The sum of the total number of service representatives who spoke with all the customers who called during a minute.

Question

I trying to solve this problem, but I can not tell if what I am doing is right,

At some company, the customer support department gets called by $5$ customers per minute independently.

The number of service representatives that are needed to be called until a customer problem is solved is $X∼G(0.4)$ for every customer independently.

Let $A$ be the sum of the total number of service representatives who spoke with all the customers who called during a minute.

Find $Pr⁡[A≤3]$.

My attempt:

Let $Y\sim \mathrm{Pois}(5)$ be the number of customers calling per minute.

$Pr[A\le 3]=Pr[A=0]+Pr[A=1]+Pr[A=2]+Pr[A=3]$.

• $Pr[A=0]=Pr[Y=0]$
• $Pr[A=1]=Pr[Y=1]\cdot Pr[X=1]$
• $Pr[A=2]=Pr[Y=2]\cdot Pr[X=1] + Pr[Y=1]\cdot Pr[X=2]$
• $Pr[A=3]=Pr[Y=3]\cdot Pr[X=1] + Pr[Y=2]\cdot (Pr[X=1] + Pr[X=2]) + Pr[Y=1]\cdot Pr[X=3]$

Notes:

• I intentionally did not expand $Pr[X=i]$ and $Pr[X=j]$ to its numeric to keep things simple.
• I am most unsure about $Pr[A=3]$.

Answer 1

Mean number of service representatives required to solve a customer problem $=\frac{1}{.4} = 2.5$

Number of customers per minute =$ 5$

Thus mean number of total service representatives $= 5\times 2.5=12.5$

$P(A\le 3) = \sum_{t=0}^{t=3}\frac{e^{-12.5}.(12.5)^{t}}{t!}=0.001555$