Consider an M/G/k queue. The arrivals are exponential with rate $\lambda$. There are $k$ servers, but no queue - meaning that if a customer arrives and find no servers free, he simply goes home. How do I find the probability, $q$ that a customer will go home without getting served?
If the general case is too difficult, I'm wondering if it's possible for an M/M/k queue or when the G distribution is deterministic (let's call it $D$).
And if a closed form is not possible, I'd really like an efficient simulation scheme to get this probability (assume I can simulate from the $G$ distribution efficiently). Or even for the M/M/k or M/D/k cases.
I managed to work this out for the special case of $k=1$ and the serving distribution is deterministic. In this case, we consider one arrival who gets served. All arrivals within the time $D$ will go home without getting served. The number of arrivals, $N$ within this period are Poisson distributed with parameter, $\lambda D$. Given $N$, the probability of getting served becomes $\frac{1}{1+N}$.
Taking expectation with respect to $N$ over the Poisson, we get the probability of getting served:
$$q = 1-\frac{1-e^{-\lambda D}}{\lambda D}$$