Determination of the laplace transform for the first falling moment to zero level of a semi-Markov random process

Abstract

One of the important problems of stochastic process theory is to define the Laplace transformations for the distribution of this process. With this purpose, we will investigate a semi-Markov random processes with positive tendency and negative jump in this article. The first falling moment into the zero-level of this process is constructed as mathematically and the Laplace transformation of this random variable is obtained.

Keywords

• Semi-Markov process,Laplace transform

References

1. Afanas’eva, L. G., Bulinskaya, E. V., ’Some asymptotical results for random walks in a strip’ Teor. Veroyatn. Primen. 29, 4, 658-668 (1983).
2. Borovkov, A. A., Stochastic Processes in The Theory of Queues, Nauka, Moscow 1972.
3. Borovkov A. A., ’On the asymptotic behaviour of the distributions of the first passage’, Mat. Zametki, Vol.75, No.1, pp.24-39, (2004).
4. Feller, W., An Introduction to Probability Theory and Its Applications, Vol. I, Wiley, New York 1968.
5. Gihman, I. I., Skorohod, A. V., The Theory of Stochastic Processes II, Springer-Verlag, 1975.
6. Lotov, V. I., On some boundary crossing problems for gaussian random walks, The Annals of Probab., 24, 1996, pp.2154-2171, (1996).
7. Lotov, V. I., ’On the asymptotics of distributions in two-sided boundary problems for random walks defined on a Markov chain’, Sib. Adv. Math. 1, 3, 26-51 (1991).
8. Nasirova, T. H, Karimova ,U. Y., ’Definition of Laplace transform of the first passage of zero level of the semimarkov random process with positive tendency and negative jump2, Applied mathematics, 2, pp. 908-912, (2011).

9. Nasirova, T. I., and Shamilova, B. G., ’Investigation of Some Probabilistic Characteristics of One Class of Semi Markov Wandering with Delaying Screens’, Automatic Control and Computer Sciences, Vol. 48, No. 2, 109-119 (2014).
10. Nasirova, T. I., Sadikova, R. I., and Ibaev, E. A., ’Determination of the Mean and Mean Square Deviations of the System Level’, Automatic Control and Computer Sciences, Vol. 49, No. 1, 37-45 (2015).
11. Omarova, K. K., Bakhshiev, S. B., ’The Laplace Transform for the Distribution of the Lower Bound Functional in a Semi Markov Walk Process with a Delay Screen at Zero’, Automatic Control and Computer Sciences, Vol. 44, No. 4, 246-252 (2010).
12. Omarova, K. K., ’Laplace transformation of ergodic distribution of the step process of semi-markov random walk with delaying screen at positive point’, The Third International Conference "Problems of Cybernetics and Informatics", (2010).
13. Prabhu, N. U., Stochastic Storage Processes, New York, Springer-Verlag, 1981.
14. Spitzer, F., Principles of Random Walk, Van Nostrand, Princeton, 1969.