# Order relations in the set of probability distribution functions and their applications in queueing theory

• Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1976

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## Abstract

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CONTENTSIntroduction......................................................................................................................................... 51. n-Monotonic functions on (— ∞, ∞)........................................................................................... 62. Order relations in the set of probability distribution functions....................................................... 12 2.1. Preliminary concepts............................................................................................................ 12 2.2. Relations ${\le }_{1.n},{\le }_{2.n}$............................................................................................. 13 2.3. Extremal probability distribution functions........................................................................ 17 2.4. Relations ${\le }_{2.0},{\le }_{2.0}$............................................................................................. 18 2.5. Isotonic operators................................................................................................................. 22 2.6. Remarks about quasi-ordering relations in the set of random variables.................. 26 3. Order relationship between queueing systems................................................................. 26 3.1. Preliminary concepts, $G{I}^{\left(x\right)}/{G}^{\left(y\right)}/1$ queues.......................................................... 26 3.2. $G{I}^{\left(x\right)}/G/1$ queues........................................................................................................... 27 3.3. Order relationship between $G{I}^{\left(x\right)}/{M}^{\left(y\right)}/1$ and ${M}^{\left(x\right)}/{G}^{\left(y\right)}/1$ queues...... 30 4. Bounds for $G{I}^{\left(x\right)}/{G}^{\left(y\right)}/1$ queues................................................................................ 32 4.1. Introduction............................................................................................................................. 32 4.2. Bounds for $G{I}^{\left(x\right)}/{G}^{\left(y\right)}/1$ queues ........................................................................... 33 4.3. Bounds for $G{I}^{\left(x\right)}/{M}^{\left(y\right)}/1,{M}^{\left(x\right)}/{G}^{\left(y\right)}/1$ queues............................................... 36 4.4. Application of the relations ${\le }_{1.n}{\le }_{2.n}$ in queues............................................ 37Appendix...................................................................................................................................................... 38References.................................................................................................................................................. 46

## How to cite

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Tomasz Rolski. Order relations in the set of probability distribution functions and their applications in queueing theory. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1976. <http://eudml.org/doc/268532>.

@book{TomaszRolski1976,
abstract = {CONTENTSIntroduction......................................................................................................................................... 51. n-Monotonic functions on (— ∞, ∞)........................................................................................... 62. Order relations in the set of probability distribution functions....................................................... 12 2.1. Preliminary concepts............................................................................................................ 12 2.2. Relations $≤_\{1.n\}, ≤_\{2.n\}$............................................................................................. 13 2.3. Extremal probability distribution functions........................................................................ 17 2.4. Relations $≤_\{2.0\}, ≤_\{2.0\}$............................................................................................. 18 2.5. Isotonic operators................................................................................................................. 22 2.6. Remarks about quasi-ordering relations in the set of random variables.................. 26 3. Order relationship between queueing systems................................................................. 26 3.1. Preliminary concepts, $GI^\{(x)\}/G^\{(y)\}/1$ queues.......................................................... 26 3.2. $GI^\{(x)\}/G/1$ queues........................................................................................................... 27 3.3. Order relationship between $GI^\{(x)\}/M^\{(y)\}/1$ and $M^\{(x)\}/G^\{(y)\}/1$ queues...... 30 4. Bounds for $GI^\{(x)\}/G^\{(y)\}/1$ queues................................................................................ 32 4.1. Introduction............................................................................................................................. 32 4.2. Bounds for $GI^\{(x)\}/G^\{(y)\}/1$ queues ........................................................................... 33 4.3. Bounds for $GI^\{(x)\}/M^\{(y)\}/1, M^\{(x)\}/G^\{(y)\}/1$ queues............................................... 36 4.4. Application of the relations $≤_\{1.n\} ≤_\{2.n\}$ in queues............................................ 37Appendix...................................................................................................................................................... 38References.................................................................................................................................................. 46},
author = {Tomasz Rolski},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Order relations in the set of probability distribution functions and their applications in queueing theory},
url = {http://eudml.org/doc/268532},
year = {1976},
}

TY - BOOK
AU - Tomasz Rolski
TI - Order relations in the set of probability distribution functions and their applications in queueing theory
PY - 1976
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - CONTENTSIntroduction......................................................................................................................................... 51. n-Monotonic functions on (— ∞, ∞)........................................................................................... 62. Order relations in the set of probability distribution functions....................................................... 12 2.1. Preliminary concepts............................................................................................................ 12 2.2. Relations $≤_{1.n}, ≤_{2.n}$............................................................................................. 13 2.3. Extremal probability distribution functions........................................................................ 17 2.4. Relations $≤_{2.0}, ≤_{2.0}$............................................................................................. 18 2.5. Isotonic operators................................................................................................................. 22 2.6. Remarks about quasi-ordering relations in the set of random variables.................. 26 3. Order relationship between queueing systems................................................................. 26 3.1. Preliminary concepts, $GI^{(x)}/G^{(y)}/1$ queues.......................................................... 26 3.2. $GI^{(x)}/G/1$ queues........................................................................................................... 27 3.3. Order relationship between $GI^{(x)}/M^{(y)}/1$ and $M^{(x)}/G^{(y)}/1$ queues...... 30 4. Bounds for $GI^{(x)}/G^{(y)}/1$ queues................................................................................ 32 4.1. Introduction............................................................................................................................. 32 4.2. Bounds for $GI^{(x)}/G^{(y)}/1$ queues ........................................................................... 33 4.3. Bounds for $GI^{(x)}/M^{(y)}/1, M^{(x)}/G^{(y)}/1$ queues............................................... 36 4.4. Application of the relations $≤_{1.n} ≤_{2.n}$ in queues............................................ 37Appendix...................................................................................................................................................... 38References.................................................................................................................................................. 46
LA - eng
UR - http://eudml.org/doc/268532
ER -

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