On extreme values in queues in series

Saulius Minkevičius,Edvinas Greičius

doi:10.21595/mme.2017.17808

Abstract

A model of queues in series under conditions of heavy traffic is developed in this paper. This is a mathematical model to measure performance of complex computer networks operating under conditions of heavy traffic. Two limit theorems are derived by investigating extreme values of a virtual waiting time of customers in queues in series. Due to serious technical difficulties, research does not often consider intermediate models of queues in series. Note that the research of extreme values in more specific systems than the classical example GI/G/N (multiserver queue, queues in series, etc.) was introduced only 20 years ago [1]. There are many real-world applications at various hierarchical levels for both queues in series and tandem queues, for example, in high-speed communication networks (from architecture of the router to protocol stacks [2]).

Highlights

A brief review of the latest achievements in the theory of queues in series after the year of 2005 is provided in this paper
A model of queues in series under conditions of heavy traffic is developed in this paper
Two limit theorems are derived by investigating extreme values of a virtual waiting time of customers in queues in series

Summary

Introduction

A brief review of the latest achievements in the theory of queues in series after the year of 2005 is provided in this paper. Functional limit theorems for extreme values of the main characteristics of queues in series are proved in heavy traffic (maximum of the total virtual waiting time of a customer, maximum of the virtual waiting time of a customer). The main tool used in this paper for the analysis of queues in series in heavy traffic is functional limit theorems for renewal and compound renewal processes (the proof can be found in Billingsley, [7]). The natural setting for functional limit theorems is the weak convergence of probability measures on the function space [0,1](≡ ). When is a constant function (not random), the convergence in probability is equivalent to a weak convergence In such cases, we write ( , ) ⇒ 0 or ⇒.

Statement of the problem

The main results

Final note on the theorems

Computational simulation

Findings

Conclusions