Computation of the steady-state probability of Markov chain evolving on a mixed state space

References

[1] J. Aitchison, On the distribution of a positive random variable having a discrete probability mass at the origin, J. Amer. Statist. Assoc. 50 (1955), 901–908. 10.1080/01621459.1955.10501976Search in Google Scholar

[2] D. R. Barr and M. U. Thomas, An eigenvector condition for Markov chain lumpability, Oper. Res. 25 (1977), no. 6, 1028–1031. 10.1287/opre.25.6.1028Search in Google Scholar

[3] R. Besenczi, N. Bátfai, P. Jeszenszky, R. Major, F. Monori and M. Ispány, Large-scale simulation of traffic flow using Markov model, PLoS ONE 16 (2021), no. 2, Article ID e0246062. 10.1371/journal.pone.0246062Search in Google Scholar PubMed PubMed Central

[4] G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi, Queueing Networks and Markov Chains Modelling and Performance Evaluation with Computer Science Applications, John Wiley & Sons, Hoboken, 2006. 10.1002/0471791571Search in Google Scholar

[5] Y. Cai, A non-monotone CFTP perfect simulation method, Statist. Sinica 15 (2005), no. 4, 927–943. Search in Google Scholar

[6] R. W. Cottle, Manifestations of the Schur complement, Linear Algebra Appl. 8 (1974), 189–211. 10.1016/0024-3795(74)90066-4Search in Google Scholar

[7] L. Dai, Sensitivity analysis of stationary performance measures for Markov chains, Math. Comput. Model. 23 (1996), 143–160. 10.1016/0895-7177(96)00069-6Search in Google Scholar

[8] J. P. Demailly, Analyse numérique et equations différentielles, 3rd. ed, EPP Sciences, Les Ulis, 2006. Search in Google Scholar

[9] H. Fakhouri and A. Nasroallah, On the simulation of Markov chain steady-state distribution using CFTP algorithm, Monte Carlo Methods Appl. 15 (2009), no. 2, 91–105. 10.1515/MCMA.2009.005Search in Google Scholar

[10] A. Feuerverger, On some methods of analysis for weather experiments, Biometrika 66 (1979), no. 3, 655–658. 10.1093/biomet/66.3.655Search in Google Scholar

[11] S. G. Foss and R. L. Tweedie, Perfect simulation and backward coupling, Stoch. Models 14 (1998), 187–203. 10.1080/15326349808807466Search in Google Scholar

[12] P. A. Gagniuc, Markov Chains: From Theory to Implementation and Experimentation, John Wiley & Sons, Hoboken, 2017. 10.1002/9781119387596Search in Google Scholar

[13] R. G. Gallager, Stochastic Processes: Theory for Applications, Cambridge University, Cambridge, 2013. 10.1017/CBO9781139626514Search in Google Scholar

[14] H. D. Herr and R. Krzysztofowicz, Generic probability distribution of rainfall in space: The bivariate model, J. Hydrology 306 (2005), no. 1–4, 234–263. 10.1016/j.jhydrol.2004.09.011Search in Google Scholar

[15] F. S. Hillier and R. W. Boling, Finite queues in series with exponential or Erlang services - A numerical approach, Oper. Res. 15 (1967), no. 2, 286–303. 10.1287/opre.15.2.286Search in Google Scholar

[16] R. J. Hyndman and G. K. Grunwald, Applications: Generalized additive modelling of mixed distribution Markov models with application to Melbourne’s rainfall. Australian and New Zealand, J. Stat. 42 (2000), no. 2, 145–158. 10.1111/1467-842X.00115Search in Google Scholar

[17] J. G. Kemeny and J. L. Snell, Finite Markov Chains, D. Van Nostrand, Princeton, 1960. Search in Google Scholar

[18] D. P. Kroese and R. Y. Rubinstein, Monte carlo methods, Comput. Statist. 4 (2012), no. 1, 48–58. 10.1002/wics.194Search in Google Scholar

[19] C. Li, V. P. Singh and A. K. Mishra, A bivariate mixed distribution with a heavy-tailed component and its application to single-site daily rainfall simulation, Water Resources Res. 49 (2013), no. 2, 767–789. 10.1002/wrcr.20063Search in Google Scholar

[20] S. Mahévas and G. Rubino, Bound computation of dependability and performance measures, IEEE Trans. Comput. 50 (2001), no. 5, 399–413. 10.1109/12.926156Search in Google Scholar

[21] J. Møller, Perfect simulation of conditionally specified models, J. R. Stat. Soc. Ser. B Stat. Methodol. 61 (1999), no. 1, 251–264. 10.1111/1467-9868.00175Search in Google Scholar

[22] D. J. Murdoch and P. J. Green, Exact sampling from a continuous state space, Scand. J. Statist. 25 (1998), no. 3, 483–502. 10.1111/1467-9469.00116Search in Google Scholar

[23] D. J. Murdoch and J. S. Rosenthal, Efficient use of exact samples, Stat. Comput. 10 (2000), no. 3, 237–243. 10.1023/A:1008991527785Search in Google Scholar

[24] J. A. Ogola, Numerical solutions of Fredholm integral equations of the second kind, Research Report in Mathematics No. 30, University of Nairobi, 2020. Search in Google Scholar

[25] D. P. O’Leary and Y.-J. J. Wu, A block-GTH algorithm for finding the stationary vector of a Markov chain, SIAM J. Matrix Anal. Appl. 17 (1996), no. 3, 470–488. 10.1137/S0895479894262534Search in Google Scholar

[26] W. J. Owen and T. A. DeRouen, Estimation of the mean for lognormal data containing zeroes and left-censored values, with applications to the measurement of worker exposure to air contaminants, Biometrics 36 (1980), no. 4, 707–719. 10.2307/2556125Search in Google Scholar

[27] A. Patra and G. K. Panda, Appearance of balancing and related number sequences in steady state probabilities of some Markov chains, Integers 20A (2020), Paper No. A11. Search in Google Scholar

[28] J. G. Propp and D. B. Wilson, Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures Algorithms 9 (1996), no. 1–2, 223–252. 10.1002/(SICI)1098-2418(199608/09)9:1/2<223::AID-RSA14>3.0.CO;2-OSearch in Google Scholar

[29] J. G. Propp and D. B. Wilson, How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph, J. Algorithms 27 (1998), 170–217. 10.1006/jagm.1997.0917Search in Google Scholar

[30] L. Quan-Lin, Markov chains on continuous state space, Constructive Computation in Stochastic Models with Applications, Springer, Berlin (2010), 216–287. 10.1007/978-3-642-11492-2_5Search in Google Scholar

[31] J. L. Romeu, A Markov chain model for Covid-19 survival analysis, Ph.D. thesis, 2020. Search in Google Scholar

[32] J. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. Reine Angew. Math. 148 (1918), 122–145. 10.1515/crll.1918.148.122Search in Google Scholar

[33] B. Sericola, Discrete-time Markov chains, Markov Chains, Wiley-Blackwell, New York (2013), 1–87. 10.1002/9781118731543.ch1Search in Google Scholar

[34] F. Serinaldi, A multisite daily rainfall generator driven by bivariate copula-based mixed distributions, J. Geophys. Res. Atmospheres 114 (2009), Paper No. D10. 10.1029/2008JD011258Search in Google Scholar

[35] S. P. Sharma and Y. Vishwakarma, Application of Markov process in performance analysis of feeding system of sugar industry, J. Indust. Math. 2014 (2014), Article ID 593176. 10.1155/2014/593176Search in Google Scholar

[36] T. J. Sheskin, Allocation of interstage storage along an automatic production line, AIIE Trans. 8 (1976), no. 1, 146–152. Search in Google Scholar

[37] T. J. Sheskin, A Markov chain partitioning algorithm for computing steady state probabilities, Oper. Res. 33 (1985), no. 1, 228–235. 10.1287/opre.33.1.228Search in Google Scholar

[38] K. Shimizu, A bivariate mixed log normal distribution with an analysis of rainfall data, J. Appl. Meteorol. 32 (1993), no. 2, 161–171. 10.1175/1520-0450(1993)032<0161:ABMLDW>2.0.CO;2Search in Google Scholar

[39] T.-K. Siu, W.-K. Ching, E. S. Fung and M. K. Ng, On a multivariate Markov chain model for credit risk measurement, Quant. Finance 5 (2005), no. 6, 543–556. 10.1080/14697680500383714Search in Google Scholar

[40] D. Spieler and V. Wolf, Efficient steady state analysis of multimodal Markov chains, Analytical and Stochastic Modeling Techniques and Applications, Springer, Berlin (2013), 380–395. 10.1007/978-3-642-39408-9_27Search in Google Scholar

[41] J. van der Hoek and R. J. Elliott, Asset pricing using finite state Markov chain stochastic discount functions, Stoch. Anal. Appl. 30 (2012), no. 5, 865–894. 10.1080/07362994.2012.704852Search in Google Scholar

[42] C. Yoo, K.-S. Jung and T.-W. Kim, Rainfall frequency analysis using a mixed gamma distribution: Evaluation of the global warming effect on daily rainfall, Hydrol. Proc. 19 (2005), no. 19, 3851–3861. 10.1002/hyp.5985Search in Google Scholar

[43] A.-E. Zakrad and A. Nasroallah, Perfect simulation of steady-state Markov chain on mixed state space, Comm. Statist. Theory Methods 51 (2022), no. 6, 1569–1587. 10.1080/03610926.2021.1924783Search in Google Scholar

[44] A. E. Zakrad and A. Nasroallah, General partitioning algorithm for computing Markov chain steady state probability, Monte Carlo Methods Appl., to appear. Search in Google Scholar

[45] F. Zhang, Matrix Theory: Basic Results and Techniques, Springer, New York, 2011. 10.1007/978-1-4614-1099-7Search in Google Scholar