Abstract
We study a system of two queues with boundary assistance, represented as a continuous-time Quasi-Birth-and-Death process (QBD). Under our formulation, this QBD has a ‘doubly infinite’ number of phases. We determine the convergence norm of Neuts’ R-matrix and, consequently, the interval in which the decay rate of the infinite system can lie.
We next consider four sequences of finite-phase approximations to the original system in which the Nth approximation has 2N+1 phases; one is derived by truncating the infinite system without augmentation, the others are obtained by using different augmentation schemes that ensure 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.
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Latouche, G., Nguyen, G.T. & Taylor, P.G. Queues with boundary assistance: the effects of truncation. Queueing Syst 69, 175–197 (2011). https://doi.org/10.1007/s11134-011-9255-9
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DOI: https://doi.org/10.1007/s11134-011-9255-9