Asymptotic expansions for the moments of a semi-Markovian random walk with exponential distributed interference of chance

Limit distribution for a semi-Markovian random walk with Weibull distributed interference of chance

“…Many applied problems of the queueing, reliability, inventory control, insurance and other theories are formulated in the terms of random walks with various types of barrier. Some important studies on this topic exist in the literature (see, for example, [1][2][3][4][5][6][7][8][9]). Let us consider the following model before stating the problem mathematically.…”

“…Note that, in the study [8], [2] and [1] the random variable ζ1, which describes the discrete interference of chance, has an exponential, triangular and gamma distribution, respectively and the stationary moments of ergodic distribution were investigated when the delaying time is zero, i.e. θ1 = 0.…”

“…But the delaying time is necessary for many real systems. So, in this study, unlike [8], [2] and [1], the asymptotic expansions with three terms for the first four moments of the ergodic distribution of the process X(t) will be investigated by taking into account the delaying time (θ1). Also in this study we assume that the random variable ζ1 has Weibull distribution with parameters (α, λ), α > 1, λ > 0.…”

On the rate of convergence of the asymptotic expansion for the ergodic distribution of a semi-Markov (s, S) inventory model

“…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.…”

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

“…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.…”