Let $X$ be a random walk on $\mathbb{Z}_{\ge 0}$ starting at $0$, with step size $1$, and there is a barrier at $0$ so that if one tries to move to $-1$ it stays at $0$ (non-reflecting). If we fix the number of steps, I think there is a way to calculate the expectation of the position in this case. What I am interested is: what if we modify so that if $X$ increases $3$ times in a row, then the step size increases to $2$ until $X$ decreases. Can we calculate the expectation of the position in this case? It seems to be close to $N/8$ where $N$ is the number of steps, but the barrier condition makes me struggle.
Thank you!