(Solved): Question 1 has 5 parts (a-e)Please answer all.Thank you. 1. (50 points) The stylish hair salon has t ...

1. (50 points) The stylish hair salon has three stylists who provide services to its customers. After checking in with the receptionist, the customer's hair is washed, dried, and styled. Each stylist has a different level of experience, so their service times differ from each other. Stylist 1 has an average service time of 20 minutes, Stylist 2 has an average service time of 40 minutes, and Stylist 3 has an average service time of 30 minutes, respectively. Suppose also that the service times are exponentially distributed and independent from each other. Let T be the service time of Stylist 1, Tz be the service time of Stylist 2, and T3 be the service time of Stylist 3 (please use this notation in your solutions). What is the rate of each Stylist to take care of their customers? Write down the rates as customers/minute as the unit. Write down also rates in terms of customers/hour as the unit. a) (10 points) A customer enters the hair salon and is being assigned to Stylist 2 since Stylist 2 is available. What is the probability that the customer will spend more than 30 minutes in the salon? b) (10 points) A customer arriving at the salon finds all stylists busy and no one else is waiting. Let T be the time the customer's waiting time before taking care by a stylist. b1) (5 points) Express Twin terms of T, Ty, and Tz. b2) (5 points) How long the waiting time will be on average? c) (10 points) A customer arriving at the salon sees all stylists are busy and no one else is waiting. What is the probability that the customer will be served by Stylist 1? d) (10 points) A customer arriving at the salon finds all stylists taking care of other customers and no one else is waiting. The customer waits for 10 minutes and the stylists are still taking care of other customers. What is the probability that the customer will wait more than 30 minutes to get taken care of? e) (10 points) Suppose on a given day Stylist 2 is on leave, and both Stylist 1 and Stylist 3 are taking care of their customers. A customer arriving at the salon finds both stylists taking care of other customers and no one else is waiting. What is the expected service time for the customer?