29. Potential customers arrive to a single-server hair salon according to a Poisson process with rate λ. A potential customer who finds the server free enters the system; a potential customer who finds the server busy goes away. Each potential customer is type i with probability pi , where p1 + p2 + p3 = 1. Type 1 customers have their hair washed by the server; type 2 customers have their hair cut by the server;

and type 3 customers have their hair first washed and then cut by the server. The time that it takes the server to wash hair is exponentially distributed with rate μ1, and the time that it takes the server to cut hair is exponentially distributed with rate μ2.

(a) Explain how this system can be analyzed with four states.

(b) Give the equations whose solution yields the proportion of time the system is in each state.

In terms of the solution of the equations of (b), find

(c) the proportion of time the server is cutting hair;

(d) the average arrival rate of entering customers.