### Abstract

A wide range of real life systems are modeled by queueing systems with finite capacity buffers. There are well established numerical procedures for the analysis of these queueing models when the load islower or higher than the system capacity, but these numerical methods become unstable as the loadgets close to the system capacity. We present simple modifications of the standard computational methodswhich remain numerically stable at saturation as well. We consider two specific Markov models: finite quasi birth death (QBD) processand finite Markov fluid queue (MFQ). The first one describes the behavior of queueing systems with discretebuffer content, while the second one describes the behavior of queueing systems with continuous buffer content. The stationary solution of a finite QBD process is a combination of two matrix geometric serieswhile the stationary fluid density of a finite MFQ is a combination of two matrix exponential functions. Apart of this there are several further similarities between the discrete and continuous buffer modelsat saturation. The proposed solution method exploits the similarities of the models.

Original language | English |
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Title of host publication | Proceedings - 2012 9th International Conference on Quantitative Evaluation of Systems, QEST 2012 |

Pages | 33-42 |

Number of pages | 10 |

DOIs | |

Publication status | Published - 2012 |

Event | 2012 9th International Conference on Quantitative Evaluation of Systems, QEST 2012 - London, United Kingdom Duration: Sep 17 2012 → Sep 20 2012 |

### Other

Other | 2012 9th International Conference on Quantitative Evaluation of Systems, QEST 2012 |
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Country | United Kingdom |

City | London |

Period | 9/17/12 → 9/20/12 |

### Fingerprint

### Keywords

- finite quasi birth death processes
- Markov fluid model
- matrix analytic methods
- matrix geometric solution

### ASJC Scopus subject areas

- Control and Systems Engineering

### Cite this

*Proceedings - 2012 9th International Conference on Quantitative Evaluation of Systems, QEST 2012*(pp. 33-42). [6354631] https://doi.org/10.1109/QEST.2012.20

**Finite queues at the limit of saturation.** / Telek, M.; Vécsei, Miklós.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings - 2012 9th International Conference on Quantitative Evaluation of Systems, QEST 2012.*, 6354631, pp. 33-42, 2012 9th International Conference on Quantitative Evaluation of Systems, QEST 2012, London, United Kingdom, 9/17/12. https://doi.org/10.1109/QEST.2012.20

}

TY - GEN

T1 - Finite queues at the limit of saturation

AU - Telek, M.

AU - Vécsei, Miklós

PY - 2012

Y1 - 2012

N2 - A wide range of real life systems are modeled by queueing systems with finite capacity buffers. There are well established numerical procedures for the analysis of these queueing models when the load islower or higher than the system capacity, but these numerical methods become unstable as the loadgets close to the system capacity. We present simple modifications of the standard computational methodswhich remain numerically stable at saturation as well. We consider two specific Markov models: finite quasi birth death (QBD) processand finite Markov fluid queue (MFQ). The first one describes the behavior of queueing systems with discretebuffer content, while the second one describes the behavior of queueing systems with continuous buffer content. The stationary solution of a finite QBD process is a combination of two matrix geometric serieswhile the stationary fluid density of a finite MFQ is a combination of two matrix exponential functions. Apart of this there are several further similarities between the discrete and continuous buffer modelsat saturation. The proposed solution method exploits the similarities of the models.

AB - A wide range of real life systems are modeled by queueing systems with finite capacity buffers. There are well established numerical procedures for the analysis of these queueing models when the load islower or higher than the system capacity, but these numerical methods become unstable as the loadgets close to the system capacity. We present simple modifications of the standard computational methodswhich remain numerically stable at saturation as well. We consider two specific Markov models: finite quasi birth death (QBD) processand finite Markov fluid queue (MFQ). The first one describes the behavior of queueing systems with discretebuffer content, while the second one describes the behavior of queueing systems with continuous buffer content. The stationary solution of a finite QBD process is a combination of two matrix geometric serieswhile the stationary fluid density of a finite MFQ is a combination of two matrix exponential functions. Apart of this there are several further similarities between the discrete and continuous buffer modelsat saturation. The proposed solution method exploits the similarities of the models.

KW - finite quasi birth death processes

KW - Markov fluid model

KW - matrix analytic methods

KW - matrix geometric solution

UR - http://www.scopus.com/inward/record.url?scp=84870734420&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870734420&partnerID=8YFLogxK

U2 - 10.1109/QEST.2012.20

DO - 10.1109/QEST.2012.20

M3 - Conference contribution

SN - 9780769547817

SP - 33

EP - 42

BT - Proceedings - 2012 9th International Conference on Quantitative Evaluation of Systems, QEST 2012

ER -