“…Mitov and Omey [18] investigated some asymptotic properties of the class of Γ(g) . When tail distribution belongs to the extended class of Γ(g) , i.e.F (x) = P {η 1 > x} ∈ Γ(g) they also provided intuitive approximation for the renewal function generated by the random variables {η n } n≥1 as follows:…”
Section: Definition 11 a Positive And Measurable Function H Belongs To The Class Of γ(G) With Auxiliary Function G If And Only If Every Fmentioning
confidence: 99%
“…Geluk [11] took the results of Anderson and Athreya [5] one step further and obtained asymptotic expansion for renewal function generated by regularly varying random variables with infinite variance. For more recent work we refer the interested reader to [9][10][11]18] and the references therein.…”
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) .
“…Mitov and Omey [18] investigated some asymptotic properties of the class of Γ(g) . When tail distribution belongs to the extended class of Γ(g) , i.e.F (x) = P {η 1 > x} ∈ Γ(g) they also provided intuitive approximation for the renewal function generated by the random variables {η n } n≥1 as follows:…”
Section: Definition 11 a Positive And Measurable Function H Belongs To The Class Of γ(G) With Auxiliary Function G If And Only If Every Fmentioning
confidence: 99%
“…Geluk [11] took the results of Anderson and Athreya [5] one step further and obtained asymptotic expansion for renewal function generated by regularly varying random variables with infinite variance. For more recent work we refer the interested reader to [9][10][11]18] and the references therein.…”
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) .
“…More recently [13] introduced a new method to obtain approximations for univariate renewal functions and these approximations cover the known results. In this paper we generalize these approximations to bivariate renewal processes.…”
“…It appears that the only case where closed form expansions are available are when F is Matrix exponential distributed, in which case an expression of U (x) is given in Asmussen and Bladt [3]. Mitov and Omey [18] provide heuristics on many terms asymptotics of U (x) that are verified on the already known cases. However, as the authors point out, those interesting expansions are only given formally and are not proved.…”
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