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  • Print publication year: 2017
  • Online publication date: August 2017

10 - Continuous-Time Markov Chain: Reliability Models

from Part III - State-Space Models with Exponential Distributions
[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.
[3] R., Marie, A., Reibman, and K., Trivedi, “Transient solution of acyclic Markov chains,Performance Evaluation, vol. 7, pp. 175–194, 1987.
[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.
[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.
[9] D., Aldous and L., Shepp, “The least variable phase type distribution is Erlang,Stochastic Models, vol. 3, pp. 467–473, 1987.
[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.
[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.
[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.
[14] M., Beaudry, “Performance-related reliability measures for computing systems,IEEE Transactions on Computers, vol. C-27, pp. 540–547, 1978.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[33] R., Marie, “Transient numerical solutions of stiff Markov chains,” in Proc. 20th ISATA Symp., 1989, pp. 255–270.
[34] A., Reibman and K., Trivedi, “Numerical transient analysis of Markov models,Computers and Operations Research, vol. 15, pp. 19–36, 1988.
[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.
[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.
[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.
[43] B. L., Fox and P. W., Glynn, “Computing Poisson probabilities,Communications of the ACM, vol. 31, no. 4, pp. 440–445, Apr. 1988.
[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.
[45] W. S. A. van, Moorsel, “Adaptive uniformization,Communications in Statistics: Stochastic Models, vol. 10, no. 3, pp. 619–648, 1994.
[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.
[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.
[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.
[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.
[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.
[54] O., Axelsson, “A class of A-stable methods,BIT, vol. 9, no. 3, pp. 185–199, 1969.
[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.
[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.
[57] M., Malhotra, “A computationally efficient technique for transient analysis of repairable Markovian systems,Performance Evaluation, vol. 24, no. 4, pp. 311–331, 1996.
[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.