On the Boundary Functional of a Semi-Markov Process

Yıl 2024, Cilt: 5 Sayı: 2, 123 - 133, 31.07.2024

Elshan Ibayev

https://doi.org/10.54974/fcmathsci.1387316

Öz

In this paper, we consider the semi-Markov random walk process with negative drift, positive
jumps. An integral equation for the Laplace transform of the conditional distribution of the boundary
functional is obtained. In this work, we define the residence time of the system by generalized exponential
distributions with different parameters via fractional order integral equation. The purpose of this paper
is to reduce an integral equation for the Laplace transform of the conditional distribution of a boundary
functional of the semi-Markov random walk processes to fractional order differential equation with constant
coefficients.

Anahtar Kelimeler

Laplace transform, random variable, semi-Markov random walk process, Riemann-Liouville fractional derivative

Etik Beyan

The author declares that the materials and methods used in her study do not require ethical committee

Destekleyen Kurum

Institute of Control Systems

Teşekkür

We wish to express our thanks to Associate professor R. A Bandaliev for the formulation of the common problems in connection with fractional order differential equation

Kaynakça

• Atangana A., Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505, 688-706, 2018.
• Atangana A., Bonyah E., Fractional stochastic modeling: New approach to capture more heterogeneity, Chaos, 29, 013118, 2019.
• Abdel-Rehim E.A., Hassan R.M., El-Sayed A.M.A., On simulating the short and long memoryof ergodic Markov and non-Markov genetic diffusion processes on the long run, Chaos, 142, 110478, 2021.
• Borovkov A.A., Probability Theory, Gordon and Breach Science Publishers, 1998.
• Bandaliyev R.A., Nasirova T.H., Omarova K.K., Mathematical modeling of the semi-Markovian random walk processes with jumps and delaying screen by means of a fractional order differential equation, Mathematical Methods in the Applied Sciences, 41(18), 9301-9311, 2018.
• Cinlar E., Markov renewal theory, Advances in Applied Probability, 1(2), 123-187, 1969.
• Feller W., On semi-Markov processes, Proceedings of the National Academy of Sciences, 51(4), 653- 659, 1964.
• Grabski F., Semi-Markov Processes: Applications in Systems Reliability and Maintenance, Elsevier, 2014.
• Gikhman I.I., Skorokhod A.V., The Theory of Stochastic Processes II, Springer, 1975.
• Ibayev E.A., Laplace transform of the distribution of the first moment reaching positive level, Transactions of National Academy of Sciences of Azerbaijan. Series of Physical-Technical and Mathematical Sciences, 6(4), 115-120, 2006.
• Kazem S., Exact solution of some linear fractional differential equations by Laplace transform, International Journal of Nonlinear Science, 16(1), 3-11, 2013.
• Khaniev T.A., Ünver I., The study of the level zero crossing time of a semi-Markov random walk with delaying screen, Turkish Journal of Mathematics, 21, 257-268, 1997.
• Lebowitz J.L., Percus J.K., Asymptotic behavior of the radial distribution function, Journal of Mathematical Physics, 4(2), 248-254, 1963.
• Levy P., Proceesus Semi-Markoviens, Proceedings of the International Congress of Mathematicians, 1954.
• Limnios N., Oprisan G., Semi-Markov Processes and Reliability, Birkhauser, 2001.
• Lotov V.I., Orlova N.G., Factorization representations in the boundary crossing problems for random walks on a Markov chain, Siberian Mathematical Journal, 46(4), 661-667, 2005.
• Mainardi F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010.
• Miller K.S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, 1993.
• Nasirova T.H., Kerimova U.Y., Definition of Laplace transform of the first passage of zero level of the semi-Markov random process with positive tendency and negative jump, Applied Mathematics, 2(7), 908-911, 2011.
• Nasirova T.I., Omarova K.K., Distribution of the lower boundary functional of the step process of semi- Markov random walk with delaying screen at zero, Ukrainian Mathematical Journal, 59(7), 1010-1018, 2007.
• Pyke R., Markov renewal processes: Definitions and preliminary properties, Annals of Mathematical Statistics, 32(4), 1231-1242, 1961.
• Smith W.L., Regenerative stochastic processes, Proceedings of the Royal Society Series A, 232(1188), 6-31, 1955.
• Tak’acs L., Some investigations concerning recurrent stochastic processes of a certain type, Maygyar Tud Akad Matematikai Kutat ´o Int ´ezet ´enek Közlem ´enyei, 3, 115-128, 1954.