A hairdresser and her assistant operate a salon. There are two types of customers: those of type I prefer to have their hair cut by the hairdresser but are willing to be served by her assistant, while those of type II want to be served by the assistant only. The type I (respectively, type II) customers arrive at the salon according to a Poisson process with rate λ1 (resp., λ2). Moreover, the two Poisson processes are independent. Finally, the hairdresser (resp., the assistant) serves in a random time having an exponential distribution with parameter μ1 (resp., 𝜇2), and the service times are independent random variables. Answer the following questions, supposing that there is no room where potential customers can wait until being served:
(a) define a state space that enables you to answer part (c);
(b) write the balance equations;
(c) give, in terms of the limiting probabilities,
(i) the average number of customers in the system;
(ii) the average time that an arbitrary entering customer will spend in the system.