
Simple queue
Consider the queue: Customer $\displaystyle \rightarrow S_1 \rightarrow S_2 \rightarrow$ leave.
Customers arrive at rate $\displaystyle 2;$ $\displaystyle S_1$ serves at rate $\displaystyle 4;$ $\displaystyle S_2$ serves at rate $\displaystyle 2.$
If a customer comes to an occupied server he leaves the system.
Formulate a markov chain model for this system with state space $\displaystyle \{0,1,2,12\}$ where the states indicate the servers who are busy.
$\displaystyle (a)$In the long run what proportion of customers enter the system? (If a customer leaves before being served by $\displaystyle S_1$ he does not enter)
$\displaystyle (b)$What proportion of customers visit server $\displaystyle 2?$