# Queuing theory for task 1

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.