A customer who comes to a fast food restaurant will get a $10\%$ discount if
he arrived within $4$ minutes after the previous customer.
If the time between the arrival is between $4$ and $5$ minutes, the discount obtained is $6\%$.
If the time is between the arrival is more than $5$ minutes, the discount obtained is $2\%$.
Time of arrival has an exponential distribution with an average of $6$ minutes.
- Determine the opportunity that a customer who comes to the restaurant will get a $10\%$ discount.
- Determine the average discount obtained by a customer who comes to the restaurant.
The restaurant has three cashiers. The service time is exponentially distributed with an average of $10$ minutes. The waiting room at the restaurant is limited. But, because food in the restaurant is delicious, customers are willing to wait before being served.
- Determine the capacity of the restaurant's waiting room (excluding current customers being serviced) so that the opportunity for someone to get into the restaurant is at least $0.9$.
I am unable to think of an appropriate approach. Any hint/help will be appreciated, because I honestly don't have any clue.