Consider an M/M/1 queue where arrivals occur at rate $\lambda(t)$ according to a Poisson process at time $t$ and move the process from state $i$ to $i+1$, and service times have an exponential distribution with rate parameter $\mu$. Assume $\lambda(t) < \mu, \forall t \geq 0$. I am wondering if we can upper bound the $\lambda(t)$ by $\lambda$ and show the stationary distribution of the queue with arrival rate $\lambda$ and service rate $\mu$ exists, can we say the stationary distribution of the original system (i.e. a system with arrival rate $\lambda(t)$ and service rate $\mu$) exists?
I did not find any book about the ergodic theory of the Markov Chain with a non-homogeneous Poisson process. I am quite confused about it.