Bektaş Kamışlık, Alakoç, Kesemen, Khaniyev JTOM(7)1, 1483-1492, 2023 1483 Journal of Turkish Operations Management Investigation the ergodic distribution of a semi-Markovian inventory model of type (s,S) with intuitive approximation approach Aslı Bektaş Kamışlık 1 , Büşra Alakoç 2 , Tülay Kesemen 3 , Tahir Khaniyev 4,5* 1 Department of Mathematics, Recep Tayyip Erdoğan University, Rize, Turkey e-mail:asli.bektas@erdogan.edu.tr, ORCID No: https://orcid.org/0000-0002-9776-2145 2 Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey e-mail: busraalakoc@gmail.com, ORCID No: https://orcid.org/0000-0001-8975-5968 3 Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey e-mail:tkesemen@ktu.edu.tr, ORCID No: https://orcid.org/0000-0002-8807-5677 4 Department of Industrial Engineering, TOBB University of Economics and Technology, Ankara,Turkey 5 The Center of Digital Economics, Azerbaijan State University of Economics, Baku, Azerbaijan. e-mail: tahirkhaniyev@etu.edu.tr, ORCID No: http://orcid.org/0000-0003-1974-0140 * Corresponding author: Tahir Khaniyev Article Info Abstract Article History: Received: 29.12.2022 Revised: 01.02.2023 Accepted: 08.02.2023 Keywords: Intuitive approximation, Inventory model of type (s, S), Renewal reward process, Ergodic distribution, Γ() class of distributions 1. Introduction In areas like stock control, queuing, stochastic finance, and reliability, semi-Markov processes have been applied, along with renewal theory. Through the use of these processes, many important stock control problems are also expressed (see, Smith, 1959; Brown and Solomon, 1975; Gikhman and Skorohod, 1975; Borovkov, 1984; Chen and Zheng, 1997; Csenki, 2000; Asmussen, 2000). For instance, it is typical to assume that the demand sequences creates a renewal process when analyzing inventory processes. Also, common inventory systems use renewal sequences to express stock replenishment times. The stochastic control model of type (s, S) is one of the inventory models where in the formula of the ergodic distribution the renewal function is encountered. There has been a substantial amount of research in the literature on the stationary characteristics of (s, S) type inventory models with different modifications. These modifications are mostly given with some kinds of discrete interference of chances or by using different distributions for inter-arrival times (see, Khaniyev et. al., 2011; Khaniyev et. al., 2013; Kesemen et. al., 2016; Hanalioglu and Khaniyev, 2019). In recent years, these studies have moved towards examining (s, S) type inventory models with heavy-tailed distributions and the literature on this subject has been enriched considerably (see, Aliyev, 2017; Bektaş et. al., 2018; Bektaş et. al., 2019). But some important distributions for example the gamma distribution and the exponential distribution are excluded from the heavy- tailed situation. The () class is encountered in many applications, especially in extreme value theory. For more This paper concerns a stochastic process () expressing (, ) type inventory system with intuitive approximation approach. The ergodic distributions of the process () can be analyzed with the help of the renewal function. Obtaining an explicit formula for renewal function () 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 (). Obtained results can be used in many situations where demand random variables have different distributions from different classes such as the () class.