**Definitions:** Let $T$ (for "time") be a random variable $T \sim \text{Exp}(\lambda)$ and $\Delta t$ is a realization (or called an observed value) of $T$. Let $D$ (for "delay") be a random variable $D \sim \text{Exp}(\mu)$. All the random variables (including those involved below) are mutually independent.

**Timed Balls-into-Bins Model:** There are $n$ bins. Consider two robots $R_1$ and $R_2$ which can produce multiple balls instantaneously.

At time 0, robot $R_1$ *(1)* produces $n$ balls instantaneously; *(2)* Immediately, these $n$ balls are independently sent to the $n$ bins, one ball per bin; *(3)* The delays for the balls going from the robot to its destination bin are IID with the same distribution as $D$ defined above.

At time $\Delta t$ (defined above), robot $R_2$ independently does exactly the same thing as robot $R_1$ does (i.e., (1), (2), and (3) for robot $R_1$ above).

Consider the time point $t$ when half (more precisely, $\lfloor n/2 \rfloor + 1$) of the $n$ bins have received the balls from $R_2$ (*no* constraints on the balls from $R_1$), and denote the set of these $\lfloor n/2 \rfloor + 1$ bins by $B$.

Question:What is the probability $\mathbb{P}$ of the event $E$ that there exists a bin $b_{\Box} \in B$ which receives a ball from $R_1$ before receiving a ball from $R_2$?

Have similar balls-into-bins models been studied in the literature?

Any suggestions (not a complete solution) towards a **closed form/approximation/numerical results (by using mathematical systems)/(even) simulations** of the probability $\mathbb{P}$?

It is OK to simplify this model for possible tractability, for example, by using $D \sim U(a,b)$.