My textbook (pg 121, formula 4.39) says:
The average number of customers at the service facility: $N_S = P[k=0]E[N_S|k=0] + P[k>0]E[N_S|N>0] = 1 - P_0 = \rho(1-P_S)$
But I can't understand why this formula give this result (and I think that there is an error in it). Can someone help me and explain please? Many thanks
The number of customers at the service facility is always either $0$ or $1$. It is $0$ if there are no customers in the system at all; otherwise, it is $1$.
I think the formula should read $$ \color{red}{\mathbb{E}}[N_s] = \Pr[\color{red}{N}=0] \cdot \mathbb E[N_s \mid \color{red}{N}=0] + \Pr[\color{red}{N}>0] \cdot \mathbb E[N_s \mid N>0] $$ where $ \mathbb E[N_s \mid N=0]$ simplifies to $0$ and $\mathbb E[N_s \mid N>0]$ simplifies to $1$. The conclusion is that $$\mathbb E[N_s] = \Pr[N>0] = 1 - \Pr[N=0] = 1 - P_0.$$ Presumably, the relation $1-P_0 = \rho(1-P_S)$ is proven earlier in the text.