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!
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).