[1] A. M., Abdel-Moneim and F. W., Leysieffer, Weak lumpability in finite Markov chains, J. of Appl. Probability, 19:685–691, 1982.
[2] S., Asmussen, Applied Probability and Queues, 2nd ed. Springer-Verlag, 2003.
[3] A., Avizienis, J. C., Laprie, B., Randell, and C., Landwehr, Basic concepts and taxonomy of dependable and secure computing, IEEE Trans. on Dependable and Secure Computing, 1:11–33, January 2004.
[4] J., Banks, J. S., Carson, B. L., Nelson, and D. M., Nicol, Discrete-Event System Simulation, 5th ed. Prentice-Hall international series in industrial and systems engineering. Prentice Hall, July 2009.
[5] R. E., Barlow and L. C., Hunter, Reliability analysis of a one unit system, Operations Research, 9(2):200–208, 1961.
[6] M., Bladt, B., Meini, M., Neuts, and B., Sericola, Distribution of reward functions on continuous-time Markov chains, in Proc. of the 4th Int. Conf. on Matrix Analytic Methods in Stochastic Models (MAM4), Adelaide, Australia, July 2002.
[7] A., Bobbio and K. S., Trivedi, An aggregation technique for the transient analysis of stiff Markov chains, IEEE Trans. on Computers, C-35(9):803–814, September 1986.
[8] Z. I., Botev, P., L'Ecuyer, and B., Tuffin, Markov chain importance sampling with applications to rare event probability estimation, Statistics and Computing, 23(2):271–285, 2013.
[9] P. N., Bowerman, R. G., Nolty, and E. M., Scheuer, Calculation of the Poisson cumulative distribution function, IEEE Trans. on Reliability, 39:158–161, 1990.
[10] P., Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation and Queues. Springer, 1998.
[11] H., Cancela, M. El, Khadiri, and G., Rubino, Rare events analysis by Monte Carlo techniques in static models, in G., Rubino and B., Tufin, Eds, Rare Event Simulation Using Monte Carlo Methods, chapter 7, pages 145–170. Chichester: J. Wiley, 2009.
[12] H., Cancela, G., Rubino, and B., Tufin, MTTF Estimation by Monte Carlo Methods using Markov models, Monte Carlo Methods and Applications, 8(4):312–341, 2002.
[13] H., Cancela, G., Rubino, and M., Urquhart, Pathset-based conditioning for transient simulation of highly dependable networks, in Proc. of the 7th IFAC Symp. on Cost Oriented Automation, Ottawa, Canada, 2004. (Also presented in “5th International Conference on Monte Carlo and Quasi-Monte Carlo Methods”, Juan les Pins, France, June 2004.)
[14] J. A., Carrasco, Failure distance based on simulation of repairable fault tolerant systems, in Proc. of the 5th Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation, pages 351–365, 1991.
[15] J. A., Carrasco, Improving availability bounds using the failure distance concept, in 4th Int. Conf. on Dependable Computing for Critical Applications (DCCA'4), San Diego, USA, January 1994.
[16] J. A., Carrasco, H., Nabli, B., Sericola, and V., Sunie, Comment on “Performability analysis: A new algorithm”, IEEE Trans. on Computers, C-59:137–138, January 2010.
[17] F., Castella, G., Dujardin, and B., Sericola, Moments' analysis in homogeneous Markov reward models, Methodology and Computing in Applied Probability, 11(4):583–601, 2009.
[18] G., Ciardo, R., Marie, B., Sericola, and K. S., Trivedi, Performability analysis using semi-Markov reward processes, IEEE Trans. on Comput., C-39:1251–1264, October 1990.
[19] B., Ciciani and V., Grassi, Performability evaluation of fault-tolerant satellite systems, IEEE Trans. on Commun., COM-35:403-409, April 1987.
[20] C. J., Colbourn, The Combinatorics of Network Reliability. New York: Oxford University Press, 1987.
[21] A. E., Conway and A., Goyal, Monte Carlo simulation of computer system availability/reliability models, in Proc. of the 17th Symp. on Fault-Tolerant Computing, pages 230–235, Pittsburg, USA, July 1987.
[22] P.-J., Courtois and P., Semal, Bounds for the positive eigenvectors of nonnegative matrices and for their approximations by decomposition, J. of the Assoc. for Computing Machinery, 31(4):804–825, October 1984.
[23] P.-J., Courtois and P., Semal, Computable bounds for conditional steady-state probabilities in large Markov chains and queueing models, IEEE Trans. on Selected Areas in Commun., 4(6):926–936, September 1986.
[24] A. P., Couto da Silva and G., Rubino, Bounding the mean cumulated reward up to absorption, in A., Langville and W. J., Stewart, Eds, Markov Anniversary Meeting, chapter 22, pages 169–188. Boson Books, 2006.
[25] A., Csenki, Dependability for Systems with a Partitioned State Space. Markov and semi-Markov theory and computational implementation, volume 90 of Lecture Notes in Statistics. Springer-Verlag, 1994.
[26] J. N., Darroch and E., Seneta, On quasi-stationary distributions in absorbing discrete-time finite Markov chains, J. of Appl. Probability, 2:88–100, 1965.
[27] H. A., David, Order Statistics. New York – London - Sidney – Toronto: John Wiley & Sons, Inc., 1981.
[28] E. de Souza e, Silva and H. R., Gail, Calculating cumulative operational time distributions of repairable computer systems, IEEE Trans. on Comput., C-35:322–332, April 1986.
[29] E. de Souza e, Silva and H. R., Gail, Calculating availability and performability measures of repairable computer systems using randomization, J. of the Assoc. for Computing Machinery, 36:171–193, January 1989.
[30] E. de Souza e, Silva and H. R., Gail, An algorithm to calculate transient distributions of cumulative rate and impulse based reward, Commun. in Stat. – Stochastic Models, 14(3), 1998.
[31] P., Diaconis, The Markov chain Monte Carlo revolution, Bulletin of the American Math. Soc., 46(2):179–205, November 2008.
[32] L., Donatiello and V., Grassi, On evaluating the cumulative performance distribution of fault-tolerant computer systems, IEEE Trans. on Comput., 40:1301–1307, November 1991.
[33] L., Donatiello and B. R., Iyer, Analysis of a composite performance measure for fault-tolerant systems, J. of the Assoc. for Computing Machinery, 34:179–189, January 1987.
[34] J. B., Dugan and K. S., Trivedi, Coverage modeling for dependability analysis of fault-tolerant systems, IEEE Trans. on Comput., 38(6):775–787, June 1989.
[35] G. S., Fishman, Discrete-Event Simulation: Modeling, Programming, and Analysis. Springer Series in Operations Research and Financial Engineering. Springer, 2012.
[36] D., Furchtgott and J. F., Meyer, A performability solution method for degradable nonre-pairable systems, IEEE Trans. on Comput., C-33:550-554, July 1984.
[37] P., Glasserman, Monte Carlo Methods in Financial Engineering. Springer, 2003.
[38] P. W., Glynn and D. L., Iglehart, Simulation methods for queues: An overview, Queueing Systems, 3(3):221–255, 1988.
[39] B. V., Gnedenko, Y. K., Belyayev, and A. D., Solovyev, Mathematical Methods ofReliability. New York: Academic Press, 1969.
[40] H., Gould, J., Tobochnik, and C., Wolfgang, An Introduction to Computer Simulation Methods: Applications to Physical Systems, 3rd ed. Boston, MA, USA: Addison-Wesley Longman Publishing Co., Inc., 2005.
[41] A., Goyal, P., Shahabuddin, P., Heidelberger, V. F., Nicola, and P. W., Glynn, A uniied framework for simulating Markovian models of highly dependable systems, IEEE Trans. on Comput., 41(1):36–51, January 1992.
[42] A., Goyal and A. N., Tantawi, Evaluation of performability for degradable computer systems, IEEE Trans. on Comput., C-36:738–744, June 1987.
[43] V., Grassi, L., Donatiello, and G., Iazeolla, Performability evaluation of multi-component fault-tolerant systems, IEEE Trans. on Rel., 37(2):216–222, June 1988.
[44] W. K., Grassmann, M. I., Taksar, and D. P., Heyman, Regenerative analysis and steady-state distributions for Markov chains, Operations Research, 33(5):1107–1116, 1985.
[45] P., Heidelberger, Fast simulation of rare events in queueing and reliability models, ACM Trans. on Modeling and Comput. Simulations, 54(1):43–85, January 1995.
[46] B. R., Iyer, L., Donatiello, and P., Heidelberger, Analysis of performability for stochastic models of fault-tolerant systems, IEEETrans. on Comput., C-35:902–907, October 1986.
[47] P., Joubert, G., Rubino, and B., Sericola, Performability analysis of two approaches to fault tolerance. Technical Report 3009, INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France, July 1996.
[48] S., Karlin and H. W., Taylor, A First Course in Stochastic Processes. NewYork – San Francisco – London: Academic Press, 1975.
[49] J., Katzmann, System architecture for non-stop computing, in 14th IEEE Comp. Soc. Int. Conf, pages 77–80, 1977.
[50] J. G., Kemeny and J. L., Snell, Finite Markov Chains. New York Heidelberg Berlin: Springer-Verlag, 1976.
[51] V. G., Kulkarni, V. F., Nicola, R. M., Smith, and K. S., Trivedi, Numerical evaluation of performability and job completion time in repairable fault-tolerant systems, in Proc. IEEE 16th Fault-Tolerant Computing Symp., (FTCS'16), pages 252–257, Vienna, Austria, July 1986.
[52] S. S., Lavenberg, Computer Performance Modeling Handbook. Notes and reports in computer science and applied mathematics. Academic Press, 1983.
[53] G. Le, Lann, Algorithms for distributed data-sharing systems which use tickets, in Proc. of the 3rd Berkeley Workshop on Distributed Data Base and Comput. Networks, Berkeley, USA, 1978.
[54] P., L'Ecuyer, M., Mandjes, and B., Tufin, Importance sampling in rare event simulation, in G., Rubino and B., Tufin, Eds, Rare Event Simulation using Monte Carlo Methods, chapter 2, pages 17–38. Chichester: J. Wiley, 2009.
[55] P., L'Ecuyer, G., Rubino, S., Saggadi, and B., Tufin, Approximate zero-variance importance sampling for static network reliability estimation, IEEE Trans. on Reliability, 60(3):590–604, 2011.
[56] J., Ledoux, On weak lumpability of denumerable Markov chains. Stat. and Probability Lett., 25:329–339, 1995.
[57] J., Ledoux, A geometric invariant in weak lumpability of inite Markov chains, J. of Appl. Probability, 34(4):847–858, 1997.
[58] J., Ledoux, G., Rubino, and B., Sericola, Exact aggregation of absorbing Markov processes using the quasi-stationary distribution, J. of Appl. Probability, 31:626–634, 1994.
[59] J., Ledoux and B., Sericola, On the probability distribution of additive functionals of jump Markov processes, in Proc. of the Int. Workshop on Appl. Probability (IWAP'08), Compiégne, France, July 2008.
[60] E. E., Lewis and F., Böhm, Monte Carlo simulation of Markov unreliability models, Nucl. Eng. and Design, 77:49–62, 1984.
[61] C., Liceaga and D., Siewiorek, Automatic specification of reliability models for fault tolerant computers. Technical Report 3301, NASA, July 1993.
[62] S., Mahévas and G., Rubino, Bound computation of dependability and performance measures, IEEE Trans. on Comput., 50(5):399–413, May 2001.
[63] M., Malhotra, J. K., Muppala, and K. S., Trivedi, Stiffness-tolerant methods for transient analysis of stiff Markov chains, Microelectronics and Rel., 34(11):1825–1841, 1994.
[64] T. H., Matheiss and D. S., Rubin, A survey and comparison of methods for finding all vertices of convex polyhedral sets, Math. of Operations Research, 5(2):167–185, 1980.
[65] C. D., Meyer, Stochastic complementation, uncoupling Markov chains and the theory of nearly reducible systems, Siam Review, 31(2):240–272, 1989.
[66] J. F., Meyer, On evaluating the performability of degradable computing systems. IEEE Trans. on Comput., C-29:720–731, August 1980.
[67] J. F., Meyer, Closed-form solutions for performability, IEEE Trans. on Comput., C-31:648–657, July 1982.
[68] J., Misra, Detecting termination of distributed computations using markers, in Proc. of the 2nd Annual ACM Symp. on Principles of Distributed Computing, Montreal, Canada, 1983.
[69] R. R., Muntz, E. de Souza e, Silva, and A., Goyal, Bounding availability of repairable computer systems, IEEE Trans. on Comput. 38(12):1714–1723, December 1989.
[70] R. R., Muntz and J. C. S., Lui, Evaluating Bounds on Steady-State Availability of Repairable Systems from Markov Models, pages 435–454. Marcel Dekker, Inc., 1991.
[71] R. R., Muntz and J. C. S., Lui, Computing bounds on steady state availability of repairable computer systems, J. of Assoc. for Computing Machinery, 41(4):676–707, July 1994.
[72] R. R., Muntz and J. C. S., Lui, Bounding the response time of a minimum expected delay routing system, IEEE Trans. on Comput., 44(5):1371–1382, December 1995.
[73] J. K., Muppala and K. S., Trivedi, Numerical Transient Solution of Finite Markovian Queueing Systems, chapter 13, pages 262–284. Oxford University Press, 1992.
[74] H., Nabli and B., Sericola, Performability analysis: a new algorithm, IEEE Trans. on Comput., C-45:491–494, April 1996.
[75] H., Nabli and B., Sericola, Performability analysis for degradable computer systems, Comput. and Math. with Applicat., 39(3-4), 217–234, 2000.
[76] M. K., Nakayama, General conditions for bounded relative error in simulations of highly reliable Markovian systems, Advances in Appl. Probability, 28:687–727, 1996.
[77] V. F., Nicola, M. K., Nakayama, P., Heidelberger, and A., Goyal, Fast simulation of highly dependable systems with general failure and repair processes, IEEE Trans. on Comput., 42(12):1440–1452, December 1993.
[78] K. R., Pattipati, Y., Li, and H. A. P., Blom, A uniied framework for the preformability evaluation of fault-tolerant computer systems, IEEE Trans. on Comput., 42:312–326, March 1993.
[79] I. G., Petrovsky, Lectures on Partial Differential Equations. New York: Interscience Publishers, 1962.
[80] O., Pourret, The Slow-Fast Approximation for Markov Processes in Reliability (text in French). PhD thesis, University of Paris-Sud Orsay, 1998.
[81] J. G., Propp and D. B., Wilson, Coupling from the past, in D. A., Levin, Y., Peres, and E. L., Wilmer, Eds, Markov Chains and Mixing Times. Oxford: American Mathematical Society, 2008.
[82] M., Raynal and G., Rubino, An algorithm to detect token loss on a logical ring and to regenerate lost tokens, in Proc. of the Int. Conf. on Parallel Process. and Applicat., L'Aquila, Italy, 1987.
[83] C. P., Robert and G., Casella, Monte Carlo Statistical Methods (Springer Texts in Statistics). Secaucus, NJ, USA: Springer-Verlag New York, Inc., 2005.
[84] R. T., Rockafellar, Convex Analysis. Princeton, New Jersey: Princeton University Press, 1970.
[85] S., Ross, Stochastic Processes, 2nd ed. Wiley, April 1996.
[86] G., Rubino, Network reliability evaluation, in K., Bagchi and J., Walrand, Eds, The State-of-the art in Performance Modeling and Simulation. Gordon and Breach Books, 1998.
[87] G., Rubino and B., Sericola, Accumulated reward over the n first operational periods in fault-tolerant computing systems. Technical Report 1028, INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France, May 1989.
[88] G., Rubino and B., Sericola, Distribution of operational times in fault-tolerant systems modeled by semi-Markov reward processes, in Proc. of the 12th Int. Conf. on Fault-Tolerant Systems and Diagnostics (FTSD'12), Prague, Czech Republic, September 1989.
[89] G., Rubino and B., Sericola, On weak lumpability in Markov chains, J. of Appl. Probability, 26:446–457, 1989.
[90] G., Rubino and B., Sericola, Sojourn times in Markov processes, J. of Appl. Probability, 27:744–756, 1989.
[91] G., Rubino and B., Sericola, A inite characterization of weak lumpable Markov processes. Part I: The discrete time case, Stochastic Processes and their Applications, 38:195–204, 1991.
[92] G., Rubino and B., Sericola, Successive operational periods as dependability measures, in A., Avizienis and J. C., Laprie, Eds, Dependable Computing and Fault-Tolerant Systems, Vol. 4, pages 239–254. Springer-Verlag, 1991. (Also presented in “1st International Conference on Dependable Computing for Critical Applications”, Santa Barbara, USA, August 1989.)
[93] G., Rubino and B., Sericola, Interval availability analysis using operational periods, Performance Evaluation, 14(3-4):257–272, 1992.
[94] G., Rubino and B., Sericola, A inite characterization of weak lumpable Markov processes. Part II: The continuous time case, Stochastic Processes and their Applications, 45:115–125, 1993.
[95] G., Rubino and B., Sericola, Interval availability distribution computation, in Proc. of the 23rd IEEE Int. Symp. on Fault Tolerant Computing (FTCS'23), Toulouse, France, June 1993.
[96] G., Rubino and B., Sericola, Sojourn times in semi-Markov reward processes. Application to fault-tolerant systems modelling, Reliability Engineering and Systems Safety, 41, 1993.
[97] G., Rubino and B., Sericola, Interval availability analysis using denumerable Markov processes. Application to multiprocessor subject to breakdowns and repair, IEEE Trans. on Comput. (Special Issue on Fault-Tolerant Computing), 44(2), February 1995.
[98] G., Rubino and B., Tufin, Markovian models for dependability analysis, in G., Rubino and B., Tufin, Eds, Rare Event Simulation using Monte Carlo Methods, chapter 6, pages 125–144. Chichester: J. Wiley, 2009.
[99] G., Rubino and B., Tufin, Eds. Rare Event Simulation using Monte Carlo Methods. Chichester: J. Wiley, 2009.
[100] P., Semal, Refinable bounds for large Markov chains, IEEE Trans. on Comput., 44(10):1216–1222, October 1995.
[101] E., Seneta, Non-Negative Matrices and Markov Chains, Springer Verlag, 1981.
[102] B., Sericola, Closed-form solution for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a inite observation period, J. of Appl. Probability, 27, 1990.
[103] B., Sericola, Interval availability of 2-states systems with exponential failures and phase-type repairs, IEEE Trans. on Rel., 43(2), June 1994.
[104] B., Sericola, Occupation times in Markov processes, Communications in Statistics – Stochastic Models, 16(5), 2000.
[105] B., Sericola, Markov Chains: Theory, Algorithms and Applications. Iste Series. Wiley, 2013.
[106] P., Shahabuddin, Importance sampling for the simulation of highly reliable Markovian systems, Management Science, 40(3):333–352, March 1994.
[107] T. J., Sheskin, A Markov partitioning algorithm for computing steady-state probabilities, Operations Research, 33(1):228–235, 1985.
[108] R. M., Smith, K. S., Trivedi, and A. V., Ramesh, Performability analysis: measures, an algorithm, and a case study, IEEE Trans. on Comput., C-37:406–417, April 1988.
[109] W. J., Stewart, Probability, Markov Chains, Queues and Simulation: The Mathematical Basis of Performance Modeling. Princeton, NJ, USA: Princeton University Press, 2009.
[110] N. M., van Dijk, A simple bounding methodology for non-product form finite capacity queueing systems, in Proc. of the 1st Int. Workshop on Queueing Networks with Blocking, Raleigh, USA, May 1988.
[111] D., Williams, Probability with Martingales. Cambridge University Press, 1991.
