Investigation the ergodic distribution of a semi-Markovian inventory model of type (s,S) with intuitive approximation approach

Year 2023, Volume: 7 Issue: 1, 1483 - 1492, 30.06.2023

Aslı Bektaş Kamışlık , Büşra Alakoç , Tulay Kesemen , Tahir Khanıyev

https://doi.org/10.56554/jtom.1226349

Abstract

This paper concerns a stochastic process expressing (s,S) type inventory system with intuitive approximation approach. The stock level in the system is modeled as a semi-Markovian renewal reward process X(t). Therefore, the ergodic distributions of this process can be analyzed with the help of the renewal function. Obtaining explicit formula for renewal function U(x) is difficult from a practical standpoint. Mitov and Omey recently present some intuitive approximations in literature for renewal function which cover a large number of existing results. Using their approach we were able to establish asymptotic approximations for ergodic distribution of a stochastic process X(t). Obtained results can be used in many situations where demand random variables have different distributions from different classes such as Γ(g) class.

Keywords

Intuitive approximation, Renewal reward process, Ergodic distribution, Γ(g) class of distributions, inventory model of type (s,S)

References

• Aliyev, R.T. (2017). On a stochastic process with a heavy tailed distributed component describing inventory model of type (s,S). Communications in Statistics-Theory and Methods, 46(5), 2571-2579. DOI:https://doi.org/10.1080/03610926.2014.1002932
• Asmussen, S. (2000). Ruin probabilities, World Scientific Publishing. Singapore.
• Bektaş, K.A., Alakoç, B., Kesemen T. and Khaniyev T. (2020). A semi-Markovian renewal reward process with Γ(g) distributed demand. Turkish Journal of Mathematics, 44, 1250-1262. DOI: https://doi.org/10.3906/mat-2002-72
• Bektaş K.A., Kesemen T. and Khaniyev T. (2019). Inventory model of type (s,S) under heavy tailed demand with infinite variance. Brazilian Journal of Probability and Statistics, 33 (1), 39-56. DOI: https://doi.org/10.1214/17-BJPS376
• Bektaş, K.A., Kesemen, T. and Khaniyev, T. (2018). On the moments for ergodic distribution of an inventory mo- del of type (s,S) with regularly varying demands having infinite variance. TWMS Journal of Applied and Enginee- ring Mathematics, 8 (1a), 318-329.Available at: http://jaem.isikun.edu.tr/web/index.php/archive/98-vol8no1a/349
• Borovkov, A.A.,(1984) Asymptotic Methods in Queuing Theory, John Wiley, New York.
• Brown, M. and Solomon, H.A. (1975). Second order approximation for the variance of a renewal reward process and their applications. Stochastic Processes and their Applications 3, 301-314. DOI: https://doi.org/10.1016/03044149(75)90029-0
• Chen, F. and Zheng, Y.S. (1997). Sensitivity analysis of an (s,S) inventory model. Operation Research and Letters, 21, 19-23. DOI: https://doi.org/10.1016/S0167-6377(97)00019-9
• Csenki, A. (2000). Asymptotics for renewal-reward processes with retrospective reward structure. Operation Research and Letters, 26, 201-209. DOI: https://doi.org/10.1016/S0167-6377(00)00035-3
• Feller, W. (1971). Introduction to probability theory and its applications II, John Wiley, New York.
• Geluk, J. L., de Haan, L. (1981). Regular variation, extensions and Tauberian theorems, CWI Track 40 Amsterdam.
• Geluk, J.L. (1997). A Renewal Theorem in the finite-mean case. Proceedings of the American Mathematical Society, 125(11), 3407-3413. DOI: http://www.jstor.org/stable/2162415.
• Gikhman, I. I. and Skorohod, A. V. (1975). Theory of Stochastic Processes II. Berlin: Springer.
• Hanalioglu, Z., Khaniyev, T. (2019). Limit theorem for a semi - Markovian stochastic model of type (s,S). Hacettepe Journal of Mathematics and Statistics, 48(2), 605-615. DOI: https://doi.org/10.15672/HJMS.2018.622
• Kesemen, T., Bektaş K.A., Küçük, Z. and Şenol E. (2016). Inventory model of type (s,S) with sub-exponential Weibull distributed demand. Journal of the Turkish Statistical Association, 9(3), 81-92. DOI: https://dergipark.org.tr/en/pub/ijtsa/issue/40955/494648
• Khaniyev, T. and Aksop, C. (2011). Asymptotic results for an inventory model of type (s,S) with generalized beta interference of chance. TWMS J.App.Eng.Math., 2, 223-236. Available at: https://dergipark.org.tr/en/pub/twmsjaem/issue/55716/761800#article_cite
• Khaniyev, T., Kokangul, A. and Aliyev, R. (2013). An asymptotic approach for a semi-Markovian inventory model of type (s,S). Applied Stochastic Models in Business and Industry, 29:5, 439-453. https://doi.org/10.1002/asmb.1918
• Levy, J.B. and Taqqu, M.S. (2000). Renewal reward processes with heavy-tailed inter-renewal times and heavy tailed rewards. Bernoulli, 6, 23-44. https://doi.org/10.2307/3318631.
• Mitov K.V. and Omey E. (2014). Intuitive approximations for the renewal function. Statistics and Probability Let- ters, 84, 72-80. https://doi.org/10.1016/j.spl.2013.09.030
• Nasirova, T. I., Yapar, C., & Khaniyev, T. A. (1998). On the probability characteristics of the stock level in the model of type (s, S). Cybernetics and System Analysis, 5, 69-76.
• Smith, W. L. (1959). On the cumulants of renewal process, Biometrika, 46, 1–29. https://doi.org/10.2307/2332804