### Abstract

The dominant pole approximation (DPA) is a classical analytic method to obtain from a generating function asymptotic estimates for its underlying coefficients. We apply DPA to a discrete queue in a critical many-sources regime, in order to obtain tail asymptotics for the stationary queue length. As it turns out, this regime leads to a clustering of the poles of the generating function, which renders the classical DPA useless, since the dominant pole is not sufficiently dominant. To resolve this, we design a new DPA method, which might also find application in other areas of mathematics, like combinatorics, particularly when Gaussian scalings related to the central limit theorem are involved.

Original language | English |
---|---|

Pages (from-to) | 211-236 |

Number of pages | 26 |

Journal | Queueing Systems |

Volume | 84 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 1 Dec 2016 |

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### Keywords

- Asymptotics
- Dominant pole approximation
- Heavy traffic
- Many sources
- QED regime
- Saddle point method

### Cite this

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*Queueing Systems*, vol. 84, no. 3-4, pp. 211-236. https://doi.org/10.1007/s11134-016-9499-5

**Dominant poles and tail asymptotics in the critical Gaussian many-sources regime.** / Janssen, A.J.E.M.; van Leeuwaarden, J.S.H.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Dominant poles and tail asymptotics in the critical Gaussian many-sources regime

AU - Janssen, A.J.E.M.

AU - van Leeuwaarden, J.S.H.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - The dominant pole approximation (DPA) is a classical analytic method to obtain from a generating function asymptotic estimates for its underlying coefficients. We apply DPA to a discrete queue in a critical many-sources regime, in order to obtain tail asymptotics for the stationary queue length. As it turns out, this regime leads to a clustering of the poles of the generating function, which renders the classical DPA useless, since the dominant pole is not sufficiently dominant. To resolve this, we design a new DPA method, which might also find application in other areas of mathematics, like combinatorics, particularly when Gaussian scalings related to the central limit theorem are involved.

AB - The dominant pole approximation (DPA) is a classical analytic method to obtain from a generating function asymptotic estimates for its underlying coefficients. We apply DPA to a discrete queue in a critical many-sources regime, in order to obtain tail asymptotics for the stationary queue length. As it turns out, this regime leads to a clustering of the poles of the generating function, which renders the classical DPA useless, since the dominant pole is not sufficiently dominant. To resolve this, we design a new DPA method, which might also find application in other areas of mathematics, like combinatorics, particularly when Gaussian scalings related to the central limit theorem are involved.

KW - Asymptotics

KW - Dominant pole approximation

KW - Heavy traffic

KW - Many sources

KW - QED regime

KW - Saddle point method

UR - http://www.scopus.com/inward/record.url?scp=84988710078&partnerID=8YFLogxK

U2 - 10.1007/s11134-016-9499-5

DO - 10.1007/s11134-016-9499-5

M3 - Article

AN - SCOPUS:84988710078

VL - 84

SP - 211

EP - 236

JO - Queueing Systems: Theory and Applications

JF - Queueing Systems: Theory and Applications

SN - 0257-0130

IS - 3-4

ER -