What is the probability of the number 1 and number 2 employees getting the bonus at a call center? Two weeks ago, a friend working at a call center told me about their staff bonus policy. Here I paraphrase it.  

Suppose employee A answers the maximum number ($N_1$) of calls among the staff, and employee B (Notice that there may be more than one such employee Ｂ if there are ties.) answers the second maximum number ($N_2$) of calls, if $N_1-N_2\le10$, then both employee A and employee B could get the same extra bonus, otherwise, only employee A could get the bonus.

It is really an interesting policy. I did try to formulize the probability that number 1 and number 2 employees (There may be more than two employees that can receive the bonus if there are ties for the second place.) that get the bonus. 

Assume that there are $m$ employers working at the call center, and the calls arrive with a homogeneous Poisson process with rate $\lambda$ at every desk (every employee) of the call center. To simplify the problem, I also assume that every incoming call can be handled immediately within a negligible short period of time. Let $N_i(t)$ denote the $i$-th maximum number of calls that have been answered, where $1\le i\le m$. That is, $N_1(t)\ge N_2(t)\ge ...\ge N_{m-1}(t)\ge N_m(t)$.  If the number 1 and number 2 of the employees can get the bonus, then $0\le N_1(t)-N_2(t)\le10$.

Trying to calculate the probability $\Pr[0\le N_1(t)-N_2(t)\le10]$, I started out like this:

Assume $N_1(t)=k$, then $N_j(t)\le k$, where $k\ge11$ and $2\le j\le m$ and there exists at least a specific $N_r (t)$, and $k-10\le N_r(t)\le k$ and $2\le r\le m$. Then I think the probability could be expressed by
  $$ \Pr[\text{Numer 1 and number 2 employees get the bonus}] = \Pr[0\le N_1(t)-N_2(t)\le10] \\ =\sum\limits_{k=11}^\infty\Pr[N_1(t)=k] (\Pr[N_r(t)\le k]^{m-1}-\Pr[N_r(t) \le k-11]^{m-1})$$

But I’m not sure I am doing this right. And even if the formulization is correct, I still could not calculate the probability. It seems that the probability has no simple expression. I tried to estimate the bound of such probability, but all I've got are very complex power series.(See question Estimate the scale of…).

Then, here are my questions:
  1) Am I right in formulating the probability? If not, then how to formulize it?
  2) What is the distribution of $N_1(t)-N_2(t)$? How to calculate the probability or estimate the scale of it?

Sorry for my poor English. And anyone that provides any clue would be highly appreciated. Thank you in advance!
 A: Here's how I would pursue something like this. Define $X$ to be the amount paid out and $B$ to be the amount that is paid out to a single person. Assume there are $n$ people at this call center. Suppose each $Y_{i}$ is the number of calls each person at this call center gets, where each $Y_i$ follows a Poisson distribution with mean $\lambda$. Also, assume that the number of calls among each person in this call center are mutually indepedent. 
Let $Y_{(1)} = \text{min}(Y_1, Y_2, \cdots, Y_n)$, $Y_{(2)} = \text{min}(Y_1, Y_2, \cdots Y_n) \setminus Y_{(1)}$, $\cdots$, $Y_{(n)} = \text{max}(Y_1, Y_2, \cdots, Y_n)$. (My notation is different from yours here - this is just what I'm used to.)
The total paid out is a discrete random variable, which takes on the value $X = \begin{cases}2B & \text{ for } Y_{(n)}-Y_{(n-1)} \leq 10 \\ B & \text{ otherwise} \end{cases}$. 
For convenience, I will assume that the total paid out to a single person, $B$, is constant. 
Then, $P(X = 2B) = P(Y_{(n)}-Y_{(n-1)} \leq 10)$. I don't have a background on discrete order statistics or joint distributions of order statistics, but you can find something here (see p. 14). 
