On property of “waiting process”

Let $X_1,X_2,\dots$ be i.i.d. integer random variables with $-\infty\le\mathbf EX_1<0$ and $\mathbf P\{X_1>0\}>0$. Consider the process $W_t$, $t=0,1,\dots$, defined by formula:
$$ W_0=0,\quad W_{t+1}=\max\{W_t+X_{t+1};0\}, $$
and its passage time $\tau(N)=\min\{t\colon W_t\ge N\}$, $N=1,2,\dots$. In this paper the existence of $\lim\sqrt[N]{\mathbf E\tau(N)}$ is proved, and its value is found.