I'm trying to solve the following question but I'm somehow confused. I know how to solve it for two parallel server but for a combination of series and parallel I don't know what should I do...
Arrivals to a 3 server system are according to a Poisson process with rate λ. Arrivals
finding server 1 free enter service with 1. Arrivals finding 1 busy but 2 free enter
service with 2. Arrivals finding both 1 and 2 busy do not join the system. After
completion of service at either 1 or 2 the customer will then either go to server 3 if
3 is free or depart the system if 3 is busy. After service at 3 customers depart the
system. The service times at i are exponential with rate μi, i = 1, 2, 3.
(a) Define states to analyze the above system.
(b) Give the balance equations. Do not attempt to solve.
(c) In terms of the solution of the balance equations, what is the average time that
an entering customer spends in the system?
(d)what is the probability that a customer who arrives when the system
is empty is served by server 3?
Thank you for your help in advance!