysis V . Edited by
P.R.Krishnaiah. North-Holland,
[1.44]. E. El-Neweihi, F. Proschan, Degradable systems: a survey of multistate system theory. Communication in Statistics. Theory and Methods .Vol.13, (1984), pp.405-432.
[1.45]. E. El-Neweihi, F. Proschan, Components
relevancy in multistate systems. In: Multivariate Analysis VI . Edited
by P.R. Krishnaiah. North-Holland,
[1.46]. D. Engel, Niezawodnosciowy model wielostanowych zlozonych obiektow technicznych o dwustanowych elementach. (Reliability model of complex multistate technical objects with binary components.) Zagadnienia Eksploatacji Maszyn , Vol.19, (1984), pp.517-525. (in Polish)
[1.47]. D. Engel On reliability models of
multistate technical objects with binary components. Zeszyty Naukowe
Akademii Ekonomicznej ,
[1.48]. M.N. Fardis, C.A. Cornell, Multistate reliability analysis. Nuclear Engineering and Design, Vol.60, (1982), pp.329-336.
[1.49]. G.F. Fishman, The distribution of maximum flow with applications to multistate reliability systems. Operations Research , Vol.35, (1987), pp.607-618.
[1.50]. J.C. Hudson The Structure and Reliability
of Multistate Systems with Multistate Components. Ph.D. Thesis,
[1.51]. J.C. Hudson, K. Kapur Multistate systems with multistate components and their reliability computations. Proceedinds of the 1981 Spring Annual Conference and World Productivity Congress, American Institut of Industrial Engineering, (1981), pp.471-477.
[1.52]. E. Korczak, Reliability model of a
multistate system with multistate components. Third International Conference
on Reliability & Exploitation of Computer Systems RELCOMEX'84 ,
[1.53]. E. Korczak, Reliability model of a
multistate system with randomly varying structure. RELECTRONIC'88, 7th
Symposium on Reliability in Electronics ,
[1.54]. E. Korczak, Calculation of stationary
reliability characteristics of multistate monotone systems. XII
International Conference on Fault-Tolerant Systems and Diagnostics ,
[1.55]. E. Korczak, Evaluation of steady state
reliability indexes of multistate monotone systems. Performance Evaluation,
Reliability and Exploitation of Computer Systems. Proceeding of the Fifth
Anniversary International Conference RELCOMEX'89 ,
[1.56].
[1.57].
[1.58].
[1.59]. E. Korczak, Reliability analysis of
non-repaired multistate systems. In: Advances in Safety and Reliability.
Proceedings of the ESREL'97 International Conference on Safety and Reliability,
17-20 June, 1997,
[1.60]. S. Kuo Generating essential primes of a boolean function with multiple-valued inputs. IEEE Transactions on Computers , Vol.C-36, No.3, (1987), pp.356-359.
[1.61]. K.V. Le, V.O.K. Li, Modeling and analysis of systems with multimode components and dependent failures. IEEE Transactions on Reliability , Vol.38, No.1 April, (1989), pp.68-75.
[1.62]. J. de Mare, B. Rosander, The production flow in the motor industry: A case study of multistate reliability theory. Scandinavian Journal of Statistics , Vol.15, (1988), pp.51-54.
[1.63]. K. Nakashima, K. Yamato Some properties of multi-state monotone systems and their Boolean structure functions. The Transactions of the IECE of Japan , Vol.E66, No.9, (1983), pp.535-542.
[1.64]. B. Natvig,
[1.65]. E.I. Ogunbiyi, E.J. Henley Irredundant forms and prime implicants of a function with multistate variables. IEEE Transactions on Reliability , Vol.R-30, No.1, (1981), pp.39-42.
[1.66]. F. Ohi, T. Nishida, Multistate systems in
reliability theory. In: Stochastic Models in Reliability Theory . Edited
by S.Osaki & Y.Hatoyama. Springer Verlag,
[1.67]. F. Ohi, S. Shinmori A definition of generalized k-out-of-n multistate systems and their structural and probabilistic properties. Japan Journal of Industrial and Applied Mathematics , Vol.15, No.2, (1988), pp.263-277.
[1.68].
[1.69]. S. Popowicz, A disjunctive-conjunctive and conjunctive-disjunctive decomposition for multistate monotone systems. Postepy Cybernetyki , Vol.11, No.4, (1988), pp.43-52.
[1.70]. F. Proshan, G. Caturjan, Mnogoznachnye
monotonnye sistemy. Chast' 1. Nadezhnost' i kontrol' kachestva , No. 1,
pp.6-16. [Proschan F., Tsaturyan G. Zh.: Multistate monotone systems
[1.71]. F. Proshan, G. Caturjan, Mnogoznachnye monotonnye sistemy. Chast' 2. Nadezhnost' i kontrol' kachestva , No. 2, pp.3-11. [Proschan F., Tsaturyan G. Zh.: Multistate monotone systems II. Reliability and Quality Control, 1990, No.2] (in Russian)
[1.72]. S. Shinmori, F. Ohi, H. Hagihara, T.
Nishida, Modules for two classes of multi-state systems. The Transactions of
the IEICE (
[1.73].
[1.74]. S. Shinmori, F. Ohi, On structural relations for component sets of multi-state systems. Mathematica Japonica , Vol.40, No.1, (1994), pp.135-142.
[1.75]. T. van Dac, Multistate system reliability analysis by binary variables. Periodica Polytechnica. Mechanical Engineering , Vol.31, No.4, (1987), pp.319-329.
[1.76]. L.V. Utkin, Ocenka znachimosti po nechetkosti ehlementov monotonnych mnogoznachnych sistem. Nadezhnost' i kontrol' kachestva , No.1, pp.8-12. [Utkin L.V.: Evaluation of significance by indistinctness of elements of monotonous multiple-value systems. Reliability and Quality Control , 1992, No.1, pp.8-12] (in Russian)
[1.77]. J. K. Vaurio Reliability and availability equations for multi-state components. Reliability Engineering , Vol.7, (1984), pp.1-19.
[1.78]. A. P. Wood, Multistate Reliability. PhD thesis , Stanford University, Department of Operations Research and Department of Statistics, Stanford, California, 1983.
[1.79]. Xue Jianan, New approach for multistate system analysis. Reliability Engineering , Vol.10, (1985), pp.245-256.
[1.80]. Xue Jianan, On multistate system analysis. IEEE Transactions on Reliability, Vol.R-34, No.4 October, (1985), pp.329-337. Corrections: 1986, Vol.R-35, No.1.
[1.81]. Xue Jianan, Yang Kai, Symmetric relations in multistate systems. IEEE Transactions on Reliability , Vol.44, No.4 (1995), pp.689-693.
[1.82]. Yang Kai, Xue Jianan, Dynamic reliability measures and life distribution models for multistate systems. Int. J. Reliability, Quality and Safety Engineering , Vol.2, (1995), pp.79-102.
[1.83]. A. Bendell, S. Humble, A reliability model with states of partial operation. Naval Research Logistics Quarterly , Vol.32, 1985, pp. 509-535.
[1.84].
[1.85]. M. C. van der Heijden, A. Schornagel Interval uneffectiveness distribution for a k-out-of-n multistate reliability system with repair. European Journal of Operational Research, Vol.36, 1988, No.1, pp.66--77
[1.86]. W.M. Hirsch, M. Meisner, C.M. Boll Cannibalization in multicomponent systems and the theory of reliability. Naval Research Logistics Quarterly , Vol.15, 1968, pp.331-360.
[1.87]. M. Hochberg, Generalized multicomponent systems under cannibalization. Naval Research Logistics Quarterly, Vol.20. 1973, pp.585-606.
[1.88]. R.M. Simon, Optimal cannibalization policies for multicomponent systems. SIAM Journal of Applied Mathematics , Vol.19, 1970, pp.700-711.
[1.89]. R.M. Simon, The reliability of multicomponent systems under cannibalization. Naval Research Logistics Quarterly , Vol.19, 1972, pp.1-14.
[1.90]. F.C. Meng, Characterizing the Barlow-Wu structure functions. Naval Research Logistics, Vol.41, 1994, No.5, pp.661-668.
[1.91]. B. Natvig, A comment on the steady-state performance of multistate monotone systems, Adv. Appl. Prob. 21 (1989), 237.
[1.92].
[1.93].
[1.94]. E. Zaitseva, V. Levashenko, New Dynamic Reliability Indices for Multi-State System, in Proc. of 3 Int. Conf. on mathematical methods in reliability, MMR 2002, June 17-20, 2002, Trondheim, Norway,pp. 687-690.
[1.95]. M. Zuo, J. Huang, W. Kuo, Dominant and binary imaged multi-state systems, in Proc. of 3 Int. Conf. on mathematical methods in reliability, MMR 2002, June 17-20, 2002, Trondheim, Norway,pp. 693-696.
[1.96]. L. Utkin, I. Kozine, A reliability model of multi-state units under partial information, in Proc. of 3 Int. Conf. on mathematical methods in reliability, MMR 2002, June 17-20, 2002, Trondheim, Norway,pp. 643-646.
[1.97]. V. Melnikov, Model of reliability for structural-complicated systems, including multistate elements, in Proc. of 3 Int. Conf. on mathematical methods in reliability, MMR 2002, June 17-20, 2002, Trondheim, Norway, pp. 445-448.
[1.98]. I. Ushakov, G. Levitin, A. Lisnianski, Multi-state system reliability: from theory to practice, in Proc. of 3 Int. Conf. on mathematical methods in reliability, MMR 2002, June 17-20, 2002, Trondheim, Norway,pp. 639-642.
[1.99]. G. Gopal, Shur property of the performance function for the multistate coherent system, in Proc. of 3 Int. Conf. on mathematical methods in reliability, MMR 2002, June 17-20, 2002, Trondheim, Norway,pp. 255-258.
[1.100]. B. Anrig, P. Monney, Using propositional logic to compute the probability of diagnoses in multistate systems. Int. J. Approximate Reasoning, 1999, 20 (2), pp.113-143.
[1.101]. J-L. Bon, O. Pourret, Boolean modelling and evaluation of a multistate-component system, International Journal of Reliability, Quality and Safety Engineering. , 2002, Vol.9, No.2, pp.183-199.
[1.102].
U. Rakowsky, A. Meyna, Redundanzformen fur Mehrwertige Modelle (Redundant
Structures in Multistate Coherent Systems) , Tagung Technische
Zuverlassigkeit - TTZ 1991, Munchen/Germany. ITG-Fachbericht 116 - Technische
Zuverlassigkeit, pp.43-56. Berlin: VDE-Verlag, 1991. (in German).
[1.103]. P. Gaspar, G. Szabo, Analysis of adaptive multi-state logic in fault-tolerant systems, A. Moshley & R.A. Bari (Eds.), Probabilistic Safety Assessment and Management. PSAM4, Springer 1998, pp: 9-14 .
[1.104]. W. Kuo, M.J. Zuo, Optimal Reliability Modeling: Principles and Applications, Wiley, NY, 2002
[1.105]. A. Lisnianski, G. Levitin, Multi-state system reliability. Assessment, Optimization and Applications, World Scientific, 2003.
[1.106] Chang Y-R., Amari S.V.,
Kuo S-Y. Reliability evaluation of multi-state systems subject to imperfect
coverage using OBDD. Proceedings of the
2002
[1.107]. Levitin G., Lisnianski A., Ushakov I., Reliability of Multi-State Systems: A Historical Overview, in Mathematical and statistical methods in reliability, Lindqvist, Doksum (Eds.), World Scientific, pp. 123-137, (2003).
[1.108]. Kuo W., Zuo M.J., Multi-state system models. In: Optimal reliability modeling, John Wiley & Sons, pp. 452-503, (2003).
[1.109]. Aven T., Hjorteland A., A Predictive Bayesian Approach to Multistate Reliability Analysis, International Journal of Reliability, Quality and Safety Engineering, Vol. 10, No. 3, pp. 221-234, (2003).
[1.110]. Paroissin C., Ycart B., Zero-One Law for the Non-Availability of Multistate Repairable Systems, International Journal of Reliability, Quality and Safety Engineering, Vol. 10, No. 3, pp. 311-322, (2003).
[1.111]. A.
M. Abouammoh and M. A. Al-Kadi, Multistate coherent systems of order k, Microelectronics and Reliability,
Vol.35, No.11, pp.1415-1421, (1995).
[1.112]. M.
Bouissou, O. Pourret, A Bayesian belief network based method for performance
evaluation and troubleshooting of multistate systems, International Journal of Reliability, Quality and Safety Engineering,
Vol. 10, No. 4, pp.407-416, (2003).
[1.113] B. Natvig, H.W.H. Morch, An application
of multistate reliability theory to an offshore gas pipeline network, International Journal of Reliability,
Quality and Safety Engineering, Vol. 10, No. 4, pp.361-381, (2003).
[1.114].
V. Rykov, B. Dimitrov B., On multi-state reliability systems, Information
Processes, Vol.2, No.2, pp.246-251, (2002).
[1.115]. Zuo M.J., Huang J.,
[1.116].
B. A. Kulik, Reliability analysis of the systems with many states on the basis
of the algebra of tuples, Automation and
Remote Control, 2003, Vol.64, No.7, pp.1029-1034.
[1.117].
V. A. Mel'nikov, Developing the methods of reliability estimation for
structurally complex systems of multistate elements, Automation and Remote Control, 2003, Vol.64, No.7, pp.1046-1053.
[1.118]. Griffith W. S, A survey of some
results in multistate reliability theory, Proceedings
of the Eleventh Annual Pittsburgh Conference on Modeling and Simulation,
Instrument Society of America, 1980, pp.579-582.
[1.119]. B. Cappelle, Multistate structure
functions and possibility theory: An alternative approach. In Introduction to the Basic
Principles of Fuzzy Set Theory and Some of Its Applications (Communication
& Cognition,
[1.120]. B. Cappelle, A possibilistic uncertainty model applied to the BarlowWu class of multistate structure functions. In Proceedings IFSA'91 (IFSA'91, Brussels, July 7 10, 1991 - Daychair Mathematics), edited by M. Roubens and R. Lowen, 1991, pp.25-28.
[1.121]. B. Cappelle, Structuur en betrouwbaarheidsafbeeldingen – Een tralietheoretische benadering van betrouwbaarheidstheorie [Structure and Reliability Functions - A LatticeTheoretic Approach to Reliability Theory ,[ PhD thesis, Universiteit Gent, 1993.
[1.122]. B. Cappelle, On the notion state in
multistate structure function theory. In: Fuzzy Set Theory and Advanced
Mathematical Applications, edited by Da Ruan. Kluwer Academic,
[1.123]. B. Cappelle, Irreducible elements and
the possibilistic and necessistic reliability functions. In: Foundations and
Applications of Possibility Theory (
[1.124]. B. Cappelle and E.E. Kerre. On a possibilistic approach to reliability theory, Proceedings of Second International Symposium on Uncertainty Modeling and Analysis, 25-28 April 1993, pp.415-418.
[1.125]. G. Levitin. A universal generating function approach for the analysis of multi-state systems with dependent elements, Reliability Engineering and System Safety, 84/3, 2004, pp. 285-292.
[1.126].
Ozen T., Garibaldi J.M., Musikasuwan S., On some analogy between
multistate system and fuzzy measure based reliability models, Proceedings of the 10th International
Conference IPMU 2004 (Information Processing and Management of Uncertainty
in Knowledge-Based Systems, July, 4-9, 2004, Perugia - Italy), pp. 649-656.
[1.127].
Xie Min, Poh Kim-Leng, Dai Yuan-Shun, Computing System Reliability: Models and Analysis, Kluwer
Academic/Plenum Publishers,
[1.127]. Zaitseva E. N., Dynamic reliability indices
for multi-state system, 33rd
International Symposium on Multiple-Valued Logic,
[1.128]. Huang,
J., Zuo, M.J,
[1.129]. Giglio
B., Wynn H.P., Monomial ideals and the Scarf complex for coherent systems in
reliability theory, Annals of Statistics,
Vol,32, No.3, 2004, pp.1289-1311.
[1.130]. Giglio
B., Wynn H.P., Alexander duality and moments in reliability modelling, Applicable Algebra in Engineering,
Communication and Computing, Vol.14, No. 3, 2003, pp. 153-174.
[1.131]. Levitin
G., Reliability and performance analysis for fault-tolerant programs consisting
of versions with different characteristics, Reliability
Engineering & System Safety,
vol. 86/1 pp. 75-81 (2004).
[1.132]. Grabski F. Semi-Markov Models for Reliability and Operation. Series: System
Research, Vol.30. Polish
[1.133]. B. Dimitrov, V. Rykov, On reliability
of hierarchical systems with gradual failures, Journal of Mathematical Sciences, September 2004, Vol.123, No.1,
pp.3802-3815.
[1.134]. Jing-An Li, Yue Wu, Kin Keung Lai and
Ke Liu. Reliability estimation and prediction of multi-state components and
coherent systems. Reliability Engineering
& System Safety, Vol.88, No.1, 2005, pp.93-98.
[1.135]. C. M. Rocco, M. Muselli. Approximate
multi-state reliability expressions using a new machine learning technique. Reliability Engineering & System Safety, Vol.89, No.3, 2005, pp.
261-270.
[1.136]. B. Kwiatuszewska-Sarnecka. Reliability
improvement of large multi-state series-parallel systems. In Advances in
Safety and Reliability, Kołowrocki
(ed.) 2005,
[1.137]. Wenjian Li, Hoang Pham, Reliability modeling of
multi-state degraded systems with multi-competing failures and random shocks, IEEE Transactions on Reliability, 2005,
Vol.54, No.2 June, pp.297-303.
[1.138]. G. Levitin, The
Universal Generating Function in Reliability Analysis and Optimisation.
Springer-Verlag:
Reliability
bounding and asymptotic analysis
[2.1]. D. Butler, Bounding the reliability of multistate systems, Operations Research, vol. R-30, 1982, pp 530-544.
[2.2]. J. Hudson, K. Kapur, Reliability bounds for multistate systems with multistate components, Operations Research, vol 33, 1985, pp 735-744.
[2.3]. J. Collet, Some Remarks on Rare-Event Approximation, IEEE Transactions on Reliability, vol. 45, NO. 1, 1996 March, pp 106-108.
[2.4]. O. Pourret, J. Collet, J-L. Bon, Evaluation of the unavailability of a multistate-component system using a binary model, Reliability Engineering and System Safety, 64 (1999) 13-17.
[2.5]. K. Kolowrocki K. An asymtotic approach to Multi-state systems reliability evaluation, Recent Advances in Reliability Theory, Methodology, Practice and Inference, Limnios N., Nikulin M. (eds.), Birkhauser, 2000, pp. 163-180.
[2.6]. K. Reinshke,
[2.7]. K. Kolowrocki, An availability modeling of a transportation systems, 2-nd International Conference on Mathematical Methods in Reliability, Methodology, Practice and Inference, MMR'2000, Bordeaux, France, July 4-7, 2000, pp.607-610.
[2.8]. K. Kolowrocki, Reliability and Risk Evaluation of Large Scale Multistate System, 17th International System Safety Conference, August 16-21, 1999, Orlando, Florida
[2.9]. K. Kolowrocki, Asymtotic approach to reliability evaluation of rope transportation system, Reliability Engineering and System Safety, 71, (2000), pp. 57-64.
[2.10]. K. Kolowrocki, On a class of limit reliability functions of some regular homogeneous series-parallel systems. Reliability Engineering and System Safety 39, No 1, 1993, 11-23.
[2.11]. K. Kolowrocki, On asymptotic reliability functions of series-parallel and parallel-series systems with identical components. Reliability Engineering and System Safety 41, 1993, 251-257.
[2.12]. K. Kolowrocki, Limit reliability functions of some series-parallel and parallel-series systems. Applied Mathematics and Computation 62, 1994, 129-151.
[2.13]. K. Kolowrocki, The classes of asymptotic reliability functions for series-parallel and parallel-series systems. Reliability Engineering and System Safety 46, No 2, 1994, 179-188.
[2.14]. K. Kolowrocki, On a class of limit reliability functions for series-parallel and parallel-series systems. International Journal of Pressure Vessels and Piping 61, 1995, 541-569.
[2.15]. K. Kolowrocki, Asymptotic reliability functions of some nonhomogeneous series-parallel andparallel-series systems. Applied Mathematics and Computation 73, 1995, 133-151.
[2.16]. K. Kolowrocki, On application of asymptotic reliability functions to the reliability and risk evaluation of pipelines. International Journal of Pressure Vessels and Piping 75, Iss 7, 1998, 545-558.
[2.17]. K. Kolowrocki, Asymptotic approach to system
reliability analysis (in Polish),
[2.18]. K. Kolowrocki, Reliability evaluation of large scale pipeline systems, Safety and reliability, Schueller, Kafka (eds), 1999, Balkema, Rotterdam, ISBN 90 5809 109 0, pp. 323-328
[2.19]. K. Kolowrocki, Asymptotic approach to reliability and risk evaluation of large piping transportation and energy distribution systems, Foresight and Precaution, Cottam, Harvey, Pape, Tait (eds), 2000, Balkema, Rotterdam, ISBN 90 5809 140 6, pp. 1201-1207
[2.20]. K. Kolowrocki, On limit reliability
functions of large systems. In: Statistical and Probabilistic Models in
Reliability. Edited by V. Ionescu &
[2.21]. B. Lindqvist, H. Langseth. Uncertainty bounds for a monotone multistate system, Probability in the Engineering and Informational Sciences, 12, 1998, pp. 239-260.
[2.22]. E. Funnemark, B. Natvig, Bounds for the availabilities in a fixed time interval for multistate monotone systems. Adv. Appl. Prob. 17 (1985), 638-655.
[2.23]. B. Natvig, Improved upper bounds for the availabilities in a fixed time interval for multistate monotone systems. Adv. Appl. Prob. 18 (1986), 577-579.
[2.24]. B. Natvig, Strict and exact bounds for the availabilities in a fixed time interval for multistate monotone systems. Scand. J. Statist. 20 (1993), 171-175.
[2.25].
[2.26]. X. L. Li Bounds for the availability and unavailability of multistate monotone coherent systems. Chinese Journal of Applied Probability and Statistics , Vol.2, No.4, (1986), pp.347-353. (in Chinese)
[2.27]. K. Kolowrocki, On limit reliability functions of large multi-state systems with ageing components. Applied mathematics and computation 121, (2001), 313-361.
[2.28]. B. Lindqvist, Bounds for the reliability of multistate systems with partially ordered state spaces and stochastically monotone Markov transitions, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, Preprint, Statistics No.11/2002.
[2.29]. Meng F.C. A note on two reliability lower
bounds for multistate systems. Probability
in the Engineering and Informational Sciences, Vol.16, No.4, pp. 485-498. (2002).
[2.30]. Lindqvist B., Bounds for the Reliability of
Multistate Systems with Partially Ordered State Spaces and Stochastically
Monotone Markov Transitions, International
Journal of Reliability, Quality and Safety Engineering, Vol. 10, No. 3, pp.
235-248, (2003).
[2.31]. Kolowrocki K., Asymptotic Approach to Reliability
Analysis of Large Systems with Degrading Components, International Journal of Reliability,
Quality and Safety Engineering, Vol. 10, No. 3, pp. 249-288, (2003).
[2.32]. Kolowrocki
K., Reliability of Large Systems,
Elsevier,
[2.33]. Meng F.C., Comparing two reliability upper
bounds for multistate systems, Reliability
Engineering & System Safety, Vol.87, No.1, 2005, pp.31-36.
Component
importance
[3.1]. D.
Butler, A complete importance ranking for components of binary coherent systems
with extensions to multistate systems, Naval Research Logistics, vol 26,
1979 December, pp 556-578.
[3.2].
[3.3]. B.
Natvig, Recent developments in multistate reliability theory. In Probabilistic
Methods in the Mechanics of Solids and Structures. IUTAM Symposium,
[3.4]. T.
Aven, R. Ostebo, Two new component importance measures for a flow network
system, Reliability Engineering, Vo. 14, pp. 75-80, 1986
[3.5]. A.
Bossche, Calculation of Critical Importance for Multi-State Components, IEEE
Transactions on Reliability, vol. R-36, NO. 2, 1987 June, pp 247-249.
[3.6]. B.
Natvig, Reliability: Importance of components. In Encyclopedia of
Statistical Sciences, Vol. 8 (1988), 17-20.
[3.7]. V.C.
Bueno, On the importance of components for multistate monotone systems. Statistics
& Probability Letters , Vol.7, (1988), pp.51-60.
[3.8] A.M.
Abouammoh, M.A. Al-Kadi. On measure of importance for components in multistate
coherent systems. Microelectronics and Reliability, Vol.31, No.1,
(1991), pp.109-122.
[3.9]. A.D.
Dharmadhikari, U.V. Naik-Nimbalkar, The importance of an item in a multistate
system. Journal of the Operations Research Society of Japan , Vol.35,
(1992), pp.31-44.
[3.10]. M.
Finkelstein, Once more on measures of importance of system components, Microelectron.
Reliab. 34, 1994, pp. 1431-1439.
[3.11]. M.
Armstrong, Reliability-Importance and Dual Failure-Mode Components, IEEE
Transactions on Reliability, vol. 46, NO. 2, 1997 June, pp 212-221.
[3.12]. G.
Levitin, A. Lisnianski, Importance and sensitivity analysis of multi-state
systems using universal generating functions method, Reliability Engineering
& System Safety, vol. 65, pp. 271-282, 1999.
[3.13]. S.
Wu, L. Chan, Performance utility analysis of multi-state systems, IEEE
Trans. Reliability, vol. 52, pp.14-21, (2003).
[3.14]. E.
Zio, L. Podofillini, Monte Carlo simulation analysis of the effects of different
system performance levels on the importance of multi-state components, Reliability
Engineering & System Safety, vol. 82, pp.63-73, (2003).
[3.15]. Zio E., Podofillini L., Importance Measures of Multi-State Components in Multi-State Systems, International Journal of Reliability, Quality and Safety Engineering, Vol. 10, No. 3, pp. 289-310, (2003).
[3.16]. G. Levitin, L. Podofillini, E. Zio, Generalized
importance measures for multi-state elements based on performance level
restrictions, Reliability Engineering & System Safety, vol. 82, pp.
287-298, (2003).
[3.17]. V.C. Bueno. Generalizing the importance of components for multistate monotone systems. Rebrape Revista Brasileira de Probabilidade e Estatistica, 1989, Vol.3, No.1, pp.1-11.
[3.18]. V.C. Bueno and I. Norros, Component importance through compensator transforms. Rebrape Revista Brasileira de Probabilidade e Estatistica, 1992, Vol.6, No.2, pp.153-161.
[3.19]. Y. Dutuit, A. Rauzy, J-P. Signoret, P. Tomas, An
Insight into Two Methods for Performance Assessment of Multi-state Systems,
Fourth International Conference on Mathematical Methods in Reliability
Methodology and Practice,
[3.20]. Tian Hong, Chen Bao-zhi, Wu Qiong, Gao
Yong-ting, Multistate system reliability and uncertainty importance of its
components, Journal of Northeastern
University (Natural Science) [Shenyang, China], Vol.21, No.6, 2000,
pp.634-636.
[3.21]. Zio E.; Podotillini
L., A Monte Carlo approach to the estimation of importance measures of
multi-state components, 2004 Annual
Reliability and Maintainability Symposium, 2004, pp.129-134.
[3.22]. E. Zio,
L. Podofillini, G. Levitin, Estimation of the importance measures of
multi-state elements by Monte Carlo simulation, Reliability Engineering & System Safety, Vol.86, No.3, 2004,
pp.191-204.
[3.23]. G. Levitin, Protection survivability importance in systems with multilevel protection, Quality and Reliability Engineering International, No. 20, pp. 727-738 (2004).
[3.24]. J.E. Ramirez-Marquez, D.W. Coit. Multi-state component criticality analysis in multi-state
systems. In Advances in Safety and Reliability, Kołowrocki (ed.) 2005,
[3.25]. J.E. Ramirez-Marquez, D.W. Coit, Composite
importance measures for multi-state systems with multi-state components, IEEE Transactions on Reliability, 2005,
Vol.54, No 3 September, pp.517-529.
[3.26]. Sh. Wu, Joint importance of multi-state systems,
Computers and Industrial Engineering,
2005, Vol.49, No 1, pp. 63-75.
Consecutively-connected
systems and networks with multi-state components
[4.1]. P.
Doulliez, E. Jamoulle, Transportation networks with random arc capacities, RAIRO,
vol. 3, (1972), pp. 45-60.
[4.2]. J.
Evans, Maximum flow in probabilistic graphs: The discrete case, Networks,
Vol. 6, (1976), pp. 161-183.
[4.3]. A.S.
Griffith, Z. Govindarjulu, Consecutive K-out-of-N failure systems: reliability,
availability, component importance, and multistate extensions. American
Journal of Mathematical and Management Sciences , Vol.5, (1985),
pp.125-160.
[4.4]. S.N.
Chiou, V.O.K. Li, Reliability analysis of a communication network with
multimode components. IEEE Journal of Selected Areas in Communications ,
Vol.SAC-4, (1986), pp.1156-1161.
[4.5]. F.K.
Hwang , Y.C. Yao, Multistate consecutively-connected systems. IEEE
Transactions on Reliability, Vol.38, October No.4, (1989), pp.472-474.
[4.6]. C.
Yang, P. Kubat, Efficient computation of of most probable states for
communication networks with multimode components. IEEE Transactions on
Communications, Vol.37, No.5 May, (1989), pp.535-538.
[4.7]. K.V.
Le, V.O.K. Li, A path-based approach for analyzing reliability of networks with
dependent failures and multimode components. INFOCOM'90 , San Francisco,
1990, pp.495-503.
[4.8]. M.
Haim, Z. Porat, Bayes reliability modeling of a multistate consecutive
k-out-of-n: F system, Proceedings Annual Reliability And Maintainability
Symposium, 1991, pp. 582-586.
[4.9]. C.
Alexopoulos, G. Fishman, Characterizing stochastic flow networks using the
Monte Carlo method, Networks, Vol. 21, (1991), pp. 775-798.
[4.10]. C.
Alexopoulos, G. Fishman, SEnsitivity analysis in stochastic flow networks using
the Monte Carlo method, Networks, Vol. 23, (1993), pp. 605-621.
[4.11]. M. Zuo,
M. Liang, Reliability of multistate consecutively-connected systems, Reliability
Engineering & System Safety, vol. 44, 1994, pp. 173-176.
[4.12]. A.
Kossow, W. Preuss, Reliability of linear consecutively-connected systems with
multistate components, IEEE Transactions on Reliability, vol. 44, 1995,
pp. 518-522.
[4.13]. J.
Malinowski, W. Preuss, Reliability of circular consecutively-connected systems
with multistate components, IEEE Transactions on Reliability, vol. 44,
1995, pp. 532-534.
[4.14]. C.
Alexopoulos, A note on state-space decomposition methods for analyzing
stochastic flow networks. IEEE Transactions on Reliability, Vol.44,
No.2, 1995, June, pp.354-357.
[4.15]. J.S.
Lin, C.C. Jane, J. Yuan, On reliability evaluation of a capacitated-flow network
in terms of minimal pathsets. Networks, Vol.25, 1995, pp.131-138.
[4.16]. J.
Malinowski, W. Preuss, Reliability of a 2-way linear consecutively-connected
system with multistate components, Microelectronics and Reliability,
vol. 36, 1996, pp. 1483-1488.
[4.17]. J.
Malinowski, W. Preuss, Reliability increase of consecutive-k-out-of-n:F and
related systems through components' rearrangement, Microelectronics and
Reliability, vol. 36, 1996, pp. 1417-1423.
[4.18]. J.
Malinowski, W. Preuss, Reliability evaluation for tree-structured systems with
multistate components, Microelectronics and Reliability, 1996, vol. 36,
pp. 9-17.
[4.19]. J.
Malinowski, W. Preuss, Reliability of reverse-tree-structured systems with
multistate components, Microelectron. Reliab., Microelectronics and
Reliability, vol. 36, 1996, pp. 1-7.
[4.20]. J.
Malinowski, W. Preuss, Reliability of a two-way circular
consecutively-connected system with multistate components, Microelectronics
and Reliability, 1997, vol. 37, pp. 1255-1258.
[4.21]. W.
Yeh, A Revised Layered-Network Algorithm to Search for all d-Minpaths of a
Limited-Flow Acyclic Network, IEEE Transactions on Reliability, vol. 47,
No. 4, 1998, pp. 436-442.
[4.22]. V.
Rykov,
[4.23]. H.
Kotner, RAMA - a Software Tool for RAM-analysis, Second International
Conference on Mathematical Methods in Reliability: Methodology, Practice and
Inference, Volume 2, pp. 618 - 621, Bordeaux, France, July 4-7, 2000.
[4.24].
[4.25]. G.
Levitin, Reliability evaluation for acyclic consecutively connected networks
with multistate elements, Reliability Engineering & System Safety,
73 (2) 2001, pp.137-143.
[4.26]. G.
Levitin, Reliability evaluation for linear consecutively-connected systems with
multistate elements and retransmission delays, Quality and Reliability
Engineering International, 17 (5), 2001, pp. 373-378.
[4.27]. C.C.
Jane, J. S. Lin, J. Yuan, Reliability evaluation of a limited-flow network in
terms of minimal cutsets. IEEE Transactions on Reliability, vol. 42, No.
3, 1993, pp. 354-368.
[4.28]. P.
Varshney, A. Joshi, P. Chang, Reliability modeling and performance evaluation
of variable link-capacity networks, IEEE Transactions on Reliability,
vol. 43, No. 3, 1994, pp. 378-382.
[4.29]. J.
Somers, Maximum flow in networks with small number of random arc capacities, Networks,
Vol. 12, 1982, pp.241-253.
[4.30]. R.
Hassin and
[4.31]. G.
Levitin, Optimal reliability enhancement for multi-state transmission networks
with fixed transmission time, Reliability Engineering & System Safety,
vol. 76, pp.289-301, 2002.
[4.32]. G.
Levitin, Optimal allocation of multi-state elements in linear
consecutively-connected systems with delays, Int. Journal of Reliability,
Quality and Safety Engineering, vol. 9, pp.89-108, 2002.
[4.33]. G.
Levitin, Linear multi-state sliding window systems, in Proc. of 3 Int. Conf.
on mathematical methods in reliability, MMR 2002, June 17-20, 2002,
Trondheim, Norway,pp. 381-384.
[4.34]. Y.
Lin, A simple algorithm for reliability evaluation of a stochastic-flow network
with node failure, Computers and Oper. Research, 28 (2001), 1277-1285.
[4.35]. Y.
Lin, Two-commodity reliaability evaluation for a stochastic-flow network with
node failure, Computers and Operations Research, 29 (2002), 1927-1934.
[4.36]. Y.
Lin, Using minimal cuts to evaluate the system reliability of a stochastic-flow
network with failures at nodes and arcs, Reliability Engineering &
System Safety, 75 (2002), 41-46.
[4.37]. Y.
Lin, Study on the system capacity for a multicommodity stochastic-flow network
with node failure, Reliability Engineering & System Safety, 78
(2002), 57-62.
[4.38].
[4.39]. G.
Levitin, Maximizing survivability of acyclic transmission networks with
multi-state retransmitters and vulnerable nodes, Reliability Engineering
& System Safety, vol. 77, pp.189-199, (2002).
[4.40]. G.
Levitin, Reliability of acyclic transmission networks with constant transmission
characteristics of lines, Reliability Engineering & System Safety,
vol. 78, pp.297-305, (2002).
[4.41]. W.
Yeh, Search for all d-Mincuts of a limited-flow network, Computers &
Operations Research, 2002, Vol.29, No.13, pp.1843-1858.
[4.42]. G. Levitin,
Optimal allocation of multi-state elements in a linear consecutively-connected
system, IEEE Trans. Reliability, vol. 52, pp.192-199, (2003).
[4.43]. G.
Levitin, Optimal allocation of multi-state elements in linear
consecutively-connected systems with vulnerable nodes, European Journal of
Operational Research, vol. 150, pp. 406-419, (2003).
[4.44]. G.
Levitin, Reliability evaluation for acyclic transmission networks of
multi-state elements with delays, IEEE Trans. Reliability, vol. 52,
pp.231-237, (2003).
[4.45]. G.
Levitin, Linear multi-state sliding window systems, IEEE Trans. Reliability,
vol. 52, pp.263-269, (2003).
[4.46].
Lin Y-K. Extend the
quickest path problem to the system reliability evaluation for a
stochastic-flow network. Computers and
Operations Research, Vol.30, No.4, pp. 567-575, (2003).
[4.47].
Lin Y-K. Using minimal
cuts to study the system capacity for a stochastic-flow network in
two-commodity case. Computers and
Operations Research, Vol.30, No.11, pp. 1595-1607, (2003).
[4.48].
Yeh W-C. An evaluation
of the multi-state node networks reliability using the traditional binary-state
networks reliability algorithm. Reliability
Engineering & System Safety, Vol.81, No.1, pp.1-7, (2003).
[4.49].
Yeh W-C.
Multistate-node acyclic networks reliability evaluation based on MC. Reliability Engineering & System Safety,
Vol.81, No.2, pp. 225-231, (2003).
[4.50]. Zuo M., Fang Z., Huang J., Xu X., Performance Evaluation of Decreasing Multi-State Consecutive-k-out-of-n: G Systems, International Journal of Reliability, Quality and Safety Engineering, Vol. 10, No. 3, pp. 345-358, (2003).
[4.51].
J.P. Jarvis, D.R. Shier, An improved algorithm for approximating the
performance of stochastic flow networks, INFORMS
Journal on Computing, Vol.8, No.4, pp.355-360, (1996).
[4.52]. Y-K. Lin, Study on the multicommodity
reliability of a capacitated-flow network, Computers
and Mathematics with Applications, Vol.42, No.1-2, pp. 255-264, (2001).
[4.53]. Y-K. Lin, J. Yuan, A new algorithm to
generate d-minimal paths in a multistate flow network with noninteger arc
capacities, International Journal of
Reliability, Quality, and Safety Engineering, Vol.5, No.3, pp.269-285,
(1998).
[4.54]. Y-K.
Lin, J. Yuan, Flow reliability of a probabilistic capacitated-flow
network in multiple node pairs case, Computers
and Industrial Engineering, Vol.45, No.3, pp.417-428, (2003).
[4.55] S.
Patra, R.B. Misra, Evaluation of probability mass function of flow in a
communication network considering a multistate model of network links, Microelectronics and Reliability,
Vol.36, No.3, pp.415-421, (1996).
[4.56] H. Pham, A. Suprasad, R. B. Misra,
Reliability analysis of k-out-of-n systems with partially repairable
multi-state components, Microelectronics
and Reliability, Vol.36, No.10, pp.1407-1415, (1996).
[4.57].
C. R. Tripathy, S. Patra, R. B. Misra, R. N. Mahapatra, Reliability evaluation
of multistage interconnection networks with multistate elements, Microelectronics
and Reliability, Vol. 36,
No.3, pp.423-428, (1996).
[4.58].
M.O. Ball, C.J. Colbourn and J.S. Provan, Network reliability, Chapter 11 in: Handbooks in Operations Research and
Management Science, Vol.7: Network
Models, edited by M.O. Ball, T.L. Magnanti, C.L. Monma and G.L. Nemhauser,
North-Holland, Amsterdam, pp. 673-762 (1995).
[4.59]. J. Huang,
M. J. Zuo, Zh. Fang,
Multi-state consecutive-k-out-of-n systems, IIE
Transactions, 2003, Vol.35, No.6, pp.527-534.
[4.60]. G. Levitin, Element availability
importance in generalized k-out-of-r-from-n systems, IIE Transactions,
2003, vol. 35, pp. 1125-1131.
[4.61]. Yeh W.-C., Multistate network
reliability evaluation under the maintenance cost constraint, International
Journal of Production Economics, 2004, Vol.88, pp. 73-83.
[4.62]. W.
Gaul and J. Hartung, Multistate reliability problems for GSP-digraphs.
In: Contributions to
Operations Research. Proceedings of the Conference on Operations Research held in
Oberwolfach, February 26-
[4.63]. G. Shmueli, Computing Reliabilities of Large
Consecutive-Type Systems, Fourth International Conference on Mathematical
Methods in Reliability Methodology and Practice,
[4.64]. M. Zuo, Zh. Fang, J. Huang, X. Xu, Performance Evaluation of Multi-State Consecutive-k-out-of-n:G, Fourth International Conference on Mathematical Methods in Reliability Methodology and Practice, June 21-25, 2004 Santa Fe, New Mexico.
[4.65]. Shmueli
G., Computing consecutive-type reliabilities non-recursively, IEEE Transactions on Reliability,
Vol.52, No.3 September 2003, pp. 367-372.
[4.66]. Yeh
W.-C., A path-based algorithm for evaluating the k-out-of-n flow network
reliability, Reliability Engineering
& System Safety, Vol.87, No.2, 2005, pp.243-251.
[4.67]. Ramirez-Marquez J.E., Coit D.W, Alternative approach for analyzing
multistate network reliability, Proceedings
of the Industrial Engineering Research Conference (IERC), Portland, OR, May
2003.
[4.67].
Ramirez-Marquez J.E., Coit D.W, A Monte-Carlo simulation approach for approximating multi-state
two-terminal reliability, Reliability
Engineering & System Safety, Vol.87, No.2, 2005, pp. 253-264.
[4.68]. G. Levitin, Reliability
of linear multi-state multiple sliding window systems, Naval Research
Logistics Vol. 52(3), 2005, pp.212-223.
[4.69]. L. Podofillini1, E. Zio, M. Marella. A multi-state
[4.70]. G. Levitin, Uneven allocation of elements in
linear multi-state sliding window system, European
Journal of Operational Research, 2005, Vol.163, No.2 June, pp.418-433.
[4.71]. Y. Chen, Q. Yang, Reliability of two-stage
weighted-k-out-of-n systems with
components in common, IEEE Transactions
on Reliability, 2005, Vol.54, No 3 September, pp.431-440.
Extensions of binary techniques
[5.1]. S.
Salem, G. Apostolakis, D. Okrent, A New Methodology for the Computer-Aided
Construction of Fault Trees, Annals of Nuclear Energy, 4:417433, 1977.
[5.2]. L.
Caldarola, Fault-tree analysis of multistate systems with multistate
components. Proceedings of the American Nuclear Society Topical Meting on
Probability Analysis of Nuclear Reactors Safety , Vol.3, Los Angeles, La
Grande Park, CA, May 1978, Paper No VIII-1.
[5.3].
[5.4]. L.
Caldarola, Fault-tree analysis with multistate components, Synthesis and
Analysis Methods for Safety and Reliability Studies, G. Apostolakis, S.
Garriba, and G. Volta eds., Plenum Press, 1980.
[5.5]. S.
Garriba, P. Mussio, F. Naldi, Multivalued logic in the representation of
engineering systems, Synthesis and Analysis Methods for Safety and
Reliability Studies, G. Apostolakis, S. Garriba, and G. Volta eds., Plenum
Press, 1980, pp 183-197.
[5.6].
[5.7]. Xizhi
Huang Fault tree analysis method of a system having components of multiple
failure modes. Microelectronics and Reliability, Vol.23, (1983),
pp.325-328.
[5.8]. Xizhi
Huang The generic method of the multistate fault-tree analysis. Microelectronics
and Reliability, Vol.24, (1984), pp.617-622.
[5.9]. A.
Wood, Multistate Block Diagrams and Fault Trees, IEEE Transactions on
Reliability, vol. R-34, NO. 3, 1985 August, pp 236-240.
[5.10]. S.
Garriba,
[5.11]. A.
Bossche, Top-frequency calculation of multi-state fault trees including
inter-state frequencies. Microelectronics and Reliability, Vol.26, No.3,
(1986), pp.481-482.
[5.12]. A.
Bossche, Fault Tree Analysis and Synthesis.
[5.13]. G.T.
Hogasen, About multistate fault-trees. Statistical Research Report ,
No.3,
[5.14]. Y.
Kai, Multistate fault-tree analysis. Reliability Engineering & System
Safety , Vol.28, No.1, (1990), pp.1-7.
[5.15]. W.W.
Charlesworth, S.S. Rao, Reliability analysis of continuous mechanical systems
using multistate fault trees. Reliability Engineering and System Safety
, Vol.37, (1992), pp.195-206.
[5.16]. E.
Shields, G. Apostolakis,
[5.17]. C.
Garrett, S. Guarro, G. Apostolakis, The Dynamic Flowgraph Methodology for
Assessing the Dependability of Embedded Software Systems,IEEE Transactions
on Systems, Man, and Cybernetics, 25:824-840, 1995.
[5.18]. M.
Yau, G. Apostolakis, S. Guarro, The use of prime imlicants in dependability
analysis of software controlled systems, Reliability Engineering and System
Safety, 62, (1998), pp23-32.
[5.19]. C. Jun,
S. Chang, Y. Hong, H. Yang, A Bayesian approach to prediction of system failure
rates by criticalities under event trees,Int. Journal of Production
Economics, 60-61, 1999, pp.623-628.
[5.20]. X. Zang, H. Sun, K.S. Trivedi. A
BDD-based algorithm for analysis of multistate systems with multistate
components. Technical Report.
Center for Advanced Computing and Communication,
[5.21]. Xinyu Zang, Dazhi Wang, Hairong Sun,
K.S. Trivedi, A BDD-based algorithm for analysis of multistate systems with
multistate components, IEEE Transactions
on Computers, Vol.52, No.12, pp.1608-1618, (2003).
[5.22]. E.
Korczak, Binary Representations of Multistate Monotone Systems, Fourth International
Conference on Mathematical Methods in Reliability Methodology and Practice,
[5.23].
[5.24]. J.E. Ramirez-Marquez, D.W. Coit, M.
Tortorella. A multistate based generalized path vector approach for multistate
two-terminal reliability. In Advances in Safety and Reliability, Kołowrocki (ed.) 2005,
[5.25]. E. Korczak, Binary representations of
multi-state systems. Chapter 21 in: Modern
Statistical and Mathematical Methods in Reliability. Alyson Wilson,
Nikolaos Limnios, Sallie Keller-McNulty & Yvonne Armijo (editors). World
Scientific Publishing Co.,
Multi-state
systems with several failure modes
[6.1]. J.
Wu, R. Chen, An algorithm for computing the reliability of weighted-k-out-of-n
systems, IEEE Transactions on Reliability, vol. 43, No. 2, 1994, pp.
327-328.
[6.2]. L.
Nordmann, H. Pham, Weighted voting systems, IEEE Transactions on Reliability,
vol. 48, No. 1, 1999, pp. 42-49.
[6.3]. G.
Levitin, A. Lisnianski, Reliability optimization for weighted voting system, Reliability
Engineering & System Safety, vol. 71, pp. 131-138, 2001.
[6.4]. G.
Levitin, Optimal unit grouping in weighted voting systems, Reliability
Engineering & System Safety, vol. 72, pp. 179-191, 2001.
[6.5]. G.
Levitin, A. Lisnianski, Structure Optimization of Multi-state System with Two
Failure Modes, Reliability Engineering & System Safety, vol. 72, pp.
75-89, 2001.
[6.6]. G.
Levitin, Analysis and optimization of weighted voting systems consisting of
voting units with limited availability, Reliability Engineering & System
Safety, vol. 73, pp. 91-100, 2001.
[6.7]. G.
Levitin, Asymmetric weighted voting systems, Reliability Engineering &
System Safety, vol. 76, pp.199-206, 2002.
[6.8]. G.
Levitin, Optimal series-parallel topology of multi-state system with two
failure modes, Reliability Engineering & System Safety, vol. 77,
pp.93-107, (2002).
[6.9]. G.
Levitin, Evaluating correct classification probability for weighted voting
classifiers with plurality voting, European Journal of Operational Research,
vol. 141, pp.596-607, (2002).
[6.10]. G.
Levitin, Threshold optimization for weighted voting classifiers, Naval
Research Logistics, vol. 50 (4), pp.322-344, (2003).
[6.11].
Yacoub S. Analyzing the
behavior and reliability of voting systems comprising tri-state units using
enumerated simulation. Reliability
Engineering & System Safety, Vol.81, No.2, pp.133-145, (2003).
[6.12]. Levitin G. Reliability of multi-state systems with two failure modes, IEEE Transactions on Reliability, vol. 52, No. 3, pp. 340-348 (2003).
[6.13].
M. Xie, H. Pham, Modelling the
reliability of threshold weighted voting systems, Reliability Engineering & System Safety, Vol.87, No.1, 2005,
pp.53-63.
[6.14]. G. Levitin, Weighted voting systems:
reliability versus rapidity, Reliability
Engineering & System Safety, 89(2) pp. 177-184 (2005).
[6.15]. G. Levitin. Weighted
voting systems: The decision making time aspect. In Advances in
Safety and Reliability, Kołowrocki
(ed.) 2005,
Optimization
[8.1]. I.
Ushakov, Optimal standby problem and a universal generating function. Soviet Journal
Computer and System Science. 1987, 25, pp. 61-73.
[8.2].
[8.3]. E. El-Neweihi,
F. Proschan, J. Sethuraman, Optimal allocation of multistate components. In: Handbook
of Statistics, Vol.7: Quality Control and Reliability. Edited by
P.R.Krishnaiah, C.R.Rao. North-Holland,
[8.4]. A.
Lisnianski, H. Ben Haim, D. Elmakis, Redundancy optimization for power station,
Proceedings 10th International Confer. of
[8.5].
A.Lisnianski, G. Levitin, H. Ben-Haim, D. Elmakis, Power system structure optimization
subject to reliability constraints, Electric Power Systems Research,
vol. 39, No.2, pp.145-152, 1996.
[8.6]. A.
Lisnianski, G. Levitin, H. Ben-Haim, D. Elmakis, Multi-level technical system
structure optimization subject to reliability constraints. Proceedings of 11
International Conference of the
[8.7]. F.
Meng, More on optimal allocation of components in coherent systems. Journal
of Applied Probability , Vol.33, 1996, pp.548-556.
[8.8]. G.
Levitin, A. Lisnianski, D. Elmakis, Structure optimization of power system with
different redundant elements, Electric Power Systems Research, vol. 43,
No.1, pp.19-27, 1997.
[8.9]. G.
Levitin, A. Lisnianski, D. Elmakis, H. Ben-Haim, Genetic algorithm for optimal
long term power system maintenance and development planning, Th. of 1-st
Mediterranean Conference on Power Generation, Transmission and Distribution,
Lefkosia (Nicosia), Cyprus, 1998.
[8.10]. G.
Levitin, A. Lisnianski, H. Ben Haim, D. Elmakis, Redundancy optimization for
series-parallel multistate systems, IEEE Transactions on Reliability,
vol. 47, No. 2, pp. 165-172, 1998.
[8.11]. G.
Levitin, A. Lisnianski, Joint redundancy and maintenance optimization for
multistate series-parallel systems, Reliability Engineering & System
Safety, vol.64, No.1, pp.33-42, 1998.
[8.12]. G.
Levitin, A. Lisnianski, Optimal multistage modernization of power system
subject to reliability and capacity requirements, Electric Power Systems
Research, vol. 50, pp. 183-190, 1999.
[8.13]. M.
Zuo, L. Choy, R. Yam, A Model for Optimal design of Multi-state parallel-series
systems, Electrical and Computer Engineering, 1999, IEEE Canadian
Conference, Vol. 3, pp. 1751-1754.
[8.14]. C. Chen,
M. Meng, M. Zuo, Selective Maintenance Optimization for Multi-state Systems. Electrical
and Computer Engineering, 1999, IEEE Canadian Conference, Vol. 3, pp.
1477-1482.
[8.15]. G.
Levitin, A. Lisnianski, Optimization of imperfect preventive maintenance for
multi-state systems, Reliability Engineering & System Safety, vol.
67, pp. 193-203, 2000.
[8.16]. A.
Lisnianski, G. Levitin, H. Ben Haim, Structure optimization of multi-state
system with time redundancy, Reliability Engineering & System Safety,
vol. 67, pp. 103-112, 2000.
[8.17]. G.
Levitin, A. Lisnianski, Optimal replacement scheduling in multi-state
series-parallel systems (short communication), Quality and Reliability
Engineering International, vol. 16, pp. 157-162, 2000.
[8.18]. G.
Levitin, Multi-state series-parallel system expansion scheduling subject to
availability constraints, IEEE Transactions on Reliability, vol. 49, pp.
71-79, 2000.
[8.19]. G.
Levitin, A. Lisnianski, Structure optimization of power system with bridge
topology, Electric Power Systems Research, vol. 45, pp. 201-208, 1998.
[8.20]. A.
Lisnianski, G. Levitin, Universal generating function application to
multi-state system reliability analysis & optimization, Proc. of MMR
2000, Second Int. Conf. on Mathematical Methods in Reliability, 2000, pp.
715-718.
[8.21]. M.
Zuo, B. Liu, D. Murthy, Replacement-repair policy for multi-state deteriorating
products under warranty , European Journal of Operational Research, 123,
2000, pp. 519-530.
[8.22]. G.
Levitin, A. Lisnianski, H. Ben-Haim, D. Elmakis, Genetic algorithm and
universal generating function technique for solving problems of power system
reliability optimization, Proc. of DRPT 2000, Int. Conf. on Electric Utility
Deregulation and Restructuring and Power Technologies, 2000, pp. 582-586.
[8.23]. G.
Levitin, A. Lisnianski, A new approach to solving problems of multi-state
system reliability optimization, Quality and Reliability Engineering
International, vol. 47, pp. 93-104, 2001.
[8.24]. G.
Levitin, Redundancy optimization for multi-state system with fixed
resource-requirements and unreliable sources, IEEE Transactions on
Reliability, vol. 50, pp. 52-59, 2001.
[8.25]. G.
Levitin, Optimal allocation of multi-state retransmitters in acyclic
transmission network, Reliability Engineering & System Safety, vol.
75, pp.73-82, (2001).
[8.26]. G.
Levitin, Allocation of test times in multi-state systems for reliability growth
testing, IIE Transactions, vol. 34, No. 6, 2002, pp. 551-558
[8.27]. G.
Levitin, Optimal allocation of elements in linear multi-state sliding window
system, Reliability Engineering & System Safety, vol. 76, pp.247-255, 2002.
[8.28]. U.
Gurler, A. Kaya, A maintenance policy for a system with multi-state components:
an approximate solution, Reliability Engineering & System Safety,
vol. 76, pp 117-127, 2002.
[8.29]. D.
Dascalu, D. Ionescu, Optimizing the maintenance strategy of multi-state systems
with several failure modes, in Proc. of 3 Int. Conf. on mathematical methods
in reliability, MMR 2002, June 17-20, 2002, Trondheim, Norway,pp. 177-180.
[8.30]. D.
Hsieh, K. Chiu, Optimal maintenance policy in a multistate deteriorating
standby system, European Journal of Operational Research, vol. 141,
pp.689-698, (2002).
[8.31].
Levitin G., Lisnianski
A., Multi-state system reliability analysis and optimization (universal
generating function and genetic algorithm approach). In: Handbook of Reliability Engineering, H. Pham (Ed.), Springer,
pp.61-90, (2003).
[8.32]. Ramirez-Marquez J.E., Coit D.W., A
heuristic for solving the redundancy allocation problem for multi-state
series-parallel systems, Reliability Engineering & System Safety, ,
vol. 83, pp. 341-349, (2004).
[8.33]. Chao-Ton Su, Cheng-Chang Chang,
Minimization of the life cycle cost for a multistate system under periodic maintenance,
International Journal of Systems Science,
2000, Vol.31, No.2, pp.217-227.
[8.34]. M.
Nourelfath and
[8.35]. G. Levitin, Fault-tolerant Software as Multi-State System, Fourth International Conference on Mathematical Methods in Reliability Methodology and Practice, June 21-25, 2004 Santa Fe, New Mexico.
[8.36]. M. Nourelfath, Y. Dutuit, A combined approach
to solve the redundancy optimization problem for multi-state systems under
repair policies, Reliability Engineering
& System Safety, Vol.86, No.3, 2004, pp.205-213.
[8.37]. G. Levitin, Optimal version sequencing in fault-tolerant programs, Asia-Pacific Journal of Operational Research, 22(1) pp. 1-18 (2005).
[8.38]. Tian Z., Zuo M., Huang H. Reliability-redundancy allocation for multi-state series-parallel
systems. In Advances in Safety and Reliability, Kołowrocki (ed.) 2005,
[8.39]. M. Nourelfath, D. Ait-Kadi. Redundancy optimization for multi-state systems under repair
policies. In Advances in Safety and Reliability, Kołowrocki (ed.) 2005,
[8.40]. G.
Levitin, Optimal structure of fault-tolerant software systems, Reliability Engineering & System Safety,
2005, Vol.89, No.3 September, pp.286-295.
Coherency
[9.1]. L.
Caldarola, Coherent systems with multistate components, Nucl. Eng. Design,
vol 58, 1980, pp 127-139.
[9.2]. M. N.
Fardis, C. A. Cornell, Analysis of coherent multistate systems, IEEE
Transactions on Reliability, vol R-30, 1981, pp. 117-122.
[9.3]. B.
Natvig, Two suggestions of how to define a multistate coherent system, Adv.
Appl. Probability, vol 14, 1982, pp 434-455.
[9.4]. J. C.
Hudson, K. C. Kapur, Modules in Coherent Multistate Systems, IEEE
Transactions on Reliability, vol. R-32, NO. 2, June 1983, pp 183-185.
[9.5].
W.D.S. Borges, F.W. Rodrigues An axiomatic characterization of multistate
coherent systems. Mathematics of Operations Research , Vol.8, (1983),
pp.435-438.
[9.6]. F.
Ohi, T. Nishida, Generalized multistate coherent systems. Journal of Japan
Statistical Society, Vol.13, (1983), pp.165-181.
[9.7]. B.
Natvig, A. Streller, The steady-state behavior of multistate monotone systems, J.
Applied Probability, vol. 21, 1984, pp 826-835.
[9.4]. F.
Ohio, T. Nishida, On multistate coherent systems, IEEE Transactions on
Reliability, vol. R-33, 1984 October, pp 284-287.
[9.9]. J.
Ansell, A. Bendell, On alternative definitions of multistate coherent systems. Optimization
, Vol.18, (1987) pp.119-136.
[9.10]. F.
Ohi, T. Nishida, Generalized multistate coherent systems. Abstract of
Meeting of Operations Research Society of
[9.11]. A.
Abouammoh, M.Al-Kadi, Component Relevancy in Multistate Reliability Models, IEEE
Transactions on Reliability, vol. 40, NO. 3, 1991 August, pp 370-375.
[9.12]. I.
Kuhnert Comment on: "Component relevancy in multistate reliability
models". IEEE Transactions on Reliability , Vol.44, No.1 March,
(1995), pp.95-96.
[9.13]. U.
Rakowsky, Theory and Applications of Multistate Coherent Systems in
Reliability Engineering (in German). VDI 8-286 (Association of German
Engineers),
[9.14]. F.
Meng, Component-relevancy and characterization in multistate systems, IEEE
Transactions on Reliability, vol 42, 1993 September, pp 478-483.
[9.15]. U.
Rakowsky, Continuous Multistate Coherent Systems in Reliability Engineering (in
German). Automatisierungstechnik, Vol. 43, No. 4, pp. 174-180, 1995.
Non-coherent systems
[10.1]. A.
Bendell, J. Ansell, The incoherency of multistate coherent systems, Reliability
Engineering, vol 8, 1984, pp 165-178.
[10.2]. A.
Bossche, The top-event failure frequency for non-coherent multistate fault
trees, Microelectronics Reliability, 1984, 24(4), pp 707-715.
[10.3]. K.
Kolowrocki, Limit reliability functions of some nonhomogeneous series-parallel
and parallel-series systems, Reliability Engineering and System Safety,
46, 1994, pp. 171-177.
Application of Markov, semi-Markov processes and
combinatorial analysis
[11.1]. N.H.
Hjort, B. Natvig, E. Funnemark, The association in time of a Markov process
with application to multistate reliability theory. J. Appl. Prob. 22
(1985), 473-479.
[11.2]. B.H.
Lindqvist, Monotone and associated Markov chains, with applications to
reliability theory. Journal of Applied Probability, Vol.24, 1987,
pp.679-695.
[11.3]. M.
Veeraraghavan, K. S. Trivedi, A Combinatorial Algorithm for Performance and
Reliability Analysis Using Multistate Models, IEEE Transactions on Computers,
Vol. 43, 1994, pp. 226-229.
[11.4]. J.
Xue, K. Yang, Dynamic Reliability Analysis of Coherent Multistate Systems, IEEE
Transactions on Reliability, vol. 44, NO. 4, 1995, December, pp 683-688.
[11.5].
Sahner R. A., Trivedi K. S., Poliafito A., Performance and reliability analysis
of computer systems . An example-based approach using the SHARPE software
package.
[11.6]. I.N.
Kovalenko, N.Y. Kuznetsov, and P.A. Pegg, Mathematical Theory of Reliability
of Time Dependent Systems with Practical Applications, John Wiley &
Sons, Inc., New York (1997)
[11.7]. F.
Grabski, K. Kolowrocki, Asymptotic reliability of multistate systems with
semi-Markov states of components, Safety and reliability, Schueller,
Kafka (eds), 1999, Balkema, Rotterdam, ISBN 90 5809 109 0, pp. 317-322
[11.8]. X.
Zang, H. Sun and K. Trivedi, A BDD approach to dependability analysis of
distributed computer systems with imperfect coverage, Dependable Network
Computing, D. R. Avresky (ed.), Kluwer Academic Publishers, The
Netherlands, Dec. 1999, pp. 167-190.
[11.9]. X.
Zang, H. Sun and K. Trivedi, A BDD-based Algorithm for Reliability Analysis of
Phased-Mission Systems, IEEE Transactions on Reliability, Vol. 48, No.
1, pp. 50--60, March 1999.
[11.10]. B.
Dimitrov, V. Rykov, P. Stanchev, On multi-state reliability systems, in Proc.
of 3 Int. Conf. on mathematical methods in reliability, MMR 2002, June
17-20, 2002, Trondheim, Norway,pp. 201-204.
[11.11]. U.
Gurler, and A. Kayab, A maintenance policy for a system with multi-state
components: an approximate solution, Reliability Engineering & System
Safety, Vol. 76, pp. 117-127
[11.12]. M.
Lanus, L. Yin, K. Trivedi, Hierarchical composition and aggregation of
state-based availability and performability models, IEEE Trans. Reliability,
vol. 52, pp.44-52, (2003).
[11.13]. Chen D., Cao Y., Trivedi K., Hong Y., Preventive Maintenance of Multi-State System with Phase-Type Failure Time Distribution and Non-Zero Inspection Time, International Journal of Reliability, Quality and Safety Engineering, Vol. 10, No. 3, pp. 323-344, (2003).
[11.14]. J. Gasemyr, B. Natvig, Probabilistic modelling of monitoring and maintenance
of multistate monotone systems with dependent components, Dept. of Math.,
University of Oslo, Statistical Research
Report, Preprint 12, (2003).
[11.15]. Y. Lefebvre, Using the phase method to
model degradation and maintenance efficiency, International Journal of Reliability, Quality and Safety Engineering,
Vol. 10, No. 4, pp.383-405, (2003).
[11.16]. A.
Linianski, Application of Semi-Markov Processes and Universal Generating
Function Technique for
Multi-state System Reliability Evaluation, Fourth International Conference on
Mathematical Methods in Reliability Methodology and Practice,
[11.17]. V. Rykov, B. Dimitrov, P. Stanchev, Reliability of Complex Hierarchical Systems with Fault Tolerant Units, Fourth International Conference on Mathematical Methods in Reliability Methodology and Practice, June 21-25, 2004 Santa Fe, New Mexico.
[11.18]. V. Rykov, D. Efrosinin, On Reliability Control of Fault Tolerance Units, Fourth International Conference on Mathematical Methods in Reliability Methodology and Practice, June 21-25, 2004 Santa Fe, New Mexico.
[11.19]. M. Szczerbinski. Multi-state reliability models for lightning hazard assessment, International Journal of Reliability, Quality and Safety Engineering, Vol. 11, No. 1 (2004) 35-46.
[11.20]. J. Li, Y. Wu, K. Lai, K. Liu,
Reliability estimation and prediction of multi-state components and coherent
systems, Reliability Engineering &
System Safety, vol. 88, pp.
93-98, 2005.
[11.21]. Lisnianski A. Extended block diagram
method for multi-state system reliability assessment. In Advances in Safety and Reliability, Kołowrocki (ed.) 2005,
[11.22]. J.S. Ivy, S.M. Pollock, Marginally monotonic
maintenance policies for a multi-state deteriorating machine with probabilistic
monitoring, and silent failures, IEEE
Transactions on Reliability, 2005, Vol.54, No 3 September, pp.489-497.
Common cause failures
[12.1]. G.
Levitin, Incorporating common-cause failures into series-parallel multi-state
system analysis, IEEE Transactions on Reliability vol. 50, No. 4, 2001,
pp. 380-388.
[12.2]. G.
Levitin, A. Lisnianski, Optimal separation of elements in vulnerable
multi-state systems, Reliability Engineering & System Safety, vol.
73, pp. 55-66, 2001.
[12.3]. G.
Levitin, A. Lisnianski, Survivability maximization for vulnerable multi-state
system with bridge topology, Reliability Engineering & System Safety,
vol. 70, pp. 125-140, 2000.
[12.4]. G.
Levitin, A. Lisnianski, Optimizing survivability of vulnerable series-parallel
multi-state systems, Reliability Engineering & System Safety, vol.
79, pp.319-331, (2003).
[12.5]. G.
Levitin, Optimal multilevel protection in series-parallel systems, Reliability
Engineering & System Safety, vol. 81, pp. 93-102, (2003).
[12.6]. G. Levitin,
Y. Dai, M. Xie, K. L. Poh, Optimizing survivability of multi-state systems with
multi-level protection by multi-processor genetic algorithm, Reliability
Engineering & System Safety, vol. 82, pp.93-104, (2003).
[12.7]. G.
Levitin, Common supply failures in linear multi-state sliding window systems, Reliability
Engineering & System Safety, vol. 82, pp.55-62, (2003).
[12.8]. G.
Levitin, Maximizing survivability of vulnerable weighted voting system, Reliability Engineering & System Safety,
vol. 83, pp.17-26, (2003).
[12.9]. G. Levitin, Multi-state series-parallel systems with Multilevel Protection, in Proc. of The 8th Annual International Conference on Industrial Engineering Theory, Applications and Practice, Las Vegas, Nevada, US, pp. 980-985.
[12.10]. E. Korczak, G. Levitin, H. Ben Haim, Survivability of series-parallel systems with multilevel protection, Reliability Engineering & System Safety, vol.90/1, pp.45-54 (2005).
Continuous systems
[13.1].
Block, H. W. , Savits, T. H. (1984) Continuous multi-state structure functions.
Operations Research, 32, 703-714.
[13.2].
Baxter, L. A. (1984) Continuum structures. I. Journal of Applied Probability,
21, 802-815.
[13.3].
Baxter, L. A. (1986) Continuum structures II. Mathematical Proceedings of
the
[13.4].
Baxter, L. A. and Kim, C. (1986) Bounding the stochastic performance of
continuum structure functions I. Journal of Applied Probability, 23,
660-669.
[13.5].
Baxter, L. A. and Lee S. M., (1989) Structure Functions with Finite Minimal
Vector Sets. Journal of Applied Probability, 26, 196-201.
[13.6]. Mak,
K., (1989) A Note on Barlow-Wu Structure Functions. Operation Research
Letters, 8, pp. 43-44.
[13.7].
Iyer, S. N. and Sathe Y. S., (1990) Redundancy in Barlow-Wu Structures. Journal
of the Operational Research Society, vol. 41, No 9, pp. 843-851.
[13.8].
Montero, J., Tejada, J. and Yanez, J. (1990) Structural properties of continuum
systems. European Journal of Operational Research, 45, pp. 231-240.
[13.9].
Montero, J., Tejada, J. and Yanez, J. (1993) Multivalued Continuum Systems. European
Journal of Operational Research, 69, pp. 55-64.
[13.10].
Montero, J. (1993) Observable Structure Functions. Kybernetes, vol. 22,
No. 2, pp. 31-39.
[13.11].
Montero, J., Tejada, J. and Yanez, J. (1994) General Structure Functions. Kybernetes,
vol. 23, No. 3, pp. 10-19.
[13.12].
Cutello, V., Montero, J., Yanez, J., (1996) Structure functions with fuzzy
states. Fuzzy Sets Systems, 83, pp. 189-202.
[13.13].
[13.14].
Brunelle, R. and Kapur, K.C. (1998) Continuous-state system reliability: an
interpolation approach. IEEE Transaction on Reliability, 47, 181-187.
[13.15].
Chrlesworth. W. W. and Rao, S. S. (1992) Reliability Analysis of Continuous
Mechanical Systems Using Multistate Fault Trees. Reliability Engineering
& System Safety, vol. 37, pp. 195-206.
[13.16]. Yang,
K. and Xue, J. (1996) Continuous state reliability analysis, in Proceedings
of the Annual Reliability and Maintanability Symposium, IEEE, Piscataway,
NJ, pp. 251-257.
[13.17].
Brunelle, R. and Kapur, K. C. (1999) Review and classification of reliability
measures for multi-state and continuum models. IEE Transactions 31,
1171-1180.
[13.18].
Lisnianski, A., Jeager, A. (2000) Time-redundant System Reliability Under
Randomly Constrainrd Time Resources. Reliability Engineering & System
Safety, vol. 70, No. 2, pp. 157-166.
[13.19].
Lisnianski, A., (2001) Estimation of Boundary Points for Continuum-state System
Reliability Measures. Reliability Engineering & System Safety, vol.
74, No. 1, pp. 81-88.
[13.20]. Dieulle,
L., Berenguer, C., Grall, A., Roussignol, M., (2001) Continuous time Predictive
Maintenance Scheduling for a Deteriorating System. (2001) Proceed. Annual
Reliability and Maintainability Symposium, pp. 150-155.
[13.21].
Langer, Yu., (2001) System with Time-Redundancy. (2001) Proceed. Annual
Reliability and Maintainability Symposium, pp. 189-192.
[13.22].
Lisnianski A. (2001), Estimating of boundary points for continuum-state system
reliability measures, Reliability Engineering & System Safety, vol.
74, pp. 81-88.
[13.23].
Lisnianski A., Continuous-state system reliability models as an extension of
multi-state models, in Proc. of 3 Int. Conf. on mathematical methods in
reliability, MMR 2002, June 17-20, 2002, Trondheim, Norway,pp. 401-404.
[13.24].
Liu P.X., Zuo M.J.,
Meng M.Q-H. Using neural network function approximation for optimal design of
continuous-state parallel–series systems. Computers
& Operations Research, Vol.30, No.3, pp. 339-352, (2003).
[13.25].
Zuo M.J., Jiang R., Yam
R.C.M. (1999). Approaches for reliability modeling of continuous state devices.
IEEE Transactions on Reliability,
Vol.48, No.1 March, pp.9-18.
[13.26]. Seung Min Lee, On the characterization
of continuum structure functions, Operations
Research Letters, 2003, Vol.31, No.4, pp.268-272.
[13.27]. Griffith W. S, On redundancy in
continuum structure functions, Proceedings
of the Physical and Engineering Sciences Section of the American Statistical
Association, 1995, pp. 212-214.
[13.28]. Griffith W. S, Some redundancy results
for continuum structure functions, Arab
Journal of Mathematical Sciences, 1997, Vol.3, No.1, pp.49-57.
[13.29]. Iyer S, Identities and inequalities
for coherent life functions and continuum structure functions. Opsearch, 2003, Vol.40, No.1, pp.11-23.
[13.30]. SeungMin Lee and RakJoong Kim,
Approximation of reliability importance for continuum structure functions. Kangweon-Kyungki Math. Jour., 1997,
Vol.5, No.1, pp.55–60.
[13.31]. K. Nakashima and K. Yamato, Two
Boolean expressions of multistate monotone systems and their reliability
analysis. In: Reliability Theory and
Applications: Proceedings of the China-Japan Reliability Symposium. Shanghai,
Xia'n and
[13.32].
L.A. Baxter and S.M. Lee, Further properties of reliability importance
for continuum structure functions. Probability
in the Engineering and Informational Sciences, 1989, Vol.3, pp.237-246.
[13.33]. L.A. Baxter and S.M. Lee, A weak
convergence theorem for continuum structure functions. Journal of Mathematical Analysis and Applications, 1990, Vol.148,
No.2, pp.463-468.
[13.34].
B. Cappelle and E.E. Kerre, Computer assisted reliability analysis: An application of
possibilistic reliability theory to a subsystem of a nuclear power plant, Fuzzy
Sets and Systems, 1995, Vol.74, No.1, pp.103-113.
[13.35]. M. Finkelstein, Simple
Continuous State Systems of Continuous State Components, Fourth International
Conference on Mathematical Methods in Reliability Methodology and Practice,
June 21-25, 2004 Santa Fe, New Mexico.
[13.36]. Montero J., Reliability bounds for
multicriteria systems. Journal of the
Operational Research Society, Vol.44, No.10, 1993, pp.1025-1034.