Roots, symmetry and contour integrals in queueing systems

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Abstract

Many queueing systems are analysed using the probability-generating-
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 languageEnglish
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages23
Publication statusPublished - May 2019

Publication series

NameTW-memoranda
PublisherUniversity of Twente, Department of Applied Mathematics
No.2067
ISSN (Print)1874-4850

Fingerprint

Contour integral
Queueing System
Probability generating function
Roots
Symmetry
Root-finding
Additive Function
Numerator
Necessary
Symmetric Functions
Random variable
Sufficient
Necessary Conditions
Polynomial
Sufficient Conditions

Cite this

Oblakova, A., Al Hanbali, A., Boucherie, R., van Ommeren, J. C. W., & Zijm, W. H. M. (2019). Roots, symmetry and contour integrals in queueing systems. (TW-memoranda; No. 2067). University of Twente, Department of Applied Mathematics.
Oblakova, Anna ; Al Hanbali, Ahmad ; Boucherie, Richard ; van Ommeren, Jan C.W. ; Zijm, Willem Hendrik Maria. / Roots, symmetry and contour integrals in queueing systems. University of Twente, Department of Applied Mathematics, 2019. 23 p. (TW-memoranda; 2067).
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abstract = "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.",
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Oblakova, A, Al Hanbali, A, Boucherie, R, van Ommeren, JCW & Zijm, WHM 2019, 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.

University of Twente, Department of Applied Mathematics, 2019. 23 p. (TW-memoranda; No. 2067).

Research output: Book/ReportReportOther research output

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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

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Oblakova A, Al Hanbali A, Boucherie R, van Ommeren JCW, Zijm WHM. Roots, symmetry and contour integrals in queueing systems. University of Twente, Department of Applied Mathematics, 2019. 23 p. (TW-memoranda; 2067).