NONE 20260723124853068 wseas/wseas content-registration-form@crossref.org content-registration-form International Journal of Applied Mathematics Computational Science and Systems Engineering 2766-9823 U. Kerimova Azerbaijan Technical University, AZERBAIJAN V. Poladova Institute of Radiation Problems, AZERBAIJAN In recent years, semi-Markov random walks have been employed to address numerous intriguing problems across fields such as inventory management, risk management, insurance, and reliability theories. Particularly, many significant problems in stock control, queue management, and reliability theory can be represented using random walks constrained by two barriers. These barriers, depending on the specific problem, may be reflecting, delaying, absorbing, elastic, or of other types. A key problem in stochastic process theory is determining the Laplace transforms for the distribution of semi-Markov random processes. In this context, this article explores semi-Markov random processes characterized by a positive tendency and negative jumps. 07 23 2026 07 23 2026 141 14 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/232026.2026.8.14 https://wseas.com/journals/amcse/2026/a28amcse-014(2026).pdf Levy P. Proceesus semi-markoviens. Proceedings of the International Congress of Mathematicians. Amsterdam. 1954; 3: 416- 426. Smith W.L. Regenerative stochastic processes. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 1955; 232(1188): 6-31. Takacs L. Some investigations concerning recurrent stochastic processes of a certain type. Magyar Tudom_anyos Akad_emia Matematikai Kutat_o Int_ezet. 1954; 3: 115- 128. Pyke R. Markov renewal processes: de_nitions and preliminary properties. The Annals of Mathematical Statistics. 1961; 32(4): 1231-1242. Pyke R. Markov renewal processes with _nitely many states. The Annals of Mathematical Statistics. 1961; 32(4): 1243-1259. Pyke R., Schaufele R. Limit theorems for Markov renewal processes. The Annals of Mathematical Statistics. 1964; 35(4):1746- 1764. Çinlar E. Markov renewal theory. Advances in Applied Probability. 1969; 1 (2): 123-187. Korolyuk V. S., Turbin A. F. Semi-Markov Processes and Their Applications. Naukova Dumka: Kiev, 1976.(In Russian) Limnios N., Oprisan G. Semi-Markov Processes and Reliability. Birkhauser: Boston, 2001. Lotov, V. I., On random walks in a band. Probability Theory and its Application, 1991 Silvestrov D. C. Semi-Markov Processes With a Discrete State Space. Sovetskoe Radio: Moscow 1980. (in Russian). Grabski F. Semi-Markov Processes: Applications in Systems Reliability and Main- tenance. Elsevier: Amsterdam, 2014. Harlamov B.P. Continuous SemiMarkov Processes. Nauka: St. Petersburg, 2001. (In Russian). Borovkov A.A. Probability Theory. Gordon and Breach Science Publishers: Amsterdam, 1998. Busarov, V. A. On asymptotic behavior of random wanderings in random medium with delaying screen, Vest. Mos. Gos. Univ., 2004 Nasirova T.I. The Processes of SemiMarkov Random Walk. Elm: Baku, 1984. (In Russian). Nasirova T.I. Complex Semi-Markov Walk Processes With Delaying Screen. Elm: Baku, 1988. (In Russian). 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., Kerimova. U.Y. 2011. Definition of Laplace -Stieltjes transforms for distribution of the first passage of zero level of the semi-Markov random process with positive tendency, negative jump, Applied Mathematics, 2(1): 908-911. Nasirova, T. I., Shamilova, B. G., Kerimova. U.Y. 2011. Definition of Laplace transforms for ergodic distribution of the semi-Markov random process, SYLWAN, 158(6): 50-58. 69 Nasirova, T. I., Kerimova. U.Y., Shamilova, B. G. 2014. Laplace transformation of the time of system sojourns within a band, Mathematica Aeterna, 4(5): 437 444.