Queueing Systems

, 69:175 | Cite as

Queues with boundary assistance: the effects of truncation

  • Guy Latouche
  • Giang T. NguyenEmail author
  • Peter G. Taylor


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.


QBD processes Infinite-phase QBD Truncation and augmentation Queueing theory Boundary assistance 

Mathematics Subject Classification (2000)

60J27 68M20 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Guy Latouche
    • 1
  • Giang T. Nguyen
    • 1
    Email author
  • Peter G. Taylor
    • 2
  1. 1.Département d’InformatiqueUniversité Libre de BruxellesBruxellesBelgium
  2. 2.Department of Mathematics and StatisticsUniversity of MelbourneVictoriaAustralia

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