On a given GNV$-$ATL flight, there are $200$ seats, and each seat is sold at an average ticket price of $ $500.$ It has been observed that on most days in the past month, the flight rarely approaches $100\%$ occupancy since there are passenger no$-$shows. Hence, the airline decides to overbook flights. The airline manager assigned the task of determining the number of seats to overbook on this flight starts by estimating no$-$shows using historical information. However, she encounters an interesting conundrum: the mean no$-$shows changes based on the rate offered to customers with a confirmed booking who are denied a seat on the flight $($referred to as the service denial rate$).$

She would like to start by analyzing the impact of an overbooking policy by considering three specific service denial rates