Suppose there is a stocastic random recursive sequence $\{b_n\}$, $n=1,2,...$, it evolves as follows: \begin{equation} b_i=\begin{cases} b_{i-1}+1, & \text{w.p. } \lambda, \\ 0, & \text{w.p. } 1-\lambda. \end{cases} \end{equation} Here $i\geq2$ and suppose $b_1=0$.

Then I construct another recursive sequence $\{a_n\}$, $n=1,2,...$, it evolves as follows: \begin{equation} a_i=\begin{cases} a_{i-1}+1, & \text{if } a_{i-1}-b_i<\gamma, \\ (1-\epsilon)(b_i+1)+\epsilon(a_{i-1}+1), & \text{if } a_{i-1}-b_i\geq \gamma. \end{cases} \end{equation} Here $i\geq2$ and suppose $a_1=1$. $\lambda$ and $\epsilon$ are both constant ranging from $[0,1]$. $\gamma$ is also a non-negative constant number, you can regard it as a "threshold" to influence the recursion.

What I want to find is the long-term summation of $\{a_n\}$, i.e., $\lim_{N\rightarrow\infty}{\frac{1}{N}\sum_{i=1}^Na_i}$, can I derive a closed form of it? Or can I find some references to solve it?