Queues with boundary assistance: The effects of truncation (Q660149)

The paper deals with a system of two queues with boundary assistance, whose dynamics is given by a continuous-time Quasi-Birth-and-Death process (QBD). According to the used state description this QBD has a `doubly infinite' number of phases. The authors determine the convergence norm of Neuts' \(R\) matrix and consequently, the interval in which the decay rate of the infinite system can lie. Further four sequences of finite-phase approximations to the original system in which the \(N-\)th approximation has \(2N+1\) phases are considered. One is derived by truncating the infinite system without augmentation, the others are obtained by using different augmentation schemes ensuring that the generator of the QBD remains conservative. The sequences of matrices \(R_N\) for the truncated system without augmentation and one of the sequences with augmentation have monotonically increasing spectral radii that approach the convergence norm of the infinite-phase \(R\) as the truncation point tends to infinity; the two other sequences of matrices \(\{R_N\}\) have spectral radii that are constant irrespective of the truncation size, and not equal to the convergence norm of the infinite \(R\).