Entropy Analysis of a Flexible Markovian Queue with Server Breakdowns.

Messaoud Bounkhel,Ramdane Hedjar,Lotfi Tadj

doi:10.3390/e22090979

Abstract

In this paper, a versatile Markovian queueing system is considered. Given a fixed threshold level c, the server serves customers one a time when the queue length is less than c, and in batches of fixed size c when the queue length is greater than or equal to c. The server is subject to failure when serving either a single or a batch of customers. Service rates, failure rates, and repair rates, depend on whether the server is serving a single customer or a batch of customers. While the analytical method provides the initial probability vector, we use the entropy principle to obtain both the initial probability vector (for comparison) and the tail probability vector. The comparison shows the results obtained analytically and approximately are in good agreement, especially when the first two moments are used in the entropy approach.

Highlights

The concept of entropy was introduced by Shannon in his seminal papers, Shannon [1].In information theory, entropy refers to a basic quantity associated to a random variable
Among a number of different probability distributions that express the current state of knowledge, the maximum entropy principle allows to choose the best one, that is the one with maximum entropy
The maximum entropy principle is used to derive the initial steady state probability vector and make sure it is in agreement with the one obtained by Bounkhel et al [10]

Summary

Introduction

The concept of entropy was introduced by Shannon in his seminal papers, Shannon [1]. In information theory, entropy refers to a basic quantity associated to a random variable. In queueing theory, a large number of papers has used the maximum entropy principle to determine the steady-state probability distribution of some process. The intention of this paper is to resume work on a paper started by Bounkhel et al [10], who studied a flexible queueing system and used an analytical method to obtain the initial steady state probability vector. The maximum entropy principle is used to derive the initial steady state probability vector and make sure it is in agreement with the one obtained by Bounkhel et al [10]. We use the maximum entropy principle to obtain the tail.

Model Formulation and Previous Results

Entropy Approach

Entropy Solution Using the First Moment

Entropy Solution Using the Second Moment

Entropy Solution Using Both First and Second Moments

Conclusions