Consider a G/G/1 queue, with sequence of i.i.d inter-arrival time $T_1, T_2, T_3,\cdots$ and i.i.d service time $S_1, S_2, S_3, \cdots$. Suppose system is initialized with workload $w>0$ and consider the first time that the workload hit zero. The workload process could be formally defined as
\begin{equation}
W(t) = w - t + \sum_{i = 1}^{N(t)} S_i \text{ where } N(t) = \inf\{n\in\mathbb{N}: T_1 + T_2 + \cdots + T_n \leq t\}
\end{equation}
and the first time hitting zero is
\begin{equation}
\tau = \inf\{t\geq 0, W(t) = 0\}
\end{equation}
I am looking for a bound for $E(\tau)$ in terms of $w$, $E(T_1)$ and $E(S_1)$.
I thought this could be related to constructing some martingale and using the optional stopping theorem, but I don't know how to construct this martingale. Or maybe other method could be helpful for this? I am really appreciated if you offer some help for this.