Level–phase independent stationary distributions for GI/M/1-type Markov chains with infinitely-many phases

GI/M/1-type Markov chains are two-dimensional processes, with one dimension called the level and the other the phase. When such a chain is positive-recurrent and the phase space is finite, much is known about the properties of the stationary distribution. In particular, it is known how to derive the decay rate as the level variable approaches infinity. However, when the phase space is infinite, the properties are more complicated. A range of decay rates may be achievable and the transition structure at the boundary has a strong influence on the stationary distribution.In this paper, we show that, for given transition probabilities in the interior states, it is possible under mild conditions to change the boundary transition probabilities in such a way that the level and phase are independent under the stationary distribution and furthermore, one may choose the decay rate of the stationary distribution from the entire range compatible with the interior transition rates.