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

A linear programming approach to error bounds for random walks in the quarter-plane

“…Proof: For t = 0 (19) holds since F 0 (n) = 0 for all n ∈ S and u * ≤ µ. The proof now follows from a simple induction on t by verifying all eight cases in (13) and (14).…”

“…It was shown in [1] that for continuous-time Markov processes in the quarter plane, such perturbations can always be found. A related result was presented in [14,12] for a (discrete-time) QBD processes that satisfy a technical condition. In [4] the existence of such perturbations is demonstrated for all random walks in the quarter-plane.…”

On quadratic matrix equations with infinite size coefficients encountered in QBD stochastic processes

“…Assume that the Markov chain having the transition matrix P of Equation 1 is irreducible. According to the works by Latouche et al 5 and Tweedie, 11 such Markov chain is positive recurrent if and only if…”

“…Here, we are interested in the important case from the applications where the process is positive recurrent. In this case, the vector π exists and is unique, the matrix G is stochastic, that is, G 1 = 1 , where 1 is the vector of all ones, and the series is convergent …”

“…In this case, the vector exists and is unique, the matrix G is stochastic, that is, G1 = 1, where 1 is the vector of all ones, and the series ∑ ∞ i=0 R i is convergent. 5 In this paper, we consider matrix equations of the kinds from Equations 3 and 4, where the matrix coefficients are semi-infinite matrices belonging to a Banach algebra. A case of interest is the one in which the matrices are quasi-Toeplitz.…”