Suppose we have $M$ independent, simple random walks, each taking value $+1$ with probability $p$ and $-1$ otherwise. Call these $S_j^t$ for $j\in\{1,\dots,M\}$. Suppose $p > 1/2$ (so they drift upward), and define the stopping time $\tau$ for $A > 0$ as
$$\tau = \inf\left\{ t : \max_{j \in \{1,\dots,M\}} S_j^t \geq A\right\}$$
i.e. $\tau$ is the first time that the maximum of the random walks exceeds $A$. My question is: what is $\mathbb{E}[\tau]$ as function of $M$ and $A$? Or rather, what is a good upper bound on $\mathbb{E}[\tau]$? I'd appreciate any references on how to handle a stopping time for the maximum of $M$ random walks as well, I haven't seen much of it before.