This paper considers a single-server queue with two independent input streams: MAP/G and M/GI streams. MAP (Markovian arrival process) is a class of semi-Markovian arrival processes and it is weakly dense in simple stationary point processes. In this paper, the authors analytically show the decomposition formula for the conditional virtual waiting time in the single-server queue with independent MAP/G and M/GI input streams, given the state of the underlying Markov chain which governs the MAP/G stream. Using the result, they also show the decomposition formulas for the actual waiting times in the queue with independent MAP/G and M/GI input streams and in a certain priority queue.
