This is the answer I got but I'm not confident with it. So the question is find the mean length of time a customer spends in the shop (Ws).
We know that a customer enters a shop at random at a rate of 20 per hour and the capacity of the shop is 4 customers, and any additional customers are turned away. Ans the mean service time is 3 mins.
In a previous question I worked out that Ls equals 2
So this is what I think you do
Ws= $\frac{Ls}{λeff}$
From research I know that λeff= $\sum_{n=o}^N λnPn$ = $\sum_{n=o}^{N-1} λ(1-PN)$ (since λN=0) because previously we worked out that ρ=1,we now have to use that Pn=$\frac{1}{N+1}$
Therefore λeff= 20(1-PN) (since λN=0)
λeff = 20(1- $\frac{1}{N + 1}$)
λeff = 20(1- $\frac{1}{4+ 1}$)
λeff = 20(1- $\frac{1}{5}$)
λeff = 20(1- $\frac{4}{5}$)
λeff = 16
Therefore Ws= $\frac{Ls}{λeff}$
Ws= $\frac{2}{16}$
OR Because we don't know what PN stands for can we just do it this way?
λeff= $\sum_{n=o}^{N-1} λnPn$
λeff= $\sum_{n=o}^{N-1} 20Pn$
Pn = $\frac{1}{N+1}$ because ρ=1 therefore Pn= $\frac{1}{5}$
λeff= $\sum_{n=o}^{N-1}(20)(1/5)$
λeff= $\sum_{n=o}^3 (20)(1/5)$
λeff= $\sum_{n=o}^3 4$
λeff= 12
Thus Ws= $\frac{2}{12}$
Some help would be much appreicated.
Since $\lambda=\mu$, the detailed balance equations $\lambda \pi_j=\mu\pi_{j+1}$ imply that $\pi_j=\pi_{j+1}$, and hence $\pi_j=\frac15$ for $j=0,1,2,3,4$. The mean number of customers in the system is then $$ L = \sum_{j=0}^n j\pi_j = \frac15 \sum_{j=1}^4 j = 2, $$ and the effective arrival rate $$ \lambda_{\mathrm{eff}} = \lambda(1-\pi_4) = 20\left(1-\frac15\right) = 16. $$ We use Little's Law to compute the mean sojourn time: $$ W = \frac L{\lambda_{\mathrm{eff}}} = \frac2{16}=\frac18. $$