We study positive recurrence and transience of a two-station network in which the behavior of the server in each station is governed by a Markov chain with a finite number of server states; this service process can represent various service disciplines such as a non-preemptive priority service and K-limited service. Assuming that exogenous customers arrive according to independent Markovian arrival processes (MAPs), we represent the behavior of the whole network as a continuous-time Markov chain and, by the uniformization technique, obtain the corresponding discrete-time Markov chain, which is positive recurrent (transient) if and only if the original continuous-time Markov chain is positive recurrent (resp. transient). This discrete-time Markov chain is a four-dimensional skip-free Markov modulated reflecting random walk (MMRRW) and, applying several existing results of MMRRWs to the Markov chain, we obtain conditions on which the Markov chain is positive recurrent and on which it is transient. The conditions are represented in terms of the difference of the input rate and output rate of each queue in each induced Markov chain. In order to demonstrate how our results work in two-station networks, we give several examples.