Abstract
A survey of the main results of algorithmic studies of Markov processes and related stochastic models is shown. It consists of stationary solutions, transient solutions, first passage times, and multidimensional denumerable state Markov processes. In conclusion, some remarks on further works are presented.
Similar content being viewed by others
References
Tijms, H. C.,Stochastic Models: An Algorithmic Approach, New York: Wiley, 1994.
Grassmann, W. K., Computational methods in probability theory, inHandbooks in Operations Research and Management Science, Vo1..2,Stochastic Models, Amsterdam: North-Holland, 1990, 199–254.
Hsu, G. H.,Stochastic Service Systems (in Chinese), 2nd ed., Beijing: Science Press, 1988.
Neuts, M. F.,Matrix-Geometric Solutions in Stochastic Models, Baltimore: Johns Hopkins Univ. Press, 1981.
Neuts, M. F.,Structured Stochastic Matrices of M/ G/l Type and Their Applications, New York: Dekker, 1989.
Grassmann, W. K., Heyman, D. P., Computation of steady-state probabilities for infinite Markov chains with repeating rows,ORSA J. Comput., 1993, 5: 292.
Heyman, D. P., A decomposition theorem for infinite stochastic matrices,J. Appl. Prob., 1995, 32: 893.
Bailey, N. T. J., A continuous time treatment of a simple queue using generating functions,J. Roy. Statist. Soc., 1954, B16: 288.
Conolly, B. W., A difference equation technique applied to the simple quene,J. Roy. Statist. Soc., 1958, B20: 165.
Cox, D. R., Smith, W. L.,Queues, London: Methuen & Co. Ltd., 1961.
Karlin, S., McGregor, J., Many-server queueing process with Poisson input and exponential senrice times,Pacific J. Math., 1958, 8: 87.
Yue, M. I., On the problemM/ M/s in the theory of queues,Acta Math. Sinica (in Chinese), 1959, 9: 494.
Saaty, T. L., Time-dependent solution of the many-server Poission queue,Opns. Res., 1960, 8: 755.
Takacs, L., The transient behavior of a single server queueing process with a Poisson input, inProc. Fourth Berkeley Symp. on Math. Statist. and Prob., Berkeley: University of California Press, 1961, 2: 535.
Conolly, B. W., A difference equation technique applied to the simple queue with arbitrary arrival interval distribution,J. Roy. Statist. Soc., 1958, B20: 168.
Takacs, L., Transient behavior of single-server queueing processes with recurrent input and exponential distributed service times,Opns. Res., 1960, 8: 231.
Wu, F., On theGI/M/n queueing process,Acta Math. Sinica (in Chinese), 1961, 11: 295.
Hsu, G. H., The transient behavior of the queueing processGI/M/n, Chinese Math., 1965, 6: 393.
Bhat, U. N., Transient behavior of multiserver queues with recurrent input and exponential service times,J. Appl. Prob., 1968, 5: 158.
de Smit, J. H. A., On the many-server queue with exponential service times,Adv. Appl. Prob., 1973, 5: 170.
Grassmann, W. K., Transient solutions in Markovian queueing systems,Comput. Opns. Res., 1977, 4: 47.
Kohlas, J.,Stochastic Methods of Operations Research, London: Cambridge University Press, 1982.
Gross, D., Miller, D. R., The randomization technique as a modelling tool and solution procedure for transient Markov processes,Opns. Res., 1984, 32: 343.
Reibman, A., Trivedi, K., Numerical transient analysis of Markov models,Comput. Opns. Res., 1988, 15: 19.
Grassmann, W. K., Transient solutions in Markovian queues,Eur. J. Opns. Res., 1977, 1: 396.
Zhang, J., Coyle, E. J., Transient analysis of quasi-birth-death processes,Stoch Models, 1989, 5: 459.
Hsu, G. H., Yuan, X. M., Transient solutions for denumerable-state Markov processes,J. Appl. Prob., 1994, 31: 635.
Ramaswami, V., The busy period of queues which have a matrix-geometric steady state probability vector,Opsearch, 1982, 9: 238.
Hsu, G. H., He, Q. M., The distribution of the first passage time for the Markov processes ofGI/M/1 type,Stoch. Models, 1991, 7: 397.
Melamed, B., Yadin, M., Randomization procedures in the computation of comulative-time distributions over discrete state Markov processes,Opns. Res., 1984, 32: 926.
Melamed, B., Yadin, M., Numerical computation of sojourn-time distributions in queueing networks,JACM, 1984, 31: 839.
Hsu, G. H., Yuan X. M., The first passage times and their algorithms for Makrov processes,Stoch. Models, 1995, 11: 195.
Hsu, G. H., Yuan X. M., First passage times: busy periods and waiting times,Science in China, Ser. A, 1995, 38: 1187.
Latouche, G., Ramaswami, V., A logarithmic reduction algorithm for quasi-birth-death processes,J. Appl. Prob., 1993, 30: 650.
Ramaswami, V., Taylor, P. G., Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases,Stoch. Models, 1996, 12: 143.
Hsu, G. H., He, Q. M., Liu, X. S., The matched queueing system with a double input,Acta Math. Appl. Sinica, 1990, 13: 39.
Hsu, G. H., Xu, D. J., Transient solutions for multidimensional denumerable state Markov processes.,Queueing Syst., 1996, 23: 317.
Hsu, G. H., Xu, D. J., First passage times for two-deimensional denumerable state Markov processes,Chinese Sci. Bulletin (in Chinese), 1997, 42(1): 9.
Hsu, G. H., Xu, D. J., First passage times for multidimensional denumerable state Markov processes,Chinese Science Bulletin (to appear).
Hsu, G. H. Jensen, U., The matched queueing networkPH/M/c→∘PH/PH/1, Queueing System, 1993, 13: 315.
Author information
Authors and Affiliations
About this article
Cite this article
Xu, G. Algorithms for Markov stochastic models. Chin. Sci. Bull. 44, 97–100 (1999). https://doi.org/10.1007/BF02884726
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02884726