I'm trying to work out this problem - any hints would be appreciated.
Taxis arrive at a taxi stand according to a Poisson process with parameter $\lambda$. Customers arrive, independently of taxis, at the rate $\mu$. If there are no taxis when a customer arrives at the stand, they will leave. Assume that $\lambda < \mu$. What is the long-term probability that an arriving customer gets a taxi?
Attempt at solution
I should obviously find the limiting distribution of this markov chain. I can find the limiting distribution by first finding the chain transition matrix $\tilde P$.
I'm thinking that this is a birth and death process with birth date $\mu$ and death rate $\lambda$... from here, I'm not even sure how to approach the problem
Any tips would be greatly aprpeciated