Transient queue size distribution solution of Geom/G/1 queue with feedback – a recursive method.

*(English)*Zbl 1300.90011Summary: This paper considers the Geom/G/1 queueing model with feedback according to a late arrival system with delayed access (LASDA). Using recursive method, this paper studies the transient property of the queue size from the initial state \(N(0^{+}) = i\). Some new results about the recursive expression of the transient queue size distribution at any epoch \(n^{+}\) and the recursive formulae of the equilibrium distribution are obtained. Furthermore, the recursive formulae of the equilibrium queue size distribution at epoch \(n^{-}\), and \(n\) are obtained, too. The important relations between stationary queue size distributions at different epochs are discovered (being different from the relations given in M/G/1 queueing system). The model discussed in this paper can be widely applied in all kinds of communications and computer network.

##### MSC:

90B22 | Queues and service in operations research |

93C55 | Discrete-time control/observation systems |

68M20 | Performance evaluation, queueing, and scheduling in the context of computer systems |

##### Keywords:

discrete time queue; equilibrium distribution; feedback; recursive expression; transient distribution
PDF
BibTeX
XML
Cite

\textit{C. Luo} et al., J. Syst. Sci. Complex. 22, No. 2, 303--312 (2009; Zbl 1300.90011)

Full Text:
DOI

##### References:

[1] | F. J. Li and R. H. Sun, Measurement-estimation approach to efficiency evaluation of bandwidth allocation scheme in ATM networks, Conference Record IEEE Internation Conference on Conmunications, 1997, 6(2): 131–142. |

[2] | H. Kobayashi and A. G. Konheim, Queueing models for computer communications system analysis, IEEE Trans, Com., 1977, COM-25(1): 2–28. · Zbl 0365.68064 · doi:10.1109/TCOM.1977.1093702 |

[3] | M. Hassan and M. Atiquzzaman, A delayed vacation model of an queue with setup time and its application to SVCC-based ATM networks, IEICE Trans, Commun., 1997, E80-B(2): 317–323. |

[4] | J. J. Hunter, Mathematical Techniques of Applied Probability (II), Academic Press, New York, 1983. · Zbl 0539.60065 |

[5] | L. Takacs, A single-server queue with feedback, Bell System Technique Journal, 1963, 6(42): 509–519. · Zbl 0117.36004 |

[6] | N. Tian, The queueing system Geometric / G / 1 with vacations, Appl. Math. and Comput., 1993, 6(7): 71–78. |

[7] | H. Takagi, Analysis of a discrete time queueing system with time-limited service, Queueing System Theory Appl., 1994, 6(18): 183–197. · Zbl 0812.90052 · doi:10.1007/BF01158781 |

[8] | M. M. Yu and Y. H. Tang, The queue-length distribution of M / G / 1 queueing system with feedback follows geometric distribution, Acta Electronia Sinica (in Chinese), 2007, 6(35): 275–278. |

[9] | Y. H. Tang and X. W. Tang, Queueing Theory-Foundation and Analysis Techniques, Science Press, Beijing, 2006. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.