### Abstract

function (pgf) technique. This approach often leads to expressions in

terms of the (complex) roots of a certain equation. In this paper, we

show that it is not necessary to compute the roots in order to evaluate

these expressions. We focus on a certain class of pgfs with a rational form

and represent it explicitly using symmetric functions of the roots. These

functions can be computed using contour integrals.

We also study when the mean of the random variable corresponding

to the considered pgf is an additive function of the roots. In this case,

it may be found using one contour integral, which is more reliable than

the root-finding approach. We give a necessary and sufficient condition

for an additive mean. For example, the mean is an additive function

when the numerator of the pgf has a polynomial-like structure of a certain

degree, which means that the pgf can be represented in a special product

form. We also give a necessary and sufficient condition for the mean to

be independent of the roots.

Original language | English |
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Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 23 |

Publication status | Published - May 2019 |

### Publication series

Name | TW-memoranda |
---|---|

Publisher | University of Twente, Department of Applied Mathematics |

No. | 2067 |

ISSN (Print) | 1874-4850 |

### Fingerprint

### Cite this

*Roots, symmetry and contour integrals in queueing systems*. (TW-memoranda; No. 2067). University of Twente, Department of Applied Mathematics.

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*Roots, symmetry and contour integrals in queueing systems*. TW-memoranda, no. 2067, University of Twente, Department of Applied Mathematics.

**Roots, symmetry and contour integrals in queueing systems.** / Oblakova, Anna ; Al Hanbali, Ahmad ; Boucherie, Richard; van Ommeren, Jan C.W.; Zijm, Willem Hendrik Maria.

Research output: Book/Report › Report › Other research output

TY - BOOK

T1 - Roots, symmetry and contour integrals in queueing systems

AU - Oblakova, Anna

AU - Al Hanbali, Ahmad

AU - Boucherie, Richard

AU - van Ommeren, Jan C.W.

AU - Zijm, Willem Hendrik Maria

PY - 2019/5

Y1 - 2019/5

N2 - Many queueing systems are analysed using the probability-generating-function (pgf) technique. This approach often leads to expressions interms of the (complex) roots of a certain equation. In this paper, weshow that it is not necessary to compute the roots in order to evaluatethese expressions. We focus on a certain class of pgfs with a rational formand represent it explicitly using symmetric functions of the roots. Thesefunctions can be computed using contour integrals.We also study when the mean of the random variable correspondingto the considered pgf is an additive function of the roots. In this case,it may be found using one contour integral, which is more reliable thanthe root-finding approach. We give a necessary and sufficient conditionfor an additive mean. For example, the mean is an additive functionwhen the numerator of the pgf has a polynomial-like structure of a certaindegree, which means that the pgf can be represented in a special productform. We also give a necessary and sufficient condition for the mean tobe independent of the roots.

AB - Many queueing systems are analysed using the probability-generating-function (pgf) technique. This approach often leads to expressions interms of the (complex) roots of a certain equation. In this paper, weshow that it is not necessary to compute the roots in order to evaluatethese expressions. We focus on a certain class of pgfs with a rational formand represent it explicitly using symmetric functions of the roots. Thesefunctions can be computed using contour integrals.We also study when the mean of the random variable correspondingto the considered pgf is an additive function of the roots. In this case,it may be found using one contour integral, which is more reliable thanthe root-finding approach. We give a necessary and sufficient conditionfor an additive mean. For example, the mean is an additive functionwhen the numerator of the pgf has a polynomial-like structure of a certaindegree, which means that the pgf can be represented in a special productform. We also give a necessary and sufficient condition for the mean tobe independent of the roots.

M3 - Report

T3 - TW-memoranda

BT - Roots, symmetry and contour integrals in queueing systems

PB - University of Twente, Department of Applied Mathematics

ER -