How to analyze this type of queue

Question

The setup is as follows:

Families arrive at a taxi stand according to a Poisson process with rate $\lambda$. An arriving family finding $N$ other families waiting for a taxi does not wait. Taxis arrive at the taxi stand according to a Poisson process with rate $\mu$. A taxi finding $M$ other taxis waiting does not wait. Derive expressions for the proportion of time are there no families waiting, and the proportion of time are there no taxis waiting.

From what I have learned so far, this appears to me to be a $M/M/c/K$ queue, where there are $c$ servers (taxis) and system capacity $K$ (since there are at most $N$ families in line). Then when finding the proportion of time are there no taxis waiting, the taxis would become the "customers" and the families the "servers". However, the $M/M/c/K$ queue has the condition that $c\leq K$, which is not specified in this question. I am wondering if I may proceed as if it were an $M/M/c/K$ queue in both scenarios, then find the balance equations to derive $\pi_0$ in both case.

Answer 1

Hint:

I think you could rewrite the question to an $M/M/1$ queue with capacity $N+M$ and baulking, saying something like:

Customers arrive at a queue according to a Poisson process with rate $\lambda$. A customer finding $N+M$ other customers waiting does not wait. Service times are exponentially distributed with rate parameter $\mu$. Derive expressions for the proportion of time are there are $M$ or fewer customers waiting, and the proportion of time there are $M$ or more customers waiting.