Line crossing problem for biased monotonic random walks in the plane

In this paper, we study the problem of finding the probability that the two-dimensional (biased) monotonic random walk crosses the line $y=\alpha x+d$, where $\alpha ,d\ge 0$. A $\beta$-biased monotonic random walk moves from $(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1\mathrm{/}(\beta +1)$ and $\beta \mathrm{/}(\beta +1)$, respectively. Among our results, we show that if $\beta \geq \lceil \alpha \rceil$,thenthe$\beta$-biased monotonic random walk, starting from the origin, crosses the line $y=\alpha x+d$ for all $d\ge 0$ with probability 1.