Level-phase independence for \(\text{GI}/\text{M}/1\)-type Markov chains (Q2725295)

A two-dimensional Markov chain corresponding GI/M/1-type queue is considered with level and phase components. The connection with the embedded Markov chain for the queue GI/M/1 with phase-type input distribution means that transitions from states in level \(k\) are restricted to states in levels less than or equal to \(k+1\). The question is discussed when the level is asymptotically independent of the phase as level \(n\) becomes large. This property holds for quasi-birth-and-death processes. It is shown that it is possible to define boundary transition probabilities in such a way that the Markov chain with a given transition probability in the interior of the state space (matrix \(A\)) has independent (phase and level) components under the stationary distribution. The uniqueness of the solution is discussed. The authors claim that reducibility/irreducibility of matrix \(A\) should not be a fundamental property of a GI/M/1-type Markov chain. The proofs are based on the matrix analysis.