In this situation the Foster-Lyapunov criterion still works. Let the state of the system be $n$ when there are $n>0$ customers in the system and the server is working, and $0$ when the server is under maintenance (regardless of the number of waiting customers). Then, $f(n)=n$ still proves positive recurrence, since you have negative drift outside $\{0\}$, and $\mathbb{E}_0f(X_1)<\infty$. See Theorems 2.6.4 (discrete time) and 7.3.4 (continuous time) of this book: http://www.ime.unicamp.br/~popov/book_lyapunov.pdf.

Let me stress, though, that there is no "easy" way to transfer this argument to multiple servers, because you need the negative drift for all states outside a finite set (and finite mean jump w.r.t. the Lyapunov function from that set). One needs to modify the Lyapunov function in some way (e.g., in such a that the drift towards the "origin" is large when at least one queue is large). Or use the Foster-Lyapunov criterion "in several steps", see Theorem 2.2.4 of [Fayolle, Malyshev, Menshikov, "Topics in the constructive theory of countable Markov chains"].

In this situation the Foster-Lyapunov criterion still works. Let the state of the system be $n$ when there are $n>0$ customers in the system and the server is working, and $0$ when the server is under maintenance (regardless of the number of waiting customers). Then, $f(n)=n$ still proves positive recurrence, since you have negative drift outside $\{0\}$, and $\mathbb{E}_0f(X_1)<\infty$. See Theorems 2.6.4 (discrete time) and 7.3.4 (continuous time) of this book: http://www.ime.unicamp.br/~popov/book_lyapunov.pdf.

Let me stress, though, that there is no "easy" way to transfer this argument to multiple servers, because you need the negative drift for all states outside a finite set (and finite mean jump w.r.t. the Lyapunov function from that set). One needs to modify the Lyapunov function in some way (e.g., in such a that the drift towards the "origin" is large when at least one queue is large, this could be achieved by considering e.g. "quadratic" Lyapunov functions). Or use the Foster-Lyapunov criterion "in several steps", see Theorem 2.2.4 of [Fayolle, Malyshev, Menshikov, "Topics in the constructive theory of countable Markov chains"].