# American Institute of Mathematical Sciences

March  2021, 17(2): 575-599. doi: 10.3934/jimo.2019124

## 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

Received  October 2018 Revised  March 2019 Published  October 2019

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.

Citation: Zsolt Saffer, Miklós Telek, Gábor Horváth. Analysis of Markov-modulated fluid polling systems with gated discipline. Journal of Industrial & Management Optimization, 2021, 17 (2) : 575-599. doi: 10.3934/jimo.2019124
##### References:

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##### References:
The joint distribution of the fluid level and the one-dimensional marginals
The joint distribution of the fluid level at polling and at departure epochs
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
 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
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
 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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