Generalized Pollaczek-Khinchin Formula for Markov Channels

The wireless fading channels with finite input buffer, Poisson arrivals and two-state Markov modulated service processes (MMSP) are modeled as M/MMSP/1/K queues in this paper. The existing performance analyses of Markov channels are almost all based on the matrix-geometric method, which provides little physical insights for system design. By contrast, we focus on deriving closed-form analytic expressions with physical interpretations in terms of system parameters of interest. Our main contribution is to derive the generalized Pollaczek-Khinchin (P-K) formula of M/MMSP/1/K queue from start-service probability to explore the impact of state transitions on the queueing behavior of Markov channels. This generalized P-K formula reveals that the performance of wireless channels with varying rates can be fully characterized by a newly defined system parameter, called state transition factor beta, which clearly explains the reason that the channel with slow state transition rate owns a larger delay for the same channel capacity. In the extreme case when the state transition factor beta approaches 0, we show that the channel under consideration can be approximately modeled as an M/G/1 queue. We use the Type I Hybrid ARQ system with a fixed data-rate as an example in this paper to illustrate our results.