A piecewise Markov process is a discrete-state, continuous-parameter stochastic process which is Markovian within contiguous time-segments. Starting at the beginning of a segment in some initial state, the process evolves in a Markovian manner until the segment terminates at a random time whose distribution is completely determined by the initial state. The terminal state of the segment then determines the initial state of the process in the following segment in accordance with a first order Markov chain.The stationary behavior of such a process is examined and it is shown how the results may be used to analyze a queue whose input is composed of two independent streams only one of which is Poissonian. Detailed analysis of such models appear in a separate paper.