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10 - Continuous-Time Markov Chain: Reliability Models

from Part III - State-Space Models with Exponential Distributions

Published online by Cambridge University Press:  30 August 2017

Kishor S. Trivedi
Affiliation:
Duke University, North Carolina
Andrea Bobbio
Affiliation:
Università degli Studi del Piemonte Orientale, Italy
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Chapter
Information
Reliability and Availability Engineering
Modeling, Analysis, and Applications
, pp. 357 - 422
Publisher: Cambridge University Press
Print publication year: 2017

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References

[1] W., Feller, An Introduction to Probability Theory and its Applications. John Wiley & Sons, 1968.
[2] Y., Ng and A., Avizienis, “A unified reliability model for fault-tolerant computers,IEEE Transactions on Computers, vol. C-29, pp. 1002–1011, 1980.Google Scholar
[3] R., Marie, A., Reibman, and K., Trivedi, “Transient solution of acyclic Markov chains,Performance Evaluation, vol. 7, pp. 175–194, 1987.Google Scholar
[4] C., Lindemann, M., Malhotra, and K., Trivedi, “Numerical methods for reliability evaluation of Markovian closed fault-tolerant systems,IEEE Transactions on Reliability, vol. 44, pp. 694–704, 1995.Google Scholar
[5] K., Trivedi, Probability and Statistics with Reliability, Queueing and Computer Science Applications, 2nd ed. John Wiley & Sons, 2001.
[6] IEC 61508, Functional Safety of Electrical/Electronic/Programmable Electronic Safety-Related Systems. IEC Standard No. 61508, 2011.
[7] M., Neuts, Matrix Geometric Solutions in Stochastic Models. Johns Hopkins University Press, 1981.
[8] D., Assaf and B., Levikson, “Closure of phase type distributions under operations arising in reliability theory,The Annals of Probability, vol. 10, pp. 265–269, 1982.Google Scholar
[9] D., Aldous and L., Shepp, “The least variable phase type distribution is Erlang,Stochastic Models, vol. 3, pp. 467–473, 1987.Google Scholar
[10] S., Mondal, X., Yin, J., Muppala, J. Alonso, Lopez, and K. S., Trivedi, “Defects per million computation in service-oriented environments,IEEE Transactions on Services Computing, vol. 8, no. 1, pp. 32–46, Jan. 2015.Google Scholar
[11] R., Fricks, A., Bobbio, and K., Trivedi, “Reliability models of chronic kidney disease,” in Proc. IEEE Ann. Reliability and Maintainability Symp., 2016, pp. 1–6.Google Scholar
[12] United States Renal Data System, 2014 Annual Data Report: An Overview of the Epidemiology of Kidney Disease in the United States. National Institute of Health / National Institute of Diabetes and Digestive and Kidney Diseases, 2014.
[13] G., Cosulich, P., Firpo, and S., Savio, “Power electronics reliability impact on service dependability for railway systems: A real case study,” in proc. IEEE Int. Symp. on Industrial Electronics, ISIE '96., vol. 2, Jun 1996, pp. 996–1001.Google Scholar
[14] M., Beaudry, “Performance-related reliability measures for computing systems,IEEE Transactions on Computers, vol. C-27, pp. 540–547, 1978.Google Scholar
[15] H., Choi, W., Wang, and K., Trivedi, “Analysis of conditional MTTF for fault tolerant systems,Microelectronics and Reliability, vol. 38, no. 3, pp. 393–401, 1998.Google Scholar
[16] J. C., Laprie, J., Arlat, C., Beounes, and K., Kanoun, “Architectural issues in software fault tolerance,” in Software Fault Tolerance, ed. M. R., Lyu. John Wiley & Sons, 1994, ch. 3, pp. 47–80.
[17] B., Randell and J., Xu, “The evolution of the recovery block concept,” in Software Fault Tolerance, ed. M. R., Lyu. John Wiley & Sons, 1994, ch. 1, pp. 1–22.
[18] A., Avizienis, “The methodology of n-version programming,” in Software Fault Tolerance, ed. M. R., Lyu. John Wiley & Sons, 1994, ch. 2, pp. 23–46.
[19] G.-H., Hsu and X.-M., Yuan, “First passage times and their algorithms forMarkov processes,Stochastic Models, vol. 11, no. 1, pp. 195–210, 1995.Google Scholar
[20] A., Koziolek, A., Avritzer, S., Suresh, D., Sadoc|Menasche, K., Trivedi, and L., Happe, “Design of distribution automation networks using survivability modeling and power flow equations,” in Proc. IEEE 24th Int. Symp. on Software Reliability Engineering (ISSRE), Nov. 2013, pp. 41–50.Google Scholar
[21] A., Avritzer, S., Suresh, D. S., Menasché, R. M. M., Leão, E. de Souza e, Silva, M. C., Diniz, K. S., Trivedi, L., Happe, and A., Koziolek, “Survivability models for the assessment of smart grid distribution automation network designs,” in Proc. 4th ACM/SPEC Int. Conf. on Performance Engineering. ACM, 2013, pp. 241–252.Google Scholar
[22] D. S., Menasché, R. M. Meri, Leäo, E. de Souza e, Silva, A., Avritzer, S., Suresh, K., Trivedi, R. A., Marie, L., Happe, and A., Koziolek, “Survivability analysis of power distribution in smart grids with active and reactive power modeling,SIGMETRICS Performance Evaluation Review, vol. 40, no. 3, pp. 53–57, Jan. 2012.Google Scholar
[23] IEEE 1366, IEEE Guide for Electric Power Distribution Reliability Indices. IEEE Std. 1366-2003, IEEE Standards Board, 2003.
[24] Z., Ma, “Towards a unified definition for reliability, survivability and resilience (I): The conceptual framework inspired by the handicap principle and ecological stability,” in Aerospace Conference, 2010 IEEE, Mar. 2010, pp. 1–12.Google Scholar
[25] A., Bobbio and A., Verna, “A performance oriented reliability model of a pumping station in a fire protection system,” in Proc. 5th EUREDATA Conf., ed. H., Wingender. Springer-Verlag, 1986, pp. 606–614.
[26] N., Piccinini, A., Verna, and A., Bobbio, “Optimum design of a fire extinguishing pumping installation in a chemical plant,” in Proc. World Congress III of Chemical Engineering, 1986, Vol. II, pp. 1112–1115.Google Scholar
[27] S., Avogadri, G., Bello, and V., Colombari, “The ENI reliability data bank: Scope, organization and example of report,” in Proc. 4th EUREDATA Conference, 1983, p. 7.3.Google Scholar
[28] S. J., Bavuso, J. Bechta, Dugan, K., Trivedi, E. M., Rothmann, and W. E., Smith, “Analysis of typical fault-tolerant architectures using HARP,IEEE Transactions on Reliability, vol. R-36, no. 2, pp. 176–185, Jun. 1987.Google Scholar
[29] K., Trivedi and R., Geist, “Decomposition in reliability analysis of fault-tolerant systems,IEEE Transactions on Reliability, vol. R-32, no. 5, pp. 463–468, Dec. 1983.Google Scholar
[30] A., Avizienis, J., Laprie, B., Randell, and C., Landwehr, “Basic concepts and taxonomy of dependable and secure computing,IEEE Transactions on Dependable and Secure Computing, vol. 1, no. 1, pp. 11–33, 2004.Google Scholar
[31] A., Bobbio and K. S., Trivedi, “An aggregation technique for the transient analysis of stiff Markov chains,IEEE Transactions on Computers, vol. C-35, pp. 803–814, 1986.Google Scholar
[32] J., McGough, M., Smotherman, and K., Trivedi, “The conservativeness of reliability estimates based on instantaneous coverage,IEEE Transactions on Computers, vol. C-34, pp. 602–609, 1985.Google Scholar
[33] R., Marie, “Transient numerical solutions of stiff Markov chains,” in Proc. 20th ISATA Symp., 1989, pp. 255–270.Google Scholar
[34] A., Reibman and K., Trivedi, “Numerical transient analysis of Markov models,Computers and Operations Research, vol. 15, pp. 19–36, 1988.Google Scholar
[35] W., Grassman, “Finding transient solutions in Markovian event systems through randomization,” in Numerical Solution of Markov Chains. Marcel Dekke, 1991.
[36] J., Muppala, M., Malhotra, and K., Trivedi, “Markov dependability models of complex systems: Analysis techniques,” in Reliability and Maintenance of Complex Systems, NATO ASI Series, ed. S., Özekici. Springer, 1996, vol. 154, pp. 442–486.
[37] E. de Souza e, Silva and H., Gail, “Transient solutions for Markov chains,” in Computational Probability, ed. W., Grassmann. Springer, 2000, ch. 3, pp. 49–85.
[38] W., Stewart, Introduction to the Numerical Solution of Markov Chains. Princeton University Press, 1994.
[39] W., Grassmann, “The use of eigenvalues for finding equilibrium probabilities of certain Markovian two-dimensional queueing problems,INFORMS Journal on Computing, vol. 15, no. 4, pp. 412–421, 2003.Google Scholar
[40] R., Sahner, K., Trivedi, and A., Puliafito, Performance and Reliability Analysis of Computer Systems: An Example-Based Approach Using the SHARPE Software Package. Kluwer Academic Publishers, 1996.
[41] C., Moler and C. Van, Loan, “Nineteen dubious ways to compute the exponential of a matrix,SIAM Review, vol. 20, pp. 801–835, 1978.Google Scholar
[42] A., Jensen, “Markoff chains as an aid in the study of Markoff processes,” Scandinavian Actuarial Journal, vol. 1953, Supplement 1, pp. 87–91, 1951.Google Scholar
[43] B. L., Fox and P. W., Glynn, “Computing Poisson probabilities,Communications of the ACM, vol. 31, no. 4, pp. 440–445, Apr. 1988.Google Scholar
[44] H., Abdallah and R., Marie, “The uniformized power method for transient solutions of Markov processes,Computers and Operations Research, vol. 20, no. 5, pp. 515–526, 1993.Google Scholar
[45] W. S. A. van, Moorsel, “Adaptive uniformization,Communications in Statistics: Stochastic Models, vol. 10, no. 3, pp. 619–648, 1994.Google Scholar
[46] A. van, Moorsel and W., Sanders, “Transient solution of Markov models by combining adaptive and standard uniformization,IEEE Transactions on Reliability, vol. 46, no. 3, pp. 430–440, Sep. 1997.Google Scholar
[47] J., Carrasco, “Transient analysis of rewarded continuous time Markov models by regenerative randomization with Laplace transform inversion,The Computer Journal, vol. 46, no. 1, pp. 84–99, 2003.Google Scholar
[48] J. D., Lambert, Computational Methods in Ordinary Differential Equations. John Wiley & Sons, 1973.
[49] W., Grassmann, “Transient solution in Markovian queueing systems,Computers and Operations Research, vol. 4, pp. 47–56, 1977.Google Scholar
[50] L. F., Shampine, “Stiffness and nonstiff differential equation solvers, II: Detecting stiffness with Runge–Kutta methods,ACM Transactions on Mathematical Software, vol. 3, no. 1, pp. 44–53, 1977.Google Scholar
[51] E., Hairer, S. P., Nørsett, and G., Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn., Springer Series in Computational Mathematics. Springer, 1993, vol. 8.
[52] C., Gear, Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, 1971.
[53] R., Bank, W. M., Coughran, W., Fichtner, E., Grosse, D., Rose, and R., Smith, “Transient simulation of silicon devices and circuits,IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 4, no. 4, pp. 436–451, Oct. 1985.Google Scholar
[54] O., Axelsson, “A class of A-stable methods,BIT, vol. 9, no. 3, pp. 185–199, 1969.Google Scholar
[55] M., Malhotra, J. K., Muppala, and K. S., Trivedi, “Stiffness-tolerant methods for transient analysis of stiff Markov chains,Journal of Microelectronics and Reliability, vol. 34, pp. 1825–1841, 1994.Google Scholar
[56] M., Malhotra, J. K., Muppala, and K. S., Trivedi, “Stiffness-tolerant methods for transient analysis of stiff Markov chains,Microelectronics and Reliability, vol. 34, pp. 1825–1841, 1994.Google Scholar
[57] M., Malhotra, “A computationally efficient technique for transient analysis of repairable Markovian systems,Performance Evaluation, vol. 24, no. 4, pp. 311–331, 1996.Google Scholar
[58] A., Papoulis, Probability, Random Variables and Stochastic Processes. McGraw Hill, 1965.
[59] D., Cox and H., Miller, The Theory of Stochastic Processes. Chapman and Hall, 1965.
[60] V. G., Kulkarni, Modeling and Analysis of Stochastic Systems. Chapman and Hall, 1995.

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