A ride-sharing platform is facing a pricing problem. For a particular trip, they offer a price (p) of $13.25 to customers for which the platform keeps 20% (). The managers are looking for alternatives and they propose either increasing the price to $15, increasing the share the platform keeps to 25% or both. The platform estimates that the arrival rate of customers follows the functional form: (p) = 20 p (customers per hour). The number of cars on the platform, which is an integer, is as follows:

C(p, ) = 3p(1 ). Average trip time is set to 15 minutes ().

We can model this as an M/M/C/C queue where C is the number of cars on the platform. When there is no available car on the platform, arriving customers will leave the platform immediately. Given the problem and the queueing model, answer the following questions:

(a) What is the expected profit rate formula of the platform as a function of , p, and the blocking probability (Pblock) of the M/M/C/C queue?

(b) Create an Excel spreadsheet to calculate the platforms expected profit rate for four cases where p takes values 13.25 and 15, and takes values 0.2 and 0.25.

(c) What is the optimal p and which maximize the expected profit rate of the platform among the four possible alternatives? Explain the intuition behind your answer.

[Hint: You will need to create four sheets, one for each (, p). In each of them first compute (p), C(p, ) (round-up to the closest integer) and the utilization of the system. Then you can compute the probability of an empty state, the probability of queuing and finally the blocking probability.]