On the stability of queues with the dropping function.

Andrzej Chydzinski,Maurizio Naldi

doi:10.1371/journal.pone.0259186

Andrzej Chydzinski, Maurizio Naldi

Open Access

https://doi.org/10.1371/journal.pone.0259186

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Journal: PloS one	Publication Date: Nov 3, 2021
Citations: 1	License type: CC BY 4.0

Affiliation: Silesian University of Technology

Abstract

In this paper, the stability of the queueing system with the dropping function is studied. In such system, every incoming job may be dropped randomly, with the probability being a function of the queue length. The main objective of the work is to find an easy to use condition, sufficient for the instability of the system, under assumption of Poisson arrivals and general service time distribution. Such condition is found and proven using a boundary for the dropping function and analysis of the embedded Markov chain. Applicability of the proven condition is demonstrated on several examples of dropping functions. Additionally, its correctness is confirmed using a discrete-event simulator.

Highlights

We study the classic queueing model with a single server, which exploits a function d : N0 ! 1⁄20; 1, assigning the probability of dropping an arriving job to the length of the queue upon the new arrival
If d(M) = 1 for some M, the queueing system of interest is equivalent to a finite-buffer system with the buffer of capacity M, which is known to be stable, no matter with or without the dropping function
We have proven a condition sufficient for the instability of a queue with the dropping function, Poisson arrivals and general service time distribution, i.e. for the M/G/1 system in the Kendall’s notation, with the dropping function added

Summary

Introduction

We study the classic queueing model with a single server, which exploits a function d : N0 ! 1⁄20; 1 , assigning the probability of dropping an arriving job to the length of the queue upon the new arrival. We study the classic queueing model with a single server, which exploits a function d : N0 ! 1⁄20; 1 , assigning the probability of dropping an arriving job to the length of the queue upon the new arrival. Function d is called the dropping function. The well-known area of applications of queueing models with the dropping function is management of packet buffers in TCP/IP networks. Several types of mathematical functions have been analyzed as candidates for the dropping function. The list begins with the simple linear function [1], goes through the doubly linear [2], quadratic [3], cubic [4], the exponential one, [5], and a composition of linear and cubic functions [6]. A product of a linear function with its logarithm has been considered, [7]

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