## 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.

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 |
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Publisher | University of Twente, Department of Applied Mathematics |

No. | 2067 |

ISSN (Print) | 1874-4850 |