This is a question on my midterm practice exam, but for some reason we weren't given solutions so it's not very helpful.
There's a single-server queueing system which arrivals follow a Poisson process with rate $\lambda$.When the number of customers in the system is even, then the service time is exponentially distributed with rate $\mu$. When the number of customers in the system is odd, then the service time is exponentially distributed with rate $\mu/2$. Supposed that at some point of time, there are 10 customers in the system. What is the probability that at least two new customers arrive to the system before two customers leave the system?
How I started:
So I'm trying to find $Pr(S_{2}^{A} < S_{2}^{L})$ I think the formula for arriving less than leaving is $\frac{\lambda}{\lambda + \mu}$. This is all I thought of before getting stuck. I also drew a Markov chain, but I don't think it's necessary to solve this.