No abstract

In this study, a stochastic process X(t), which describes an inventory model of type (s,S) is considered in the presence of heavy tailed demands with infinite variance. The aim of this study is observing the impact of regularly varying demand distributions with infinite variance on the stochastic process X(t). The main motivation of this work is, the publication by Geluk (1997) where he provided a special asymptotic expansion for renewal function generated by regularly varying random variables. Two term asymptotic expansion for the ergodic distribution function of the process X(t) is obtained based on the main results proposed by Geluk (1997). Finally weak convergence theorem for the ergodic distribution of this process is proved by using Karamata theory.MSC 2010 subject classifications: Primary 60K05; secondary 60K20

We consider a classical semi-Markovian stochastic model of type (s, S) with Logistic distributed demand random variables. Logistic distribution is a member of special distribution class known as Γ(g) that encounters in many real-life applications involving extreme value theory. The objective of this study is to observe some major characteristics of a stochastic process X(t) which represents semi-Markovian renewal reward process of type (s, S) .We used new approximation results for renewal function that allow us to obtain three-term asymptotic expansion for ergodic distribution function and for n th order moments of ergodic distribution of the process X(t) .

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.

Bu çalışmada üçgensel müdahaleli (s,S) tipli yarı-Markov bir envanter(stok kontrol) model yenileme süreci ele alınmış, ve bu sürecin ergodik dağılımının momentleri için tahmin edici problemi araştırılmıştır. Yenileme süreçlerinde, yenilemeler arasında geçen sürenin dağılımı tam olarak bilinmediğinde yenileme fonksiyonu için tahmin ediciler literatürde mevcuttur. Bu çalışmanın ana motivasyonu ise, Frees (1986b) tarafından yenileme fonksiyonu için önerilmiş tahmin edicidir. Bu amaçla Frees (1986b)'in yaklaşımı kullanılarak ele alınan modelin ergodik dağılımının momentleri için istatistiksel bir tahmin edici bulunmuş ve bu tahmin edici için sırasıyla tutarlılık, asimptotik yansızlık ve asimptotik normallik gibi istatistiksel özellikler araştırılmıştır.