Analysis of queuing systems with threshold renovation mechanism and inverse service discipline

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Abstract

The paper presents a study of three queuing systems with a threshold renovation mechanism and an inverse service discipline. In the model of the first type, the threshold value is only responsible for activating the renovation mechanism (the mechanism for probabilistic reset of claims). In the second model, the threshold value not only turns on the renovation mechanism, but also determines the boundaries of the area in the queue from which claims that have entered the system cannot be dropped. In the model of the third type (generalizing the previous two models), two threshold values are used: one to activate the mechanism for dropping requests, the second - to set a safe zone in the queue. Based on the results obtained earlier, the main time-probabilistic characteristics of these models are presented. With the help of simulation modeling, the analysis and comparison of the behavior of the considered models were carried out.

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1. Introduction According to [1] the problem of congestion avoidance for communication networks does not have a satisfying solution, so the development and the analysis of new active queue management (AQM) algorithms appears to be the actual task for researches [2]-[13] and practitioners [14]-[24]. In this paper we will consider queuing systems with probabilistic renovation mechanism, which allows to adjust the number of packets in the system by dropping (resetting) them from the queue depending on the ratio of a certain control parameter with specified thresholds [25], [26] at the moment of the end of service on the device (server) [27]-[29] in contrast to standard RED algorithm, when a possible reset occurs at the time of the next packet arrival and the control parameter is an exponentially weighted average queue length [30]-[34]. In our models the renovation mechanism uses one or two thresholds (which determine as the place in the buffer from which the packets are dropped, but also the place to which the reset of packets occurs). The previous works devoted to the analysis of queuing systems with threshold based renovation are [35]-[38]. In [35], [36] some aspects of using the renovation mechanism (different types of renovation, definitions and brief overview were also given) with one or several thresholds as the mathematical models of active queue management mechanisms were considered. Some results of comparing classic RED algorithm with renovation mechanism were presented. In [37] two queuing models with threshold based renovation mechanism were presented: in the first model the threshold value is only responsible for activating the renovation mechanism (the mechanism for probabilistic reset of claims), in the second model the threshold value not only turns on the renovation mechanism, but also determines the boundaries of the area in the queue from which claims that have entered the system cannot be dropped. In [38] the queuing system with two threshold values (one to activate the mechanism for dropping requests, the second - to set a safe zone in the queue) for renovation mechanism was investigated. All three queuing systems have been studied for the service discipline FCFS (First Come First Served), and in this article we will present some results for the discipline LCFS (Last Come First Served). The study will again be carried out using embedded Markov chains. We will not consider in detail the derivation of the stationary distribution of the number of customers (which does not depend on the service discipline and presented in [37], [38]) and will focus only on the service (reset) probabilities and on time characteristics. The structure of the article is following. In the section 2 the results for the queuing model, where the threshold value is only responsible for activating the renovation mechanism, are presented; the section 3 is devoted to the queuing model, in which the threshold value not only turns on the renovation mechanism, but also determines the boundaries of the area in the queue from which claims that have entered the system cannot be dropped. In section 4 the characteristics for the queuing system with two threshold values (one to activate the mechanism for dropping requests, the second - to set a safe zone in the queue) for renovation mechanism are presented. In section 5 the results of GPSS simulation are considered. The last section 6 concludes the paper with the short discussion. 2. The first model Consider the
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About the authors

Ivan S. Zaryadov

Peoples’ Friendship University of Russia (RUDN University); Institute of Informatics Problems, FRC CSC RAS

Email: zaryadov-is@rudn.ru
ORCID iD: 0000-0002-7909-6396

Candidate of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); Senior Researcher of Institute of Informatics Problems of Federal Research Center “Computer Science and Control” Russian Academy of Sciences

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 44-2, Vavilova St., Moscow 119333, Russian Federation

Hilquias C. C. Viana

Peoples’ Friendship University of Russia (RUDN University)

Email: hilvianamat1@gmail.com
PHD student of Department of Applied Probability and Informatics 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Tatiana A. Milovanova

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: milovanova-ta@rudn.ru
ORCID iD: 0000-0002-9388-9499

Candidate of Physical and Mathematical Sciences, Lecturer of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

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Copyright (c) 2022 Zaryadov I.S., Viana H.C., Milovanova T.A.

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