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 β, 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 β 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.