**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 $R\_2$?
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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 $D \sim U(a,b)$.



Suggestions for dealing with the "timed" balls-into-bins model
(I use the notation from my comments). By symmetry, $$\mathbb{P}(E^c)={n \choose m}\,\mathbb{P}(E^c, B=\{1,\ldots,m\})\;\;.$$ 
If $D\_1=M\_m\;\;$ we have to compute $$I\_1:=\mathbb{P}(D\_1^\prime>t+M\_m,D\_2^\prime>t+D\_2,\ldots,D\_m^\prime>t+D\_m, D\_{m+1}>M\_m,\ldots,D\_n>M\_m)$$
Conditioning on $M\_m=D\_1,D\_2,\ldots,D\_m$ and using the independence assumptions gives 
$$I\_1=\int e^{-\mu\_1(t+s)}\left(\prod\_{i=2}^m e^{-\mu\_1(t+x\_i)}\right) e^{-\mu\_2(n-m)s} f(s,x\_2,\ldots,x\_m)\,dx\_2\ldots dx\_m\,ds$$ where 
$f(s,x\_2,\ldots,x\_m)=1\_{[0,\infty[}(s) \mu\_2e^{-\mu\_2s} \prod\_{i=2}^m \mu\_2e^{-\mu\_2x\_i}1\_{[0,s]}(x\_i)$.
 After carrying out the $x\_i$ integrals $$I\_1=e^{-m\mu\_1t}\mu\_2^m\int\_0^\infty e^{-(\mu\_1+\mu\_2)s}\left({1-e^{-(\mu\_1+\mu\_2)s}\over \mu\_1 +\mu\_2}\right)^{m-1} e^{-\mu\_2(n-m)s}\,ds$$ and the substitution $y=1-e^{-(\mu\_1+\mu\_2)s}$ gives
$$I\_1=e^{-m \mu\_1 t}\alpha^m\,B(m,\alpha (n-m)+1)$$ Finally, by symmetry the cases $D\_2=M\_m$,...,$D\_m=M\_m$ give the same result, so that$$\mathbb{P}(E^c)=m{n\choose m} e^{-m\mu\_1t}\alpha^m\,B(m,\alpha(n-m)+1)=e^{-m\mu\_1t}{\alpha^m\,B(m,\alpha(n-m)+1)\over B(m,n-m+1)}$$


