An asymptotic approach for a semi‐Markovian inventory model of type (s, S)

Abstract

In this study, we constructed a stochastic process (X.t// that expresses a semi-Markovian inventory model of type (s,S/ and it is shown that this process is ergodic under some weak conditions. Moreover, we obtained exact and asymptotic expressions for the n th order moments .n D 1; 2; 3; : : :/ of ergodic distribution of the process X.t/, as S s ! 1. Finally, we tested how close the obtained approximation formulas are to the exact expressions.

A novel replacement policy for a linear deteriorating system using stochastic process with dependent components

“…Many interesting real‐life problems of stock control, inventory, queuing theory, mathematical insurance, and financial mathematics can be expressed by random walk, renewal, renewal‐reward processes, and various modifications of these processes . They can also be used to model and solve many problems regarding reliability.…”

Limit Theorem for a Semi - Markovian Stochastic Model of Type (s,S)

“…A number of very interesting problems arising in the theories of inventory, stock control, queuing theory, reliability, mathematical insurance, stochastic finance etc., can be expressed by means of renewal processes, renewal-reward processes, random walk processes and their modifications. There are many important theoretical results about these subjects in literature (Borovkov (1984), Brown and Solomon (1975), Feller (1971), Gihman and Skorohod (1975), Janssen and Leuwaarden (2007), Khaniyev et al (2008Khaniyev et al ( , 2013, Lotov (1996), Rogozin (1964), etc.). The results of these studies have complex mathematical structures and they are not useful for applied problems.…”

“…To avoid this difficulty, in recent years, the asymptotic methods have started to be applied to these problems. There are several valuable studies about using asymptotic methods, as well (e.g., Janssen and Leuwaarden (2007), Lotov (1996), Khaniyev (2005), Khaniyev and Atalay (2010), Khaniyev and Mammadova (2006), Khaniyev et al (2008Khaniyev et al ( , 2013).…”

“…Lately, the inventory models of type (s,S) have been extensively considered and some of their characteristics have been investigated in the literature (see, e.g., Khaniyev and Atalay (2010), Khaniyev and Mammadova (2006), Khaniyev et al (2013)). Especially, in the studies Khaniyev and Atalay (2010) and Khaniyev et al (2013), an inventory model of type (s,S) with triangular distributed interference of chance is tackled. In Khaniyev and Atalay (2010), the weak convergence theorem for the considered process is proved and in Khaniyev et al (2013), three-term asymptotic expansions for the moments of ergodic distribution are obtained.…”

“…Especially, in the studies Khaniyev and Atalay (2010) and Khaniyev et al (2013), an inventory model of type (s,S) with triangular distributed interference of chance is tackled. In Khaniyev and Atalay (2010), the weak convergence theorem for the considered process is proved and in Khaniyev et al (2013), three-term asymptotic expansions for the moments of ergodic distribution are obtained. These results are not only remarkable from the theoretical point of view but are also very useful in the application.…”