Any help or confirmation on this question would be greatly appreciated.
So when answering the question about queues I know at for this particular scenario that there is a single sever and the capacity of the shop is 4 customers, and any additional customers are turned away.
I worked out that $\rho = 1$, therefore the steady state equation implies that $P_n=P_0$ (for all $n \leq 4$).
Therefore $P_n=P_0=\frac{1}{N+1}$ (when $\rho=1$)
So the point of the question is to find $L_s$ (The mean number of customers in the shop)
This is what I have got, but i'm not confident that I am correct.
As $N=4$,
$$P_n = \frac{1}{N+1} = \frac{1}{4+1} = \frac{1}{5}.$$
Then
\begin{align}
L_s &= \sum_{n=0}^{N} n P_n \\
&= \sum_{n=0}^N n \frac{1}{N+1} \\
&= \frac{1}{5} \sum_{n=0}^{4} n \\
&= \frac{1}{5} \cdot \frac{4(4+1)}{2} \\
&= 2.
\end{align}
Any help would be very much appreciated if this is wrong. Or if someone could tell me I am right If I am.