Let say we have a Coast Guard center wich have a emergency center hold only one rescuer. accidents comes according to a Poisson process. There is usually 16 casualties on 8 hours. Casualties are examined according to an exponential law with a mean equal to 15 minutes for each victim. Victims are examined in the order of arrival and there is no limitations of places in the emergency service. Then they are oriented to hospitals according to the severity of their injuries.
I sayed that the mean number of casualties is
$$E(L)=\frac{16}{8}*\frac{1}{4}=\frac{1}{2}$$
then I searched for the number of casualties waiting to be cured $E(L^q)$:
$$E(L^q)=\bar\lambda E(W^q)$$
$E(W^q)$ is the mean time a victim has to wait.
yet,
$E(W)=E(W^q)+\frac{1}{\mu}$ with $\frac{1}{\mu}$, the rate of healing,
$E(L)=\bar\lambda E(W)$
- and the rate of arrival of victims $\bar\lambda= 1/2$
Yet, I find that
$$E(W^q)=0$$ ,therefore $E(L^q)$=0.
which is pretty odd because, even if someone is directly healed, the mean time can't be zero as far as some people may wait!
Where did I went wrong?