Consider a single queue server system with Poisson arrival rate of jobs $\lambda$ and exponential service times with mean $1/\mu$. When $\lambda < \mu$ the system can be proven to be positive recurrent. One general methodology to prove that is using Foster-Lyapunov criterion for continuous time processes, with Lyapunov function simply being equal to the number of jobs in the queue. See section 6.9.2 of this book for more details. The particular example is simple enough to admit simpler solutions though.
Now consider that whenever the server and queue are empty, the server enters a maintenance state with exponential maintenance duration with mean $1/\mu_2$ and during that time no job is scheduled. Intuitively the system will still be positive recurrent, because no matter how many jobs will arrive during maintenance, the queue will empty eventually during regular service time. Can this intuition be expressed with the previous theorem and if not what other theorems could be used to prove the above?
While I realize the last problem admits an analytical solution I am not interested into that. I would like a methodology for a more general class of problems where e.g. there could be possibly multiple queues and multiple servers.