Asymptotic Expansions for the Stationary Moments of a Modified Renewal-Reward Process with Dependent Components

Abstract

In this paper, a modification of a renewal-reward process \(X(t)\) with dependent components is mathematically constructed and the stationary characteristics of this process are studied. Stochastic processes with dependent components have rarely been studied in the literature owing to their complex mathematical structure. We partially fill the gap by studying the effect of the dependence assumption on the stationary properties of the process \(X(t)\). To this end, first, we obtain explicit formulas for the ergodic distribution and the stationary moments of the process. Then we analyze the asymptotic behavior of the stationary moments of the process by using the basic results of the renewal theory and the Laplace transform method. Based on the analysis, we obtain two-term asymptotic expansions of the stationary moments. Moreover, we present two-term asymptotic expansions for the expectation, variance, and standard deviation of the process \(X\left(t\right)\). Finally, the asymptotic results obtained are examined in special cases.