We have $n$ bins, in each step we throw a ball in a bin chosen uniformly and independently from the $n$ bins we have.
We repeat the process $k$ times. Let $B_k$ be the number of balls in maximum-loaded bin after $k$ steps, and $b_k$, accordingly, the number of balls in minimum-loaded bin after $k$ steps.
Let $K\ge 1$ be the number of step in which we had $2b_K\ge B_K$ for the first time.
I need to find function $f(n)$ and two constants $0<c_1\le c_2$ such that
$c_1f(n)\le T\le c_2f(n)$ with ptobability tending to $1$.