Multidimensional Queueing Models in Telecommunication Networks

Transcription

1 Multidimensional Queueing Models in Telecommunication Networks

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3 Agassi Melikov Leonid Ponomarenko Multidimensional Queueing Models in Telecommunication Networks

4 Agassi Melikov Department of Teletraffic Theory Institute of Cybernetics, National Academy of Sciences of Azerbaijan Baku Azerbaijan Leonid Ponomarenko International Research and Training Center for Information Technologies and Systems of the National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine Kiev Ukraine ISBN ISBN (ebook) DOI / Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Preface The increasing complexity of telecommunication networks puts at the forefront the problem of developing adequate mathematical models for them. The main goals are finding their characteristics, solving the problems of their optimization subject to chosen criteria, and developing the corresponding control algorithms. The basic mathematical tool that allows us to build both adequate analytical and numerical models of telecommunication networks is queueing theory. The core of this theory was founded more than 100 years ago in the pioneering work of Agner Erlang. He studied only the then recently telephony systems, but since then the models and methods of queueing theory have been widely used for studying service processes in various branches of science and industry, among them economics, manufacturing systems, military science, and transportation. Remarkably, the most important reason for studying queueing theory, now as well as 100 years ago, is telecommunication networks. However, there are many distinctions between the models of the past and modern telecommunication networks. We should first note that in classical Erlang s models, it was assumed that calls do not differ from each other, i.e., calls are identical. In other words, early telecommunication networks were queueing systems with single traffic. However, in modern telecommunication networks, the calls (messages) essentially differ from each other with respect to some parameters. For example, calls can vary in arrival intensity and/or processing time, in the level of priorities, in the service mechanism, etc. These facts show that classical queueing models with single traffic can serve only as rough (approximate) mathematical models of modern telecommunication networks. The functioning of modern telecommunication networks can be described only by means of queueing models with several types of traffic adequate models of modern telecommunication networks are multidimensional ones. Such kinds of models are especially useful for studying integrated networks in which real-time calls, for voice, video, etc., and non-real-time calls, for data, fax, , etc., are handled. Queueing models with single traffic are well studied, and they are described in well-known textbooks and monographs, but multidimensional queueing models are v

6 vi Preface insufficiently studied, and there are only a few monographs on this theme. This book is devoted to the problem of applying multidimensional Markov models in modern telecommunication networks. Unlike one-dimensional models, using exact and simple formulas to calculate the quality-of-service (QoS) metrics of multidimensional models is usually impossible. This is explained by the fact that in many cases the appropriate system of global balance equations (SGBE) for the steady-state probabilities has no explicit solutions e.g., the solution in the multiplicative form. In such cases, various numerical (exact or approximate) methods must be used. The classical approach to calculating the steady-state probabilities is based on the theory of multidimensional generating functions. However, there are wellknown associated computational difficulties because we must solve systems of partial differential equations and equations for boundary states as well. The alternative and more effective approach based on the use of SGBE for calculating the QoS metrics of multidimensional Markov models contains the following stages. First of all, note that this approach is used mainly for models with finite dimension of state space. In the first stage, the state of the system is defined, and the set of all possible states (state space) is formed. As a rule, the system s state is described by a vector of corresponding dimension. In the second stage, an infinitesimal matrix (Q-matrix) of the appropriate multidimensional Markov chain (MC) is constructed. It is known that constructing the Q-matrix is enough to develop the SGBE. In the third stage, steady-state probabilities are found from the SGBE. In the final stage, the desired QoS metrics are calculated via steady-state probabilities, i.e., the QoS metrics are determined as appropriate marginal distributions of the initial multidimensional MC. By taking into account the unique property of Markov models, it is possible that the stationary probability of a state represents part of the sojourn time of the system in a corresponding state for a large supervision time interval. The main problem in this approach is solving the SGBE, i.e., realizing the third stage, since the growing traffic and the increasing number of channels, as well as the buffer sizes of the corresponding telecommunication network, rapidly lead to an increase in the dimension of the state space. In some cases, by using the specific structure of a corresponding Q-matrix, it is possible to simplify this problem. So, for instance, it becomes simpler for networks that are described by models of reversible Markov chains since for such models the analytical solution in multiplicative form can be obtained. If the analytical solution of the SGBE is not available, as mentioned above, then various numerical methods are used. In this book, we use both approaches to investigate models of telecommunication networks. The book consists of five chapters. In Chap. 1, both single-rate and multi-rate Erlang s models are considered, and we examine known computational algorithms to calculate their QoS metrics. Here also we propose hybrid access schemes in mono-service cellular networks without buffers, and we develop both exact and approximate methods to calculate the QoS metrics of such access schemes. In Chap. 2, we develop an analytical method to investigate the models of multi-rate

7 Preface vii Erlang s models with randomized access schemes. We also consider analytical methods to study the models of multiservice cellular networks with various partition schemes of common radio channels. In Chap. 3, we investigate models of mono-service cellular networks. Two types of models are considered: models with either finite or infinite buffers for both types of new and handover calls and retrial models. Numerical algorithms to calculate their QoS metrics are developed. We investigate models of multiservice cellular networks with buffers in Chap. 4. Finally, in Chap. 5, we examine models of packet-switching networks with priorities. Here we examine in detail nonclassical priority schemes with multiple space and time priorities as well as jump priorities. Each chapter of the book contains results from numerical experiments carried out using the developed algorithms, and each chapter contains comments and references that allow the reader to understand the current situation in the corresponding areas of research. This book is recommended for researchers engaged with the mathematical theory of teletraffic. It will be useful for graduate and PhD students in informatics and applied mathematics as well as other fields. Baku, Azerbaijan Kiev, Ukraine Agassi Melikov Leonid Ponomarenko

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9 List of Acronyms and Abbreviations AV BDP BS CAC CLP CP CS CTD C u E B (ν, m) e i EV FCFS GC GM GPO h-call H-cell hd-call hv-call I(A) i-call Int(x) JP L-cell MC MP MRQ Approximate value Birth death process Base station Call admission control Cell loss probability Complete partition Complete sharing Cell transfer delay Channel utilization Erlang s B-formula for the model M/M/m/0 with load of ν erl Unit vector in direction i in a Euclidean space whose dimension is specified in each particular case Exact value First come first served Guard channels Generating matrix Generalized push-out Handover call High priority cell Handover data call Handover voice call Indicator function of the event A Call of type i or call from the traffic of type i Integer (or whole) part of x Jump priorities Low priority cell Markov chain Multiple priorities Multi rate queue ix

10 x MU NPO NRT o-call od-call ov-call P(A) p(x), π(x), ρ(x) PB PB i P h P o PO PS PSN q(x, y) QoS RT SEV SGBE SLBE SP SPO TP VP x!y x + ¼ max(0,x) λ h (μ h ) λ i (μ i ) λ o (μ o ) ν h ¼ λ h /μ h ν i ¼ λ i /μ i ν o ¼ λ o /μ o δði; jþ ¼ 1 ifi ¼ j, 0 ifi 6¼ j, List of Acronyms and Abbreviations Mobile user Non-push-out Non-real-time Original (new) call Original (new) data call Original voice call Probability of the event A Stationary probability of state x Probability of blocking Probability of blocking of calls of type i Probability of blocking (or dropping) of h-calls Probability of blocking of o-calls Push-out Partial sharing Packet switching network Transition intensity from state x to state y Quality of service Real-time Set of efficient values System of global balance equations System of local balance equations Space priorities Simple push-out Time priorities Virtual partitioning Transition from state x to state y Choice a maximum 0 and x Arrival (service) rate of handover calls Arrival (service) rate of type i calls Arrival (service) rate of new calls Load offered by h-calls Load offered by calls of type i Load offered by o-calls Kronecker s symbols Empty set Approximate equality

11 Contents 1 Multidimensional Loss Models of Queueing Systems and Their Applications in Telecommunication Networks Multi-flow Loss Models with Deterministic Access Schemes Applications of Multi-flow Loss Models in Mono-service Cellular Networks Model with Guard Channels Model with Cutoff Scheme Model with Hybrid Access Scheme Selection of Optimal Values for Parameters of Hybrid Access Scheme Numerical Results Conclusion References Analytical Methods for Analysis of Integral Cellular Networks Model of Multi-rate Queue with Randomized Access Scheme Model of Integral Wireless Network with Multi-parametric Access Scheme Numerical Results Models of Integral Cellular Networks with Partition of Channels Model with Complete Partition Model with Virtual Partition Numerical Results Conclusion References Algorithmic Methods for Analysis of Mono-service Cellular Networks Models of Mono-service Cellular Networks with Buffers Models with Buffers Models with Finite Buffers xi

12 xii Contents Models with Infinite Buffers Numerical Results Models with Retrial Calls Exact Method Approximate Method Numerical Results Conclusion References Algorithmic Methods for Analysis of Integral Cellular Networks Models with Buffering of Handover Non-Real-Time Calls Various Access Schemes to Channels Selection of Effective Values of Access Scheme Parameters Numerical Results Models with Buffering of Both New and Handover Non-Real-Time Calls Integrated Access Schemes to Channels and to Buffer Numerical Results Conclusion References Priority Schemes in Packet Switching Networks Space and Time Priorities in Switch Modeling Models of Switches with Typed Output Ports Space Priorities Based on Non-Push-Out Schemes Space Priorities Based on Push-Out Schemes Numerical Results Models of Switches with Common Output Ports Multiple Priorities Based on Non-Push-Out Scheme Multiple Priorities Based on Push-Out Scheme Numerical Results State-Dependent Jump Priorities Various Schemes to Determination of State-Dependent Jump Priorities Numerical Results Conclusion References Index