Maximum number of calls received?

Question

A toll-free phone number is available from 9AM to 9PM for your customers to register a complaint with a product purchased from your company. Past history indicates that an average of 0.4 calls are received per minute.

What is the maximum number of phone calls that will be received in a one minute period 99.99% of the time?

Answer 1

For the Poisson distribution, m=0.4 calls/minute. In 12 hours (9am to 9pm) the number of calls averages at 720*0.4=288. The average number of calls per 15 minutes is 15*0.4=6. Po(m;x)=e^-6 * 6^x/x!=0.9999. We need x.

y=6^x/x!=403.39 approx. Let's make a table

x | y

3 | 36

4 | 54

5 | 64.8

6 | 64.8

7 | 55.5

9 | 27.8

We can see from this simple table that x=5 or 6 calls per quarter hour yields the best match. 

If we change the time period to 30 minutes, m=12 and y=e^-12 * 12^x/x! and 12^x/x!=162754 approx.

x   | y

10 | 17062

11 | 18613

12 | 18613

13 | 17182

x=11 or 12 is the peak. For m=24 calls an hour, x peaks at 23 and 24. So there's a pattern and the average number of calls a minute is around 0.392 calls at best. This is to be expected.