Asymptotic Rate for Weak Convergence of the Distribution of Renewal-Reward Process with a Generalized Reflecting Barrier

Abstract

In this study, a renewal-reward process (X(t)) with a generalized reflecting barrier is constructed mathematically and under some weak conditions, the ergodicity of the process is proved. The explicit form of the ergodic distribution is found and after standardization, it is shown that the ergodic distribution converges to the limit distribution R(x), when $\lambda \rightarrow \infty$, i.e.,

$$\begin{aligned} Q_{X}(\lambda x)\equiv & {} \lim _{t\rightarrow \infty }P\{X(t)\le \lambda x\}\rightarrow R(x) \\\equiv & {} \frac{2}{m_{2}}\int \limits _{0}^{x}\int \limits _{v}^{\infty }{ [1-F(u)]du}dv. \end{aligned}$$

Here, F(x) is the distribution function of the initial random variables $\{\eta _{n}\},$ $n=1,2,\ldots ,$ which express the amount of rewards and $m_{2}\equiv E(\eta _{1}^{2})$. Finally, to evaluate asymptotic rate of the weak convergence, the following inequality is obtained:

$$\begin{aligned} |Q_{X}(\lambda x)-R(x)|\le \frac{2}{\lambda }|\pi _{0}(x)-R(x)|. \nonumber \end{aligned}$$

Here,

$$\begin{aligned} \pi _{0}(x)=(\frac{1}{m_{1}})\int \limits _{0}^{x}(1-F(u))du \end{aligned}$$

is the limit distribution of residual waiting time generated by $\{\eta _{n}\},$ $n=1,2,\ldots ,$ and $m_{1}=E(\eta _{1})$.