Probability characteristics of the inventory level in (s, S) Model

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) .

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“…see Mitov and Omey [18] page (77). Ergodicity of the process X(t) is given by Nasirova et al [19] with the following proposition.…”

Section: Mathematical Construction and Ergodicity Of The Process X(t)mentioning

confidence: 99%

A semi-Markovian renewal reward process withΓ(g)distributed demand

Kamişlik¹,

Alakoç²,

Kesemen³

et al. 2020

Turk J Math

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“…are given by means of random walk with various type of barriers. There exist some important studies on this topic in the literature (see, for example, Alsmeyer, 1991;Aras and Woodroofe, 1993;Borovkov, 1976;Brown and Solomon, 1975;Khaniev et al, 1998Khaniev et al, , 2001Khaniyev et al, 2008;Khaniyev and Kucuk, 2004;Khaniyev and Mammadova, 2006;Lotov, 1996;Nasirova et al, 1998;Rogozin, 1964;Skorohod and Slobodenyuk, 1970;Spitzer, 1964, etc.). In these studies, absorbing, delaying, reflecting, and other known barriers are usually used.…”

Section: Introductionmentioning

confidence: 97%

Asymptotic Expansions for the Moments of the Semi-Markovian Random Walk with Gamma Distributed Interference of Chance

Aliyev

Khaniyev

Kesemen

2009

Communications in Statistics - Theory and Methods

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In this study, a semi-Markovian random walk with a discrete interference of chance (X t ) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of the ergodic distribution of the process X t are obtained when the random variable 1 , which describes a discrete interference of chance, has a gamma distribution with parameters > 1 > 0. Based on these results, the asymptotic expansions are obtained for the first four moments of the ergodic distribution of the process X t , as → 0. Furthermore, the asymptotic expansions for the skewness and kurtosis of the ergodic distribution of the process X t are established. Finally, it is discussed that the alternative estimations for the stationary characteristics of this process can be offered by using obtained asymptotic expansions.

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“…Numerous works (see, for example, [1][2][3][4][5][6][7][8][9][10][11]) are devoted to this model. The model is investigated with allowance for requirements of inventory theory, queuing theory, reliability theory, actuarial theory, etc.…”

Section: Introductionmentioning

confidence: 99%