Vincent’s barbershop. Vincent’s barbershop is very modest in size. It consists of a single barber’s chair where Vincent serves his customers and two other chairs for waiting customers. Prospective customers arrive at the barbershop in a Poisson manner at the rate of λ per hour. Those who find an available chair will eventually receive service in a First-In-First-Out (FIFO) manner. Those who find the barbershop full (barber’s chair and the two other chairs are all occupied) go to other barbershops and are lost forever. It takes Vincent 1/μ hours, on average, to complete service to a customer. All questions below refer to steady-state conditions. a. Assume that the service time is exponentially distributed. For the cases (i) λ = μ (ii) λ = 2μ compute L, the expected number of customers in the barbershop. b. Vincent wants to add a sufficient number of chairs for waiting customers to make sure that at least 92% of his prospective customers become actual customers. For λ=μ, what is the minimum number of chairs he will need in the shop (including the barber’s chair)? [Assume that all prospective customers who find at least one chair empty will become actual customers, while those who do not will be lost.] Return to the situation described in part a. Suppose that Vincent decided not to serve customers in a FIFO way but in random order. That is, if Vincent finishes a haircut and finds two customers waiting, he will select at random between the two customers, i.e., each waiting customer will have probability 0.5 of being the next one to be served. In which of the two cases, if any, will W be greater? Please justify your answer briefly. [W is the expected total amount of time that a customer has to spend in the barbershop.] Hint: Think about the scope of Little’s Law we discussed in class. No numerical analysis is necessary. d. Assume now that Vincent has added a sufficiently large number of waiting chairs to his shop that customers are never lost, for all practical purposes, due to lack of waiting space. For the case λ=(0.9)μ , compute L, the expected number of customers in Vincent’s shop at any randomly chosen time with the system in steady state. A numerical answer is expected for this part.