Women arrive at a hairdresser that operates in Withington, Manchester with a single haircut and single manicure services according to a Poisson process. The arrival rate of customers is 2 per hour. Service times for haircut and manicure are represented by exponential random variables with rates 4 and 6 per hour, respectively. Since getting a haircut takes longer than a manicure, the arriving customer directly sits for a haircut if the seat is not occupied. If another person is getting a haircut once the new customer arrives and manicure seat is free, she begins her manicure. If both of the seats are busy, the new arrival is cannot enter the system. A customer who completes her haircut goes for a manicure if it is empty else, leaves the hairdresser. Customers leave the hairdresser after receiving a manicure. Show all your work step-by-step while answering the following

1. f) The hairdresser modifies its operations by cancelling manicure services. Additionally, they enabled a

   queue of maximum size 2 (except the one who receives the service). Under the new circumstances, an arriving customer receives a haircut and leaves the system. However, the rate of arriving customers is reduced by half due to cancelling manicure operations. A revenue per haircut is 10 and a cost of 1 incur while providing a haircut service at rate (service times are represented by exponential random variables). What is the optimal rate of haircut service () to maximize the hourly profit?