Analysis of Markov-modulated fluid polling systems with gated discipline

Zsolt Saffer; Miklós Telek; Gábor Horváth

doi:10.3934/jimo.2019124

Article Contents

2021, Volume 17, Issue 2: 575-599. Doi: 10.3934/jimo.2019124

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Analysis of Markov-modulated fluid polling systems with gated discipline

1.
Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria
2.
MTA-BME Information Systems Research Group, Magyar Tudósok Körútja 2, 1117 Budapest, Hungary
3.
Department of Networked Systems and Services, Budapest University of Technology and Economics, Magyar Tudósok Körútja 2, 1117 Budapest, Hungary

^* Corresponding author
^* Corresponding author

Received: October 2018

Revised: March 2019

Early access: October 2019

Published: March 2021

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Abstract

In this paper we present two different analytical descriptions of the fluid polling model with Markov modulated load and gated discipline. The fluid arrival to the stations is modulated by a common continuous-time Markov chain (the special case when the modulating Markov chains are independent is also included). The fluid is removed at the stations during the service period by a station dependent constant rate.

The first analytical description is based on the relationships of steady-state fluid levels at embedded server arrival and departure epochs. We derive the steady-state vector Laplace transform of the fluid levels at the stations at arbitrary epoch and its moments. The second analytical description applies the method of supplementary variables and results in differential equations, from which the joint density function of the fluid levels can be obtained.

We also propose computational methods for both analytical descriptions and provide numerical examples to illustrate the numeric computations.

Keywords:

Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 60J25.

Citation:

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Figure 1. The joint distribution of the fluid level and the one-dimensional marginals

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Figure 2. The joint distribution of the fluid level at polling and at departure epochs

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Table 1. Steady-state vector moments of the fluid levels at polling epochs

	1st moment	1st moment	2nd moment	2nd moment
	element 0	element 1	element 0	element 1
Station 1:	1.0614	0.7386	2.1640	1.7821
Station 2:	2.1759	0.7170	8.3775	2.2387

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Table 2. Mean fluid levels of the queue in different phases of the server

	St. 1. busy	St. 1. vacation	St. 2. busy	St. 2. vacation
Station 1:	7.559	5.827	7.861	9.418
Station 2:	3.915	5.932	4.362	2.194

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References

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