Originally, this is a question with 7 different sub-questions. I send them one by one. There are answers already uploaded, but only one same answer copy pasted for every sub-question. I am assuming, experts thought that they were the same question since all 7 of them has the same story in the beginning. Please beware of the actual question at the end.

Q1. 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: ii. Calculate long-run probabilities of all states