How can I model overlapping service times in queueing model?
That's not in series.  If their times are overlapping, they're not in series.

You mentioned that in your particular example, the latency is $t_1 + t_2 - \min(t_1,t_2)$.  This quantity can be more cleanly expressed as $\max(t_1,t_2)$.

As a result, your queries are certainly not in series -- on the contrary, they are in parallel.  And, there are clean and simple ways to model two queues in parallel.

In particular, assuming the queue times are independent and letting $t=\max(t_1,t_2)$, we have

$$\Pr[t \le x] = \Pr[t_1 \le x] \times \Pr[t_2 \le x].$$

Thus, the cumulative distribution function (CDF) for $t$ is the product of the CDF's for $t_1,t_2$.  The CDF for a single queue is easy to compute given standard properties of the queue (e.g., it's easy to compute the CDF from the PDF).