# Continuous time markov chain - long-term probability

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

• You should first define what a state represents in your Markov Chain. I don't think CTMCs can be represented as a transition matrix because time spent in states is exponentially distributed. As far as I know, the transition matrix can be used for DTMCs only. – an4s Apr 15 '18 at 17:06

## 1 Answer

This is just a job queue with Poisson arrivals and exponential services times. The taxis represent jobs. The "service time" is the time to the arrival of the next customer. Since the inter-arrival times of Poisson arrivals are exponential, the service times are exponentially distributed with parameter $\mu.$