A transient non symmetric random walk

Question

Let $X_n$ be random walk starting at $M< N$. At each step $X_n = k$ it goes up a unit step with probability $1-k/N$ otherwise it stays at its place. Do we have any nice interpretation of the asymptotic distribution of the time to reach $X_n = N$

Answer 1

This is the coupon collector problem, with a started collection of $M$ coupons. The asymptotic is known to be $nH_n$, where $H_n$ is the $n^{th}$ harmonic number