Queueing Theory 1

Advanced Trends

Vladimir Anisimov, Nikolaos Limnios

• Publisher: Wiley-ISTE
• Year: 2021
• ISBN:
  9781119755425
• Edition: 1
• Topic: Matematica
• Subtopic: Probabilità e statistica

The aim of this book is to reflect the current cutting-edge thinking and established practices in the investigation of queueing systems and networks.

This first volume includes ten chapters written by experts well-known in their areas. The book studies the analysis of queues with interdependent arrival and service times, characteristics of fluid queues, modifications of retrial queueing systems and finite-source retrial queues with random breakdowns, repairs and customers' collisions. Some recent tendencies in the asymptotic analysis include the average and diffusion approximation of Markov queueing systems and networks, the diffusion and Gaussian limits of multi-channel queueing networks with rather general input flow, and the analysis of two-time-scale nonhomogenous Markov chains using the large deviations principle.

The book also analyzes transient behavior of infinite-server queueing models with a mixed arrival process, the strong stability of queueing systems and networks, and applications of fast simulation methods for solving high-dimension combinatorial problems.

1
Discrete Time Single-server Queues with Interdependent Interarrival and Service Times

Attahiru Sule ALFA

University of Manitoba, Canada and University of Pretoria, South Africa

Discrete time single-server queues in which the interarrival and service times are interdependent are presented. First, we study the simple Geo/Geo/1 system, let the interarrival times depend on the service times and then consider special cases. The idea is then extended to the PH/PH/1 system. We then consider the case where the interarrival times are constructed from a combination of a set of interarrival times driven by the service times distribution. The initiating vector for the resulting combined PH distribution for the interarrival times is constructed as a function of the service times distribution, so that any changes in the service time distributions are reflected in this initiating vector. We capitalize on the structures of discrete phase type distributions in generalizing the resulting interarrival times. Finally, we introduce a general case where there is interdependence between service and interarrival times. We present a generalized matrix version of the bivariate geometric distribution, which can be used to capture some interdependence between the interarrival and service times.

1.1. Introduction

Most of the queueing models in the literature usually assume that the interarrival times and service times are independent of each other. While this may be true in some instances, it was pointed out by Cidon et al. (1993) that there are instances where there are dependencies between the two distributions. They mentioned that in communication systems where finite speed of communications links constrain the amount of data that can be received, one finds the interarrival times and service times to be correlated. They studied communication systems in which the interarrival time between nth and

st customer depends naturally on the service time Bn of the

customer. In a subsequent paper, Cidon et al. (1996) studied queues with interarrival tim...

Table of contents

• Cover
• Table of Contents
• Title Page
• Copyright
• Preface
• 1 Discrete Time Single-server Queues with Interdependent Interarrival and Service Times
• 2 Busy Period, Congestion Analysis and Loss Probability in Fluid Queues
• 3 Diffusion Approximation of Queueing Systems and Networks
• 4 First-come First-served Retrial Queueing System by Laszlo Lakatos and its Modifications
• 5 Parameter Mixing in Infinite-server Queues
• 6 Application of Fast Simulation Methods of Queueing Theory for Solving Some High-dimension Combinatorial Problems
• 7 Diffusion and Gaussian Limits for Multichannel Queueing Networks
• 8 Recent Results in Finite-source Retrial Queues with Collisions
• 9 Strong Stability of Queueing Systems and Networks: a Survey and Perspectives
• 10 Time-varying Queues: a Two-time-scale Approach
• List of Authors
• Index
• End User License Agreement