Tra c Modelling of Cellular Mobile Communications ... - CiteSeerX

Trac Modelling of Cellular Mobile Communications Systems Operating with Code Division Multiple Access

A dissertation submitted to the University of Melbourne in partial ful llment of the requirements for the degree of Master of Engineering Science

Jamie Scott Evans Department of Electrical and Electronic Engineering University of Melbourne July 1995

Abstract This thesis concerns the trac modelling of cellular mobile communications networks operating with code division multiple access (CDMA). Such systems have come into prominence in recent years in conjunction with the development and validation of the Qualcomm CDMA cellular system. Most research has concentrated on transmission issues and existing capacity analyses inevitably assume xed and spatially uniform user distributions. However such capacity speci cation provides little information on how the network will behave under a stochastically varying load. This trac behaviour is critical in order to properly design and manage a network but to date, there have been few contributions in this area. The goal of this thesis is to develop analytic techniques for trac modelling that can be used in the performance analysis and design of CDMA cellular networks. As a rst step, each cell is modelled as an independent M=G=1 queue and trac capacity assessed based on the maximum Erlang trac that leads to acceptable link quality with a certain high probability. In practice, the network operator would control access to the network so as to maintain tighter service quality contraints and it is important to nd sensible control procedures. The second trac modelling framework of this thesis centres on a robust call admission strategy that handles variability in a similar manner to eective bandwidth techniques from ATM-based Broadband ISDN. This leads to trac models that can be taken directly from the theory of circuit-switched networks operating with xed routing. Indeed the major contribution of this thesis is the demonstration that CDMA cellular networks operating with simple power control schemes, exible call admission control, and supporting multiple classes of user with bursty bit rate and modem requirements, can be reduced to this well known circuit-switched form.

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Acknowledgements I am extremely grateful to the Commonwealth Government for supporting my study with an Australian Postgraduate Award and to Telecom Research Laboratories for their generous Postgraduate Fellowship. I would like to thank several people for contibuting to the contents of this thesis, most notably my supervisor, David Everitt. David supplied a great topic and provided many insightful comments and ideas along the way. In particular, the guts of Chapter 5 is based on a similarity between ATM and CDMA networks brought to my attention by David. Phil Whiting, with endless energy and enthusiasm, provided a lot of motivation in the early stages, supplying me with a host of useful references and introducing me to large deviations theory. Thanks also to the other students in the Communications Networks group for many useful and stimulating discussions. I would especially like to thank Jason Choong for sharing a windowless room with me and for letting me win at golf. Most importantly I would like to thank my pet dung beetle for endless hours of stimulating conversation and for putting a great deal of time into the production of the diagram on page six. Of course the city of Melbourne deserves special mention for providing golf courses, cafes, book shops, cinemas and weather unrivalled by any other Victorian city thats name starts with M. Finally, I thank my family and friends for their continual support.

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Contents Abstract

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Acknowledgements

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1 Introduction

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2 Mobile Cellular Fundamentals

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1.1 Problem Motivation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 Summary and Contributions : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.3 List of Publications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3

2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : 2.2 Evolution of Mobile Cellular Systems : : : : : : : : : : : 2.3 Trac Modelling of Mobile Cellular Systems : : : : : : 2.3.1 Performance Measures : : : : : : : : : : : : : : 2.3.2 Channel Allocation Strategies : : : : : : : : : : 2.3.3 Trac Analysis of Channel Allocation Strategies 2.4 CDMA for Mobile Cellular Systems : : : : : : : : : : : 2.4.1 Spread Spectrum Communication : : : : : : : : 2.4.2 Spread Spectrum Multiple Access : : : : : : : : : 2.4.3 Mobile Cellular CDMA : : : : : : : : : : : : : : 2.4.4 Capacity Calculations : : : : : : : : : : : : : : : 2.5 Trac Modelling for Cellular CDMA : : : : : : : : : :

3 Other-Cell Interference

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Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Propagation Issues : : : : : : : : : : : : : : : : : : : : : : System Model : : : : : : : : : : : : : : : : : : : : : : : : : Interference Calculations : : : : : : : : : : : : : : : : : : : 3.4.1 Deterministic Path Loss: One-Dimensional Network 3.4.2 Deterministic Path Loss: Two-Dimensional Network 3.4.3 Inclusion of Lognormal Shadowing : : : : : : : : : : 3.5 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : :

3.1 3.2 3.3 3.4

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4 M=G=1 Models

4.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4.2 The M=G=1 Approximation : : : : : : : : : : : : : : : : : : : : : 4.2.1 Basic Model : : : : : : : : : : : : : : : : : : : : : : : : : : : 4.2.2 Inclusion of Voice Activity Eects : : : : : : : : : : : : : : 4.3 Modi ed Blocking Probability : : : : : : : : : : : : : : : : : : : : 4.3.1 De nition : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4.3.2 Interference : : : : : : : : : : : : : : : : : : : : : : : : : : : 4.3.3 Example : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4.4 Blocking Probability Approximations : : : : : : : : : : : : : : : : : 4.4.1 Normal Approximation : : : : : : : : : : : : : : : : : : : : 4.4.2 Large Deviations Bound : : : : : : : : : : : : : : : : : : : : 4.5 Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4.5.1 Network Models : : : : : : : : : : : : : : : : : : : : : : : : 4.5.2 Comparison of Bound and Approximation with Simulation 4.5.3 Variation of Bound with System Parameters : : : : : : : : : 4.6 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

5 Eective Interference and Product Form

5.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.2 Eective Bandwith Based Admission Control : : : : : : : : 5.2.1 Calls with Constant Resource Requirement : : : : 5.2.2 Calls with Variable Resource Requirement : : : : : 5.3 Eective Interference Based Admission Control : : : : : : : 5.3.1 Single Cell Models : : : : : : : : : : : : : : : : : : 5.3.2 Multiple Cell Models: System State Feasibility : : 5.3.3 Multiple Cell Models: Network State Admissibility 5.3.4 Multiple Cell Models: Examples : : : : : : : : : : 5.4 Trac Modelling and Control Issues : : : : : : : : : : : : 5.5 Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.6 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : :

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6 Conclusion

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Bibliography

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List of Figures 2.1 A Cellular Network with a 7 Cell Frequency Reuse Pattern : : : : : : : : : 6 2.2 Baseband DS-SS Transmitter and Receiver : : : : : : : : : : : : : : : : : : 11 2.3 Diversity in Cellular CDMA : : : : : : : : : : : : : : : : : : : : : : : : : : 13 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Mobile Propagation Environment : : : : : : Standard 1-D Cellular Layout : : : : : : : : Standard 2-D Cellular Layout : : : : : : : : Interference in 1-D Network : : : : : : : : : Interference in 2-D Network : : : : : : : : : Interference in 2-D Network - Approximation Con rmation of Analytical Result : : : : : : Approximating the Hexagonal Cells : : : : : Comparison of Approximations : : : : : : : : Approximation versus Simulation : : : : : : :

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Blocking Probability vs Normalised Trac per Cell Blocking Probability vs Normalised Trac per Cell Variation of Blocking Probability with PLE : : : : Variation of Blocking Probability with ? : : : : : : Variation of Blocking Probability with PLE : : : : Variation of Blocking Probability with ? : : : : : :

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Types of Interference : : : : : : : : : : : : : : : : : : : : : : Seven Cell Network of Example : : : : : : : : : : : : : : : : Normalised Eective Interference Values for Type 2 Mobile Normalised Eective Interference Values for Type 3 Mobile Normalised Eective Interference Values for Type 2 Mobile Normalised Eective Interference Values for Type 3 Mobile

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List of Tables 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Normalised Eective Interference Values for 3 Call Types : Normalised Eective Interference Values for 3 Call Types : Normalised Eective Interference Values for 3 Call Types : Normalised Eective Interference Values for 3 Call Types : Normalised Eective Interference Values for Type 1 Mobile Normalised Eective Interference Values for Type 1 Mobile Reuse Factors : : : : : : : : : : : : : : : : : : : : : : : : : : Reuse Factors : : : : : : : : : : : : : : : : : : : : : : : : : :

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Chapter 1

Introduction 1.1 Problem Motivation Code division multiple access (CDMA) is an alternative multiple access strategy to frequency division and time division multiple access. Provided the synchronisation and power control problems can be overcome, CDMA is a very attractive technique for wireless communications. Its advantages over other multiple access schemes include higher spectral reuse eciency, greater immunity to multipath fading, gradual overload capability, simple exploitation of sectorisation and voice inactivity and more robust hando procedures. As early as 1978 a CDMA system had been proposed for mobile communications [CN78], however interest was limited until Qualcomm demonstrated the feasibility of implementing such a system in the late 1980s. Since then there has been an explosion in CDMA research mainly concentrating on the design and performance analysis of receivers, coding and modulation techniques and power control algorithms. Yet to be properly examined is the trac behaviour of a cellular mobile network employing CDMA. Important issues of trac capacity, call admission control, analysis of soft hando and the eects of gradual overload and imperfect power control on trac behaviour, are not well understood. The limited amount of published research in these areas is quite surprising considering their tremendous signi cance to the successful operation of a network. The trac modelling of orthogonally channelised reuse-based cellular systems such as those employing frequency division or time division multiple access, is well developed. The behaviour of networks employing xed channel assignment and dynamic channel assignment has been studied and several approaches to analysing hando have been put forward. Much of the success in this area results from the separation of trac analysis from transmission issues which allows the mobile network to be treated as a conventional circuit switched or open queueing network. Unfortunately in CDMA the separation be1

CHAPTER 1. INTRODUCTION

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tween trac and transmission issues is not so clear with capacity being determined by the interference caused by all transmitters in the network. The goal of this thesis is to contribute to the development of a deeper understanding of the trac behaviour of CDMA mobile cellular networks through the determination of analytic tools for performance analysis and design of these networks. Such an understanding is vital to sensible network operation under the stochastically varying loads that characterise teletrac. The focus of this thesis is on capacity calculations for systems employing

exible call admission control schemes. The frameworks developed allow us to explore the sensitivity of trac capacity to propagation and system parameters and to provide robust performance guarantees in the face of the tremendous amount of variability that arises in the mobile environment.

1.2 Summary and Contributions We begin in Chapter 2 with an introduction to mobile cellular networks and spread spectrum systems. The emphasis in this succint summary is on existing approaches to trac modelling of cellular networks. Chapter 3 examines methods of quantifying the interference produced by mobiles in other cells of the network. This naturally involves propagation models and so a short description of the important models is included. A large portion of this chapter is taken up by a thorough review of existing approaches to the problem. This is important to show the de ciencies of the other approaches and to place the new results of the chapter in context. The main result is an analytic expression for the distribution function of the interference from a mobile whose position is a random variable in another cell. This only applies for simple propagation models however it is the key to providing performance criteria that are robust to mobile positions and is continually used in later chapters. The nal section shows how the model can be extended to include lognormal shadowing although in this case the distribution function must be calculated numerically. The few attempts at trac modelling of CDMA cellular networks that are scattered through the literature are based on modelling each cell as an independent M=G=1 queue. These attempts are reviewed in Chapter 4 before a new model based on this assumption is presented. From the network operators point of view the model corresponds to a system where no calls are blocked and no calls are terminated prematurely. This scenario is reasonable for CDMA systems due to the soft overload property with no hard limit on the number of available channels and all users suering a gradual performance degradation as the load is increased. From a mathematical point of view the number of users in each


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cell becomes a Poisson random variable and the total interference can be modelled as a compound Poisson sum. This leads to a very ecient technique for assessing the reverse link trac capacity of CDMA cellular mobile networks and for investigating its sensitivity to propagation and system parameters. The development of the model in this chapter and its subsequent analysis is more thorough than any of the existing attempts, capturing the variability in the number of active users and their locations in a surprisingly simple manner. While M=G=1 models are justi able based on soft overload, they are unrealistic from the network operators viewpoint. At times when the network becomes heavily loaded it may be sensible to reject new calls in order to maintain the quality of the existing connections. A simple scheme to do this is one that allows some xed maximum number of calls in any one cell independent of the load on surrounding cells. However this is wasteful since extra calls could be tolerated in a cell with lightly loaded neighbours and we would like a capacity region or call admission scheme that exploited this exibility. This is achieved for a broad class of CDMA cellular networks in Chapter 5 which contains the most signi cant and valuable contributions of this thesis. Starting with a quality of service measure based on every mobile being able to achieve its required signal to interference ratio, techniques from the analysis of statistical multiplexing at an ATM-based broadband ISDN link are used to give performance guarantees that overcome the variability in interference levels characteristic of CDMA cellular networks. This leads to the assignment of an eective interference to a call dependent on its location and signal to interference ratio requirements and to a capacity region bounded by a set of hyperplanes. The CDMA mobile network, operating within the admissible region described above, has a very similar form to a circuit-switched network operating with xed routing. This similarity allows the existing trac modelling techniques and network management strategies for general loss networks, to be applied to CDMA mobile cellular networks. Finally, the major results and conclusions of this thesis are summarised in Chapter 6.

1.3 List of Publications The following is a list of publications related to the results presented in this thesis and either presented at, submitted to, or to be presented at conferences.

J. Evans and D. Everitt, \Analysis of Reverse Link Trac Capacity for Cellular Mobile

Communication Networks Employing Code Division Multiple Access," in Proc. Australian


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Telecommunication Networks and Applications Conference, Melbourne, Australia, December 1994, pp. 775{780.

J. Evans and D. Everitt, \Eective Interference: A Novel Approach for Interference

Modelling and Trac Analysis in CDMA Cellular Networks," in Proc. IEEE GLOBECOM'95, Singapore, November 1995, pp. 1804{1808.

D. Everitt and J. Evans, \Trac Variability and Eective Interference for CDMA

Cellular Networks," in Proc. ITC Specialists Seminar on Teletrac Modelling and Measurement, Leidschendam, The Netherlands, November 1995.

J. Evans and D. Everitt, \Call Admission Control in Multiple Service DS-CDMA Cellular Networks," in Proc. IEEE VTC'96, Atlanta, Georgia, USA, April 1996.

The following papers have been submitted to IEEE Transactions on Vehicular Technology.

J. Evans and D. Everitt, \Non-Blocking Trac Models of CDMA Cellular Mobile Networks," Submitted.

J. Evans and D. Everitt, \Eective Bandwidth Based Admission Control for Multiple Service CDMA Cellular Networks," Submitted.

Chapter 2

Mobile Cellular Fundamentals This chapter provides a basic introduction to cellular mobile communication systems. Topics discussed include the evolution to cellular, personal communication services, multiple access techniques and trac modelling of mobile cellular systems. After a brief treatment of spread spectrum communication, the important features of networks operating with code division multiple access are covered. It is the trac modelling of these systems that constitutes the main topic of this thesis.

2.1 Introduction Mobile and wireless communication systems will be an integral part of future telecommunication networks. In spite of the relative expense of mobile services, the advantages of users being freed from a xed communication tether and having a more personal communication address, have led to a rapid growth in the popularity of such systems. A plethora of existing technologies falls under the mobile communication umbrella. These include analog cellular, digital cellular, mobile satellite, cordless telephony, paging systems, mobile computing and wireless local area networks. The challenge in the next decade is to amalgamate these systems into a coherent, universal personal communication service. The catchcry of communicating \Anywhere, Anytime, Anything," is often used to indicate the ambitious goals of such a universal system. A realistic vision is of a future mobile communication system which provides global coverage for speech and low to medium bit rate services, while supporting high bit rate services over limited coverage areas [Ste94, Lip94, PGH95]. Of the aforementioned mobile technologies, this work is exclusively concerned with analog and digital cellular mobile communication systems. For a detailed account of these systems refer to the reference works [Lee90, Ste92]. 5

CHAPTER 2. MOBILE CELLULAR FUNDAMENTALS

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Figure 2.1: A Cellular Network with a 7 Cell Frequency Reuse Pattern

2.2 Evolution of Mobile Cellular Systems Original mobile telephone services operated with one central transceiver serving a large urban area. Few radio channels were available and thus few mobile subscribers could be supported while maintaining a reasonable grade of service. In the early 1980s the rst commercial cellular systems were introduced. By regularly reusing the available spectrum in a controlled manner, cellular systems provided a dramatic improvement in capacity and service quality. The rst generation cellular networks use frequency division multiple access (FDMA) with analog narrowband frequency modulation. The service area is partitioned into cells and each cell is serviced by a local transceiver - the cell site, or base station (BS). Each cell is assigned a subset of the available radio channels which can be reused in other cells of the network provided the spatial isolation is sucient to guarantee adequate signal to interference ratio (SIR). A network with a typical seven cell reuse pattern is shown in Figure 2.1. Frequencies can be reused in identically numbered cells. A consequence of employing localised service areas is the requirement to support hando. Hando occurs when an active mobile station (MS) moves from one cell to another, the current channel is released and a new channel in the destination cell is assigned. It is this overhead, along with uncertainty in the radio propagation environment, that limits the gains from continually decreasing cell sizes. As the capacity limits of the rst generation cellular systems are approached, second generation systems using digital communication techniques are being implemented. The power of digital coding allows more robust operation leading to smaller cell sizes (microcells) and greater spectral reuse. Capacity increases of between three to twenty over existing analog networks are expected.


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The main digital systems currently in existence are GSM, D-AMPS and Qualcomm CDMA. The rst two rely on hybrid FDMA and time division multiple access (TDMA) while the Qualcomm system operates with combined FDMA and code division multiple access (CDMA). The European GSM system seems likely to become a global standard for future personal communications services unless the predicted capacity and quality gains from CDMA are demonstrated to be signi cant. This thesis concerns trac modelling of cellular systems operating with CDMA and constitutes an attempt to understand the behaviour of these systems under a stochastically varying load. In the next section the trac analysis of cellular networks that use FDMA/TDMA is reviewed, followed by an examination of the important features of CDMA cellular systems.

2.3 Trac Modelling of Mobile Cellular Systems In this section issues relating to the performance of cellular systems under a stochastic load are brie y discussed. See [Eve94a] and the references therein for a more thorough engagement with these issues. The trac oered to a network is a measure of the number of call initiations and the duration of calls. It is de ned as the product of the mean arrival rate (calls/sec) and the mean call holding time (sec). The dimensionless unit of trac is the Erlang. The aim of a cellular network provider is to maximise the trac carried by the network (Erlangs per unit area) for a xed amount of resource and subject to some level of service requirement. In an FDMA/TDMA cellular network there are a xed number of radio channels (resource) that must be allocated subject to reuse constraints (the service quality restriction). In the remainder of this section, we look at performance measures for cellular mobile systems, methods of allocating radio channels and techniques for analysing these methods.

2.3.1 Performance Measures An important performance measure is the probability of a mobile requesting access to the radio link being denied access. We will associate this new call blocking probability with Grade of Service (GoS). In cellular networks there is also hando blocking, caused when an active mobile leaving one cell is unable to be allocated a channel in the destination cell. This leads to the forced termination of the call.


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The dropout probability is the overall probability that a call will suer abnormal termination, either through insucient SIR, or from hando blocking. Finally, there is speech quality. Despite its more subjective nature, the speech quality is closely related to the SIR achieved on the radio link. It is an SIR requirement that governs the reuse group size in reuse based systems. Note that this measure is related to the dropout probability, since an insucient SIR for a certain period of time usually leads to the call being dropped. Hereafter the Quality of Service (QoS) will refer to the satisfaction of SIR requirements.

2.3.2 Channel Allocation Strategies Channel allocation strategies refer to methods for assigning radio channels to mobiles in cellular systems employing orthogonal channels (FDMA/TDMA). The easiest form of channel allocation is xed channel assignment (FCA). A xed group of channels is made available to each cell in such a way that the reuse constraints will always be satis ed. If a mobile makes a call request to a cell site that has already allocated all of its channels, the call will be blocked1 . FCA can handle spatial variation in load by allocating more channels to the busier cells, however it is not capable of adjusting to short term load variations. A much more exible approach is dynamic channel assignment. Dynamic channel assignment (DCA), in its most exible form, makes no restrictions on the channels that can be used by any cell. A mobile can be allocated any channel provided the reuse constraints are not violated at the time of the assignment. Algorithms to implement DCA require large amounts of data on the state of the network and are computationally and control intensive. The predicted gains over FCA range from 10% 20% [EM89] at typical GoS operating points. The exibility of DCA schemes can be further increased by using real time measurements of interference levels to decide whether or not a channel can be allocated. This is more general than using reuse constraints, which are usually calculated under worst case assumptions. This scheme will be called dynamic resource allocation (DRA) to distinguish it from DCA. Between the extremes of FCA and DCA lie several other strategies that are simpler than DCA, but more exible than FCA. These include hybrid schemes, directed retry and directed hando. For more information refer to [Eve94a]. 1

That is unless the mobile can secure a link from another base station within its range.


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2.3.3 Trac Analysis of Channel Allocation Strategies Here we look at analytical models of cellular networks that allow the trac behaviour of the networks and various channel assignment policies to be investigated. The simplest case to analyse (not surprisingly) is FCA with no modelling of hando. In this scenario, the system is comprised of a network of independent queues. With some fairly standard assumptions, a cell with N channels is modelled as an M=G=N=N queue and the new call blocking probability calculated from the Erlang B formula [Kle75]. When DCA is introduced the independence of the queues vanishes and the problem becomes much more interesting. By using a generic DCA model, independent of implementation details, it was shown in [EM89] that the joint distribution for the number of active calls in each cell takes a product form on a truncated state space. Although this analytic formulation allows an expression for the new call blocking probability to be written down immediately, the evaluation of the expression is nontrivial. In most cases the use of approximations from circuit-switched network theory, or simulations such as a Monte-Carlo acceptance-rejection technique, are required. The above approach can be extended to analyse several variants of DCA including hybrid schemes, directed hando, equipment limits, grouped channels and multiple classes. Although cellular mobile networks employing CDMA have operational features greatly dierent from those using orthogonal channels, it was shown in [Eve94b] that the trac performance of CDMA systems can be analysed in a similar framework to that of DCA. We next look at the fundamentals of CDMA cellular systems before reviewing the existing work on trac modelling of these networks.

2.4 CDMA for Mobile Cellular Systems Code division multiple access, or CDMA, is an application of the spread spectrum communication concept. In this section we look brie y at the spread spectrum concept, discuss its use as a multiple access method and examine important features of mobile networks employing CDMA.

2.4.1 Spread Spectrum Communication A de nition of spread spectrum is as follows [PSM82]: Spread spectrum is a means of transmission in which the signal occupies a bandwidth in excess of the minimum necessary to send the information; the band spread is accomplished by means of a code which is independent of the


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data and a synchronized reception with the code at the receiver is used for despreading and subsequent data recovery.

The spread spectrum concept has been around since at least the 1920s and systems developed since the 1950s [Sch82]. Most of the applications were to military communications and exploited one or more of the following properties of spread spectrum:

Low probability of intercept Antijamming Multipath suppression Security Multiple access capability The two most common methods of accomplishing the spreading are direct sequence (DS) modulation and frequency hopping (FH). In DS modulation, a data modulated carrier is further modulated by mixing with a fast (bandwidth much greater than data bandwidth), pseudorandom code sequence. In FH, the carrier frequency is periodically shifted in a pseudorandom manner. Existing cellular spread spectrum systems use DS modulation which we focus on below. For a tutorial review of the history and operation of spread spectrum systems see [Sch82, Sch77, PSM82, CM83].

2.4.2 Spread Spectrum Multiple Access Since the 1980s spread spectrum has found an increasing number of applications outside the military [SMP91, PMS91, MNE94]. Most of these, notably satellite systems and mobile networks, make use of the multiple access capability. Other properties of spread spectrum give this technique distinct advantages over alternative multiple access methods, especially in the hostile mobile cellular environment. A simple direct sequence spread spectrum (DS-SS) transmitter-receiver pair at baseband is shown in Figure 2.2. The transmitted signal consists of a digital data sequence modulated by a much faster pseudorandom code sequence. The signal power is eectively spread over a bandwidth far in excess of the data bandwidth. The pseudorandomness properties of the spreading sequence imply the spread data has a at spectrum not unlike that of white noise. At the receiver the signal is correlated with a synchronised local version of the pseudorandom code used in the transmitter. This operation despreads the data while leaving

CHAPTER 2. MOBILE CELLULAR FUNDAMENTALS TRANSMITTER

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DATA

PSEUDORANDOM CODE

Figure 2.2: Baseband DS-SS Transmitter and Receiver noise and other interference spread over the larger bandwidth. This noise and interference can then be greatly supressed by low pass ltering. In a multiple access scenario, each user is given a unique pseudorandom code or a unique time shift of the same code. Low cross-correlation and auto-correlation properties mean that each users correlator receiver leaves other user interference spread. The narrowband lter at the data bandwidth then eliminates most of the interference power. The pseudorandom codes are generally not orthogonal, so unlike FDMA and TDMA, spread spectrum multiple access (or CDMA) has the property that all users interfere with one another. This property is not desirable in multi-user communication between a single transmitter-receiver pair where orthogonal schemes give the greatest capacity. The situation is quite dierent in the mobile cellular environment however.

2.4.3 Mobile Cellular CDMA Proposals for employing CDMA for mobile communications began to appear in the literature in the late 1970s [CN78, NCG79]. These schemes involved FH-CDMA and predicted capacity improvements over existing systems of the order of ve. More recently, the development and validation of a DS-CDMA mobile cellular system by Qualcomm Inc. [SG91, SP92, Pad94], has lead to an immense amount of interest in this area. Predicted capacity improvements of between ten and twenty over existing analog systems and from three to seven over existing digital systems, have been demonstrated. In an area where bandwidth is the prime resource, this signi cant increase in spectral eciency is making CDMA technology a prime candidate for future mobile and wireless markets. We now discuss several generic operational features of CDMA cellular systems. In what follows it is assumed that the forward and reverse links employ orthogonal frequency bands and thus do not interfere with one another. For further discussion of these issues refer to [EML91, Lee91, GJP91, KMM95].


12

Power Control The capacity of CDMA systems is limited by the cross interference from all users. In a single cell mobile scenario, with all users transmitting at the same power, a mobile close to the cell site would be received at much greater power than a mobile near the cell boundary. This gives close in users an unfair advantage and reduces the capacity of the system. This problem, known as the near-far eect, is overcome using power control. On the reverse link (mobile to cell site) the aim of power control schemes is to keep either the received signal power, or the SIR, constant for each user. Both methods are equivalent for a single cell but in the multi-cell case, power control to xed SIR gives signi cant gains. On the forward link (cell site to mobile), power control is not necessary in a single cell environment. By giving all users an equal portion of the transmitted power and noting that at each mobile the signal and interference will suer the same path loss, the received SIRs of the users will be equal and constant2 In a multi-cell environment however, mobiles near cell boundaries would be at a de nite disadvantage because of the interference emanating from other cell sites. Power control algorithms to combat this inequality allocate a larger fraction of total transmitted power to more distant mobiles.

Diversity A major source of signal degradation in the mobile environment is multipath fading caused by replicas of the signal arriving at the receiver with varying time delays. In FDMA or TDMA systems powerful equalizers are required to overcome this problem. In CDMA systems however, the autocorrelation properties of the code sequences allow paths with delay greater than the chip time to be resolved and used constructively. This in eect provides a form of internal diversity and greatly reduces fading. Another form of diversity available in CDMA cellular systems is macrodiversity where a mobile is able to connect to two or more cell sites. On the reverse link, signals3 are received at dierent base stations and can be combined using standard diversity combining techniques. On the forward link, signals from dierent cell sites can be treated just like the multipath replicas, rst resolved and then combined. This assumes that background noise is not signi cant. Inclusion of this eect would again give close in users an advantage as the magnitude of their received signal power would be greater. 3 Hopefully with a fair degree of statistical independence. 2


13

MULTIPATH DIVERSITY

CELLULAR DIVERSITY

Figure 2.3: Diversity in Cellular CDMA

Soft Hando Related to macrodiversity is soft hando. When a mobile moves from one cell to another, rather than dropping its existing connection before it starts communicating with the target cells site, it can connect to one or more base stations during the hando procedure. Only when the mobile is well established in the new cell will the other connections be dropped. This form of hando is much smoother and more reliable than the hard, or break before make hando in FDMA/TDMA cellular networks.

Voice Activity In a typical two way conversation, each speaker is active about 40% of the time4 . The detection of the onset of the inactive cycle and subsequent suppression of transmission during the period, directly reduces interference to all users. This translates to an increase in capacity of nearly the inverse of the voice activity factor. It is much more dicult to implement procedures that take advantage of voice inactivity in FDMA or TDMA. In these systems channels would need to be dropped and reassigned during every voice cycle.

Sectorization If directional antennas are used at base stations, cells can be divided into sectors. On the reverse link, interference results from only those mobiles located in the bandwidth of the antenna. Thus sectorization ideally produces a capacity increase equal to the number of This gure is expected to rise for the reverse link as more mothers-in-law discover mobile cellular technology. 4


14

sectors. Due to the spatial roll o of the antenna patterns and also to the variability in user locations, the capacity gain will not be this great. Typically, three sectors are used per cell leading to a capacity increase of around 2.5.

Gradual Overload Systems employing FDMA and TDMA, or indeed any other orthogonal multiple access technique, have a hard limit on the number of available channels for a xed available spectrum and data bandwidth. CDMA on the other hand, uses pseudo-orthogonal codes and provided the code rate is much greater than the data rate, there are a very large number of such codes. What this means is that an extra user can nearly always be accepted to the system at the cost of a slight increase in interference to all other users. This dynamic tradeo between capacity and service quality is not present in systems with hard channel limits.

No Frequency Management In CDMA cellular networks the entire available spectrum is used in each cell. There is thus no need to carry out frequency resource planning as in reuse based systems. Further, the exibility of DCA is inherently available in CDMA cellular without the centralised, control intensive task of channel allocation and reassignment.

2.4.4 Capacity Calculations Before introducing some basic capacity calculations it should be noted that the capacity of CDMA cellular networks has to be calculated separately on the forward and reverse links. Synchronous communication with orthogonal signalling is possible on the forward link due to the use of a high power pilot signal. This isolates interference from other mobiles in the same cell or sector and generally gives the forward link superior performance. For this reason we concentrate on the limiting reverse link performance. In this section capacity will refer to the static number of users that can be supported on the reverse link while maintaining an acceptable quality of service (BER) with a given probability.

Single Cell Consider a single cell with N active mobiles and perfect power control of received signal strength. The signal to interference ratio for each mobile, ignoring background noise, is

CHAPTER 2. MOBILE CELLULAR FUNDAMENTALS given by

15

SIR = N 1? 1 :

This can be related to the bit energy to interference density ratio (Eb =I0) through the system processing gain [GJP91] by

Eb = W SIR = W=R I0 R N ?1 where W is the spread spectrum bandwidth, R is the data rate and W=R is the system

processing gain - the main measure of the size of the system. Let us assume that an acceptable BER results provided a certain Eb =I0 is achieved. The number of users supported is then

N = 1 + EW=R =I : b 0

This capacity gure can be increased by using sectorization and supression of transmission during silent periods as discussed above. If is the gain due to voice activity monitoring and is the sectorization gain, then the enhanced system capacity is

N EW=R =I : b 0

With substitution of typical values the capacity is N W=R, which makes CDMA competitive with FDMA/TDMA on a single cell basis. It is in a multi-cellular environment however, where CDMA makes its real gains.

Multiple Cells Consider a two dimensional array of cells with N users per cell. The simplest way to account for interference from other cells is by including the other-cell interference factor [PMS91, SG91, VVZ94]. The multi-cell capacity becomes

N = E =IW=R (1 + ) b 0

with typically around 0.5 - 0.6. Thus in a multi-cell environment, the capacity per cell is approximately 60% of the isolated single cell capacity. In a reuse based system the per cell capacity is reduced by the reuse factor and so with a reuse pattern size of seven, the number of channels per cell is only 14% of the total number. Why does CDMA lead to such a capacity improvement in cellular systems? The simple answer is that in CDMA the total interference is composed of small contributions from a large number of users while in FDMA or TDMA interference is caused by a small number of users that produce signi cant interference unless substantial separation between cochannel users is maintained.


16

Shortcomings of Existing Capacity Analyses The rst paper that looked at capacity of multi-cellular CDMA with real emphasis on the analysis of other-cell interference was [GJP91]. It was assumed that there were an equal number of users per cell and that they were evenly (and continuously) spread over each cell. Perfect power control to a xed signal power was assumed and the propagation model accounted for independent lognormal shadowing. A Gaussian approximation, along with bounds on the rst two moments of the interference, lead to a relationship between the number of users per cell and the probability that the BER was acceptable. Analytical extensions of this work have centred on including eects of multipath fading [MRB92, TW94], imperfect power control [SH93], diversity reception [SK92] and dynamic narrowband sectorization with antenna arrays [LR94, NPK94]. The fundamental aim of this thesis is to extend the simple multi-cell analysis in another direction - one where two major sources of variation, namely the stochastic nature of the call arrivals and departures and the random locations of the mobiles, are taken into consideration. We are thus interested in quantifying the trac capacity and performance of CDMA cellular networks. Somewhat surprisingly, very little work has appeared in the literature on trac modelling or network operation for CDMA systems. In the next section we review existing analytic approaches to the problem.

2.5 Trac Modelling for Cellular CDMA As mentioned in the previous section, most existing capacity analyses for multi-cellular CDMA networks assume that all cells contain an equal number of active mobiles. Capacity is de ned as the maximum number of mobiles per cell that can be supported while maintaining an acceptable SIR with a given probability. A more useful and important measure of capacity is the trac capacity of the system measured in Erlangs per cell. Trac capacity is a more dynamic measure that incorporates how the network handles variability in the call arrival times and call durations. The most obvious procedure to arrive at gures for trac capacity of CDMA cellular systems is as follows. 1. Specify a QoS constraint in terms of the SIR exceeding some lower threshold with a certain high probability5 . b =I0 with greater than 99% probability, where E =I is an adequate For example the SIR must exceed EW=R b 0 bit energy to interference density ratio for acceptable BER and thus acceptable speech quality.

5


17

2. Use the existing capacity analyses to determine the maximum number of users per cell, N , that can be tolerated while satisfying the QoS constraint. 3. Assume that no more than N calls are accepted in any one cell. A call arriving at a cell with N calls already in progress is blocked. With other standard assumptions each cell is modelled as an independent M=G=N=N queue and the blocking probability calculated from the Erlang B formula. 4. With the GoS constraint specifying a desired new call blocking probability, the trac capacity is simply the maximum oered trac per cell that results in the satisfaction of the GoS constraint. This method of network operation and analysis is similar to FCA in channelised cellular systems. Like FCA the scheme is fairly simple and in exible. It does not exploit the

exibility inherent in CDMA cellular systems whereby a cell with lightly loaded neighbours may be able to accept more than N calls and thus lead to a reduced new call blocking probability. There are currently two approaches evident in the literature aimed at providing a more exible analysis of trac capacity. The rst centres around modelling each cell as an independent M=G=1 queue. With no new call blocking, the variability in the number of users can be handled in a simple manner and trac capacity determined based on QoS measures alone. One original contribution of this thesis is based on this approach and is detailed in Chapter 4 along with a review of related work. The second approach aims at allowing a more exible call admission control procedure similar to DCA in FDMA/TDMA cellular. A call is accepted based not only on the number of users in the target cell, but on the number of users in every cell of the network that contributes to the interference at the target cell. Thus at times when a cell nds fewer than normal users in adjacent cells it may be able to accept more calls than in the more simplistic approach that takes no notice of the state of the adjacent cells. A review of the literature taking this angle is given in Chapter 5 along with an original and fundamental extension. Before we can discuss the trac models in more detail it is important to gain an understanding of the form of the interference that a mobile user produces at the various cell sites in the network. This is the topic of the next chapter.

Chapter 3

Other-Cell Interference In this chapter we discuss methods of characterising interference from other users in CDMA cellular mobile networks along with related propagation and system issues. These methods are extremely important in the development of the trac models of later sections. We begin with a review of several existing approaches to the problem of handling other-cell interference. The standard mobile propagation environment is then discussed with treatment of multipath fading, shadowing and distance driven attenuation. Finally a novel characterisation of the interference in CDMA cellular is developed in the form of an analytic expression for the interference distibution function in the deterministic propagation environment.

3.1 Introduction In this introduction we review several approaches to the characterisation of interference in cellular CDMA networks. While simulation studies allow a great deal of complexity to be included we are solely concerned with analytical and numerical treatments of the problem. There are two important sources of variation in the interference received at a target BS from an active MS in another cell. The rst is the location of the MS within the cell, while lognormal shadowing is the second1 . Variation in the total interference is also dependent on the number of active MSs in the system, however this is a trac issue which is dealt with in Chapter 4 and Chapter 5. In all the papers we review there is no modelling of multipath fading. It is generally assumed that the use of techniques such as interleaving, diversity reception and equalization, as well as the employment of a RAKE receiver, greatly mitigate fast fading. At any 1

Voice activity is another source of variation which we include in later chapters.

18

CHAPTER 3. OTHER-CELL INTERFERENCE

19

rate it is reasonable to assume that the eects of the fast fading are encapsulated in the Eb =I0 requirements of the system. This means that the propagation models used centre on distance driven path loss like (3.1) and the inclusion of lognormal shadowing as in (3.2). Another common theme is the assumption that an MS is power controlled by the BS to which it is connected to be received at a xed signal power. Power control errors are usually not modelled. The rst paper to give an analysis of other-cell interference in spread spectrum mobile systems was [CN78]. Although the authors study a frequency hopped system their interference analysis in terms of SIR is applicable to direct sequence systems. The propagation model of (3.1) is assumed, which coupled with perfect power control leads to a simple expression for the interference at the desired BS from an MS with known position in the network. The total interference from a cell other than the desired one is calculated by integrating the above interference expression mixed with a continuous and uniform user density over a circular region approximating an hexagonal cell. An analytic result is only possible for restricted values of the path loss exponent (PLE) and so the authors use numerical integration to calculate the interference levels. The overall other-cell interference results after summing the contributions from all interfering cells apart from the desired one. This paper does not deal with the randomness of the user locations and is equivalent to calculating expected values when each user is independently and uniformly distributed over the cell of concern. In [Kim93] a very similar analysis to the above is presented with the exception that the xing of the PLE at 4 leads to analytic expressions for the interference from the circular cells. An extension of [CN78] which includes the eects of shadowing and voice activity monitoring is found in [GJP91], the most heavily referenced work in the eld of CDMA cellular. The authors begin with a simple treatment of single cell capacity in terms of maximum number of users as in Section 2.4.4. With perfect power control of received signal strength, this analysis is independent of the propagation environment. The eects of voice activity monitoring and sectorization are added in a simple multiplicative manner. Their analysis of reverse link capacity in the multiple cell environment is more interesting. A standard hexagonal cellular layout is assumed with the propagation model of (3.2) that includes lognormal shadowing taken to be independent on distinct paths. The total interference at a target BS is examined assuming that there are an equal number of users per cell, N , spread evenly and continuously over each cell. MSs are initially assumed to connect to the BS oering the least path loss. If this BS is the target then the interference is the xed, constant power speci ed by the power control, otherwise the


20

interference is a lognormal random variable (rv) with mean dependent on the position of the MS. To simplify the analysis, an MS decides between the closest BS (not including the target) and the target BS only. An expression for the interference dependent upon the MS position is then multiplied by the user density2 and integrated over the network to give the total other-cell interference. This total interference is a rv due to the lognormal shadowing and in the paper its mean and variance are calculated numerically as functions of the user density. Approximating the probability density function (pdf) as a Gaussian, the other-cell interference is fully characterised. From this point it is trivial to include the intra-cell interference and the eects of voice activity monitoring, leading to a nal expression for the total interference power received at the target BS. This expression is readily related to the SIR and Eb =I0 attained by each user and the probability of a user achieving an acceptable BER is plotted as a function of the user density. Capacity is then calculated as the maximum user density that gives a certain (say 99%) probability of acceptable BER. An analysis of the forward link is also presented in this paper, but apart from noting that the forward link capacity is found to be superior to the reverse link, we do not discuss the analysis here. An extension of the reverse link analysis of [GJP91] is discussed in [VVZ94, VVG94]3. Firstly, the propagation model is extended to take into consideration the dependence of the shadowing from an MS to dierent BSs. This is done in a somewhat arbitrary manner and does not lead to any added complexity in the analysis. Secondly, rather than choosing between the target BS and the closest BS, an MS can connect to any of the nearest M BSs. This involves a fairly straightforward extension of the analysis in [GJP91] although the computational complexity increases considerably to the extent that only mean values for the interference are calculated. The results show a dramatic drop in the mean other-cell interference from M = 1 to M = 2 for typical values of the shadowing variance, while the improvement is small for M > 3. In [RM92] there is no modelling of shadowing but more detailed and accurate versions of (3.2) are employed. The analysis assumes a circular target cell plus wedge shaped adjacent cells, this geometry allowing a fairly simple investigation into the sensitivity of other-cell interference to user density pro le variation. Power control errors are not examined and the results are all numerical. In all of the above treatments, the randomness in user location within a cell has not been dealt with. Rather some (usually uniform) continuous user density has been assumed 2 3

Not a probability density function. Despite the distinctly dierent titles, these papers are essentially identical.


21

Area mean ( distance ) Signal Strength ( dB )

Local mean ( shadowing )

Fast fading ( multipath )

Distance from base station

Figure 3.1: Mobile Propagation Environment and its product with an interference function integrated over the network. An alternative approach is to look at the interference as a function of a random position vector as in [FRG94] where the MS location is assumed uniformly distributed over each cell. In this paper however, only the mean and variance of the interference are required since a Gaussian approximation is used. This means the treatment is identical to [GJP91] and it is only because of the slightly dierent angle taken that it is mentioned here. Section 3.4 discusses a novel method of characterising other-cell interference in CDMA cellular networks. As with the above work, Raleigh fading is not studied and perfect power control of received signal strength is assumed. To begin we look at one dimensional cells and the propagation model of (3.1). Given an arbitrary df for the MS location within a cell, the df of the interference from that MS is calculated analytically as a function of the PLE and the location of the cell. In two dimensions the problem is more dicult and we assume circular cells and uniformly distributed users and arrive at the interference df as a function of the centre and radius of cells and the PLE. The calculation of this df compares to [Kim93]4 which assumes a similar geometry and propagation model yet only derives an expression for the mean for a xed PLE of 4. The results are then extended to include lognormal shadowing similarly to [GJP91], however unlike it, an expression for the df of the interference is constructed. This df must be calculated numerically.


22

3.2 Propagation Issues The simplest model for the mobile radio channel is a propagation loss inversely proportional to the distance between the transmitter and the receiver raised to an exponent [Hat80, ARY95]. If the transmitter and receiver are separated by d units, then the received power is given by PR (d) = PT P0d? (3.1) where PT is the transmit power and P0 and are independent of distance. P0 is a function of carrier frequency, antenna heights and antenna gains, and we assume it is constant for all paths between a mobile and a base station. is the path loss exponent (PLE) which varies with antenna heights and is typically in the range three to four. The fact that the exponent is greater than the free space exponent is a result of a multipath eect in the vertical plane. The simple model of (3.1) is accurate for distances from one to twenty kilometres with base station antenna heights greater than thirty metres and in areas with little terrain pro le variation. Thus the model is reasonable for conventional cellular systems in at service areas but is not accurate in city microcells which employ small cells and low antennas. Empirical results have illustrated that the deviation from (3.1) is normally distributed on a log-log plot5 [Lee90, pp. 105-107]. The errors are due primarily to variations in terrain contour and to shadowing from buildings. Incorporation of this deviation, commonly called lognormal shadowing, leads to the equation

PR (d) = PT P0 d? 10=10

(3.2)

where P0 and are as before and is a zero mean Gaussian random variable with standard deviation typically in the range six to twelve. PR (d) is now a random variable with lognormal density 1 e?(ln z?) 2 =2 2 fP (z) = 0 p z 2 0 where = ln PT P0 ? ln d and = ln 10=10. The spatial correlation between shadowing random variables is signi cant over a distance of several metres [Gud91] giving rise to a local mean over small areas. The third propagation eect is a fast fading about this local mean. The fast fading is due to the arrival of several replicas of the signal with varying time delays and is characterized by a 0

To the author's knowledge this is the only paper which gives analytical results. This lognormal modulation of the area mean is what one might expect from a series of independent multiplicative losses. 4

5


23

Base Station

Figure 3.2: Standard 1-D Cellular Layout Base Station

Figure 3.3: Standard 2-D Cellular Layout Raleigh distribution for the received signal amplitude. The fading is basically independent over distances greater than half a carrier wavelength. In this thesis we assume that all eects of fast fading are encapsulated in the Eb =I0 requirements and we do not discuss them further. Figure 3.1 illustrates the three main factors that characterize the mobile radio channel6 .

3.3 System Model Throughout this work we consider both one and two dimensional networks. The one dimensional system consists of a line of evenly spaced base stations as in Figure 3.2. The two dimensional system is simply the standard, uniform, hexagonal layout as shown in Figure 3.3 with a base station at the centre of every cell. The forward and reverse links use disjoint frequency bands and can thus be analysed independently. We only consider the reverse link as it is generally accepted to be the limiting factor in capacity calculations7 . In the sequel, all mention of path loss, SIR and capacity refers to the reverse link. Unless otherwise stated, a mobile connects to the base station that oers the least path loss at any given time. The chosen base station employs power control to maintain 6 7

The period of the fast fading is greatly exaggerated See the beginning of Section 2.4.4


24 Cell

a-

0

a

0.5

x

Target BS

a + 0.5

BS

Interfering MS

Figure 3.4: Interference in 1-D Network the received signal power at one unit. Finally it is assumed that all systems are interference limited and that background noise is negligible. In real systems the background noise provides the reference from which absolute signal powers are set.

3.4 Interference Calculations In this section the models and assumptions of the last two sections are used to develop expressions for the interference produced at a target base station by an active mobile in another cell of the network. We rst treat the one and two dimensional cases with distance driven path loss, followed by an extension to include lognormal shadowing.

3.4.1 Deterministic Path Loss: One-Dimensional Network Consider the situation shown in Figure 3.4 where a mobile station (MS) at location x is connected to the base station (BS) at location a. To be received with one unit signal power the MS must transmit with power

PT (x) = jx ?P aj ; 0

assuming the simple propagation loss of (3.1). The resulting interference to the BS at location 0 is given by

a P j x ? a j 0 (3.3) I (x) = P jxj = 1 ? x 0

Based on the standard one dimensional layout of Figure 3.2, we assume the cells are one unit in length and centered on the integers. We are interested in the distribution of the interference when the location of the MS within the cell is not known but is a random variable (rv). Let X be an rv taking values in (a ? 0:5; a + 0:5] and with distribution function (df) FX . We require the df for the rv I (X ). Firstly if a = 0 then the interference is clearly one unit regardless of FX and the df is degenerate with a step at one. Assuming a 6= 0 and noting that I is piecewise monotone,


25

Interfering MS (x,y) Target BS (0,0)

BS (-a,0)

1 3

Figure 3.5: Interference in 2-D Network standard transformation techniques give

8 > 0; z < 0 > > < g (z); 0 z < (2a + 1)? FIa(z) = > 1 g2(z); (2a + 1)? z < (2a ? 1)? > > : 1; (2a ? 1)? z

where

(3.4)

g1(z) = FX 1 ?az 1= ? FX 1 +az 1= g2(z) = 1 ? FX 1 +az 1= :

In particular, if X is uniformly distributed on (a ? 0:5; a + 0:5] then

g1(z) = 1 ?az 1= ? 1 +az 1= g2(z) = 0:5 + a ? 1 +az 1= :

3.4.2 Deterministic Path Loss: Two-Dimensional Network Consider the situation shown in Figure 3.58. An MS is located at (x; y ) within a hexagonal cell of the standard two dimensional layout of Figure 3.3. The MS is connected to the BS with coordinates (0; 0) and causes interference to the BS at (?a; 0). Similarly to the one dimensional case the interference is given by 2 2 I (x; y) = (x ?x a+)2y+ y 2

! =2

(3.5)

and we would like to be able to calculate the df of the rv I (X; Y ) given the joint df FX;Y of the rvs X and Y . 8

The orientation of the cell with respect to a line between the BSs will vary.


26

Interfering MS (x,y) Target BS (0,0)

BS (-a,0)

a>b b

Figure 3.6: Interference in 2-D Network - Approximation This is in general a complicated problem, so to simplify matters let us assume the joint density of X and Y is uniform over the hexagon. Due to the large number of possible orientations of the hexagonal cell and to the dependence of X and Y , the analysis remains exceedingly tedious. These problems can be eradicated by approximating the hexagonal cells by circles as shown in Figure 3.6. The orientation diculty clearly vanishes and by having a and b as parameters, a great deal of exibility results. Before proceeding, note that by symmetry it is only necessary to consider the upper semi-circle. It is also expedient to work in polar coordinates since the rvs R and de ned p by R = X 2 + Y 2 and = arctan(Y=X ) are independent with 2 FR; (r; ) = rb2 FR (r) = r2 =b2 F () = =

(3.6) (3.7)

where these dfs are de ned only on the domain 0 < r b and 0 . If the MS is at location (r; ) in polar coordinates and is connected to the BS at the origin, the interference caused at the BS with location (a; ) is given by

I (r; ) = p

r

!

r2 + a2 + 2ar cos and the problem now is to calculate the df of the rv I (R; ) where R and are independent

and distributed as in (3.6) and (3.7). The above problem can be solved analytically. The development is fairly straightforward but tedious and below we simply outline the steps involved. 1. Show that I (r; ) is monotone in r for xed . 2. Fix and calculate FI (z j) = P (I (R; ) z ), the conditional df of I given = . This involves standard transformation techniques for monotone functions of an rv.


27

3. Uncondition FI (z j) using the relation

Z

FI (z) =

FI (zj) F (d)

[0;] 1Z

=

0

FI (zj) d:

Carrying out the aforementioned steps leads to the df of the interference received at (a; ) from an MS that has a uniformly distributed location within the circle of radius b and centre the origin. The df is given below.

8 > 0; z < > > > > < g1(z); ? a ?0 z < a;b FI (z) = > g2(z); b + 1 z < > g3(z); 1 < > > ? : 1; ab ? 1 ? z

0

? a + 1? ? ab ? 1? ; z 6= 1 ? ab ? 1? ; z = 1

(3.8)

b

where

g1(z) =

2 2=

az ? 2 b z 2= ? 1 2

g2(z) = 1 arccos (h1(z)) + 1 g1(z) [ ? 2z 2= h1 (z)h2(z) ? ab z2= h2(z) ? arccos (h1(z)) ? arcsin z1= h2(z) ] a a p 1 g3(z) = arccos ? 2b + 4b2 4b2 ? a2 and 2 2 2 ?2= h1 (z) = ? a ? b2ab+ b z

h2 (z) =

q

1 ? h21 (z )

To con rm the validity of (3.8) we compare it with simulation for various values of a and b in Figure 3.7. We stress that the simulation is done for circular cells as assumed in the theoretical treatment and is solely used to con rm the analytical result. Excellent agreement is clearly obtained. p With the length of a side of the hexagon equal to 1= 3, the question arises as to the p value of b that should be used. Figure 3.8 illustrates some possibilities: b = 1= 3 0:58 leads to a circle that contains the hexagon, b = 0:5 gives a circle that is contained in p p the hexagon and b = 4 3= 2 0:53 makes the areas of the hexagon and circle equal. The accuracy of these approximations is examined in Figure 3.9 where for a = 1 and PLE = 4 the three circular approximations are plotted alongside results for the hexagonal


28

cell obtained via simulation. The value b = 0:53 matches the simulation very well and is thus used in the sequel. As the values of a and PLE are varied the above approximation remains excellent as shown in Figure 3.10.

3.4.3 Inclusion of Lognormal Shadowing In the following we treat the one and two dimensional cases simultaneously using the notation:

x is a position vector in 1-D or 2-D, k k is the Euclidean norm in 1-D or 2-D. (3.3) and (3.5) are generalised in the expression

? xi k Ii;j (x) = kkxx ? xj k

!

which represents the interference in the non shadowing environment to a BS at xj (BSj ) from an MS at x connected to a BS at xi (BSi ). We are interested in extending the interference results of the last sections, which were based on the propagation model of (3.1), to include shadowing eects as given in (3.2). With reference to the latter equation, we assume P0 , and are constant over all paths, and that the shadowing rvs are independent for dierent paths. Initially assume that an MS at x connects to the closest BS, BSi , and suppose we are interested in the subsequent interference produced at a target BS, BSj . If xi = xj then the interference is clearly one unit since the MS will connect to the target BS and be power controlled to one unit signal power. If xi 6= xj then

Ji;j (x) = Ii;j (x)10(j?i )=10 = Ii;j (x)10=10

(3.9)

where as the dierence of two independent, zero mean, Gaussian rvs, is a zero mean Gaussian rv with variance 2 2. Suppose however, that rather than connecting to the closest BS, an MS is linked to the BS oering the least path loss9 . It is both theoretically and practically infeasible to allow connection to any BS in a large network and it is sensible to consider choosing between only the M closest10. This is the approach taken in [VVZ94, VVG94] though as 9 10

Shadowing may cause this BS to be other than the closest. Especially since the soft hando facility makes this type of operation very practical


29

1 simulation a=2, b=0.50 a=2, b=0.58 a=1, b=0.50 a=1, b=0.58

0.9

P( Interference < z )

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -5

-4

-3

-2

-1

Log z

PLE = 4

Figure 3.7: Con rmation of Analytical Result

b = 0.58

b = 0.5

b = 0.53

Figure 3.8: Approximating the Hexagonal Cells

0


30

1 simulation approximation: b=0.50 approximation: b=0.53 approximation: b=0.58

0.9


0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -5

-4

-3

-2

-1

0

-1

0

Log z

PLE = 4, a = 1

Figure 3.9: Comparison of Approximations 1 0.9

simulation a=2, PLE=4 a=2, PLE=2 a=1, PLE=4 a=1, PLE=2


0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -5

-4

-3

-2 Log z

b = 0:53

Figure 3.10: Approximation versus Simulation


31

already mentioned, a fairly complicated analytical and numerical procedure, results only in mean values for the other cell interference. We prefer a simpler approach similar to [GJP91, FRG94] where the choice is made between the closest BS and the target BS. As most of the other cell interference to the target BS comes from MSs near its cell boundaries this seems a reasonable approximation. Once more we assume that if the target BS, BSj , is the closest BS, then the MS connects to it and causes one unit of interference. Failing that and with BSi the closest BS, the interference produced by an MS at x is given by

Ji;j (x) = min 1; Ii;j (x)10=10

(3.10)

which is simply a truncated version of (3.9). One signi cant advantage of (3.10), apart from being a more accurate model of the system operation, is that the moment generating function of the interference exists only in the truncated case11 . Given (3.10) our aim, just as in the last sections, is to allow the position vector x to be an rv taking values in the cell corresponding to the BS at xi . Denote this random position vector X and assume it has df FX de ned on cell i . Our problem then is to nd the df of Ji;j (X), which means calculating the df of Ii;j (X)10=10. Given the position vector x, the df of the rv Ii;j (x)10=10, which we treat as a df conditioned on X, is ! 1 ln z ? ln I ( x ) i;j p F (zjx) = 2 + erf (3.11) 2 where Zy 2 e?t =2 dt: erf(y ) = p1 2 0 Alternatively we can view (3.11) as a df conditioned on Ii;j (X)

F (zjIi;j ) = 12 + erf ln zp? ln Ii;j 2

!

(3.12)

p

In the above, = ln 10=10 and 2 is the standard deviation of in (3.10). (3.11) can be unconditioned on X given FX ;

Z

F (zjx)FX(dx); cell i while (3.12) can be unconditioned on Ii;j (X) given its df, FIi;j ; F (z) =

Z

F (z) = F (zjIi;j )FIi;j (dI ); I

11

(3.13)

(3.14)

The moment generating function is used for obtaining bounds on sums of interferers in later chapters.


32

In the above, FIi;j comes from (3.3) in one dimension and from (3.5) in two dimensions12 with a replaced by kxi ? xj k. The nal form for the df of Ji;j (X) is given by

8 > z < 0 > < 0; FJi;j (z) = > F (z); 0 z < 1 > : 1; 1 z

(3.15)

and results because of the min operator in equation 3.10. In summary, (3.15) gives the df of the interference produced at BSj from one MS with location in celli distributed as FX . It is equivalent to (3.3) and (3.5) when lognormal shadowing is modelled as above. In one dimension neither (3.13) nor (3.14) is advantageous, however in two dimensions the latter requires one less level of integration.

3.5 Conclusion The main purpose of this chapter was to provide a characterisation of other-cell interference in CDMA cellular networks that is crucial to later chapters. After reviewing existing approaches to the problem, some new results were developed. The end products are expressions for the distribution functions of the interference when a mobile's location is a random variable within a cell. These expressions are analytic for the deterministic propagation environment but involve numerical integration when shadowing is introduced.

12

(3.5) is an approximation and is only valid when X is uniformly distributed.

Chapter 4

M=G=1 Models In this chapter we look at extending the capacity analyses that assume a xed and equal number of users in every cell to handle the random nature of call arrivals and departures. The simplest way to do this is by modelling each cell of the network as an independent M=G=1 queue. This allows us to replace the deterministic number of users in each cell by an independent Poisson random variable for each cell. The resulting compound Poisson sums have some very nice properties that allow us to calculate a blocking probability by analysing a single random sum. This leads to a very ecient technique for assessing the reverse link trac capacity of CDMA cellular mobile networks.

4.1 Introduction Despite the abundant literature on CDMA for cellular mobile, very few contributions concern the teletrac behaviour and networking issues associated with the running of such systems. The main emphasis to date has been on lower level issues including code design, receiver structure, bit error rate calculation, multiuser detection and estimation, and power control algorithms. Almost without fail capacity analyses concern a static and equal number of users in each cell however such analyses oer scant insight into the real time operation of the network under a stochastic load. Issues that need to be addressed include calculation of the teletrac capacity of CDMA cellular networks, call admission control schemes, and behaviour of these networks in a mixed trac environment. In this chapter we present a technique that allows the teletrac capacity of CDMA cellular networks to be estimated. The simplicity of the technique arises from modelling each cell of the network as an independent M=G=1 queue and consequently the theory of this chapter provides no input to the understanding of how calls should be admitted to the system. A more advanced network model which does impact on call admission control 33

CHAPTER 4. M=G=1 MODELS

34

schemes is presented in the following chapter. We begin with a review of several papers [VV93, Fap93, LE94, FRG94] that employ the M=G=1 approximation and compare them to the model presented in this chapter and in [EE94]. Note that in these papers the generally distributed holding times are replaced by negative exponentially distributed holding times giving M=M=1 queues. However all their results apply in the general case since it is only the stationary distribution of the number in the system that is used.

[VV93] The rst of the above papers to appear was [VV93] which was a continuation of the work of Qualcomm Inc. [GJP91, SG91, SP92, Pad94, VVZ94, VVG94] aimed at convincing readers of the enormous gains of their CDMA system over existing mobile technologies. The paper in both title and introduction stresses the importance of Erlang capacity as a more meaningful gure of \economic usefulness" than the static capacity. After an interesting treatment of a single cell the authors make several unjusti ed assumptions in an attempt to extend their analysis to multiple cells that severely limit the contribution of the paper to the teletrac analysis of CDMA systems. The paper looks only at the limiting reverse link and has as its aims the development of a model that deals with: 1. variability in the number of users per cell, 2. voice activity, and 3. variable Eb=I0 requirements. Concentrating on a single cell (or sector) the authors assume that no new call requests are denied and as such model the cell as an M=M=1 queue. The number of users in the cell is thus modelled as an rv with a Poisson distribution having mean equal to the cell oered trac. So much for the rst aim. Voice activity is included simply by assuming each mobile is gated on with probability and o with probability 1 ? . Voice activity is thus modelled by a Bernoulli rv . Based on tting of empirical data on the received Eb=I0 values required to maintain frame error rates below 1%, the authors model the Eb =I0 required by each mobile as a lognormal rv. Since the data involves actual received values required for acceptable performance it must include the eects of imperfect power control and varying propagation conditions. This approach provides one of the simplest methods for incorporating imperfect power control but does rely on empirical observations.


35

With the above aims achieved the authors essentially de ne blocking to occur when the instantaneous Eb =I0 and bit rate requirements of all users cannot be met. This leads to the relation ! k X i (Eb=I0)i constant (4.1) Pblocking = P i=1

where k is a Poisson rv, the i are Bernoulli rvs, the (Eb=I0 )i are lognormal rvs, and all rvs are assumed independent. This probability is bounded for a range of cell oered tracs using a modi ed Cherno bound1 and compared to results from a Gaussian approximation and simulation. Based on this comparison and the relative ease with which it is calculated, the Gaussian approximation is used in the extension to multiple cells. The rst assumption of the extension is that each cell remains equally loaded. This is a very poor assumption made without justi cation other than an appeal to resultant simplicity. The second assumption is that the k users in every outer cell produce a combined interference equivalent to kf users in the inner cell where f is an expected outer-cell interference fraction obtained from [VVZ94]. The network model used to calculate f however assumed perfect power control of each mobiles received signal strength to some xed power and its inclusion in a model allowing variable Eb =I0 values is dubious. Accepting the assumptions the multiple cell case reduces to the single cell problem with an equivalent number of active mobiles of (1 + f )k where as before k is a Poisson rv. To apply the Gaussian approximation the mean and variance of the sum in (4.1) need only be scaled by (1 + f ). In summary, the treatment of imperfect power control (or variable Eb =I0 requirements) in a multiple cell network is a very dicult problem by itself. In [VV93] an already dubious approach is compounded by an attempt to include the stochastic nature of the call arrivals.

[Fap93] In [Fap93] a computationally intensive procedure is presented for the evaluation of the teletrac capacity of both forward and reverse links in a CDMA cellular mobile system. Each cell is modelled as an independent M=M=1 queue and the QoS criterion evaluated is the outage probability or the probability that the SIR of a link is below a certain threshold. A uniform hexagonal layout, a uniform density for the mobile location within each cell, and a propagation model including lognormal shadowing are other features of the model presented. The standard Cherno bound can not be used because the moment generating function of a lognormal rv is in nite outside zero. 1


36

The procedure for a calculation of outage probability given some oered trac per cell is as follows: 1. Suppose we are looking at some central cell and wish to evaluate the probability that the SIR for a mobile connected to the central base station is below a threshold. Suppose further that only the central cell and rst two layers of surrounding cells can contribute signi cantly to the total interference at the central base station. 2. Assuming we are given the number of interfering mobiles and their locations as well as the position of the desired mobile, calculate the area mean SIR (this is independent of the shadowing). This can be done for no power control and for area mean power control. 3. Substitute the result into an existing expression that approximates the outage probability when the interference is a sum of lognormal rvs. 4. At this stage we have an outage probability conditioned on the number of interfering mobiles and their locations and on the position of the desired mobile. We rst uncondition on the mobile locations using Monte-Carlo integration (the dimension of the integration is twice the number of mobiles). 5. Finally we can uncondition on the number of mobiles by repeating the above for all integers and weighting by the appropriate probabilities obtained from the M=M=1 approximation. The main disadvantage of the above approach is the extreme computational eort required and it is debatable whether the approach is any more valuable then a straight out simulation. To illustrate the problem we need look no further than the author's results for which it is assumed that the number of mobiles per cell is equal. Further no results on trac capacity are presented with all outage probabilities conditioned on the above number of mobiles per cell. At any rate the results are pessimistic since only area mean power control is examined and no macrodiversity is considered.

[LE94] In [LE94] a teletrac model of the reverse link is considered. The assumptions include uniform hexagonal layout, equal trac oered to every cell, uniform density for mobile locations within cell, two layers of interfering cells considered, deterministic propagation loss only, and perfect power control of received signal strength. Despite the initial discussion of a model including a nite number of modems, trunk reservation for handovers, and mobility the subsequent analysis does not allow for mobility


37

and assumes trac levels which reduce the new call blocking probability to a negligible gure. Thus the system is actually modelled as a network of independent M=M=1 queues. Once more the QoS measure concerns the probability of the SIR being below a given threshold which with the assumption that each mobile is received at a xed power level involves calculating the probability that the interference gets too large. The total interference is calculated as the sum of inner-cell interference and outer-cell interference. In line with the M=M=1 assumption the contribution from within the desired cell is taken as a Poisson rv with mean equal to the cell oered trac. The outer-cell interference is approximated as a Gaussian rv with mean and variance obtained via simulation. A numerical convolution of the Gaussian and Poisson dfs then leads to the df for the total interference. The trac capacity corresponding to two QoS values is presented as a function of the path loss exponent of the propagation model.

[FRG94] Many of the assumptions of this paper are as in [LE94]. Only the reverse link is considered, the QoS is based on a minimum SIR requirement, perfect power control of received signal strength is assumed, and each cell acts as an independent M=M=1 queue. The internal and external interference are again treated separately, the internal a Poisson rv and the external a Gaussian rv. The mean and variance of the external interference are calculated by analytical and numerical methods based on the treatment in [GJP91] which includes lognormal shadowing in the propagation model. The blocking or outage probability is then given in the form of a convolution as in [LE94]. The results presented do not shed much light on the trac behaviour of CDMA mobile systems since the above technique is used solely to compare CDMA with a packet reservation multiple access scheme. It is also not clear how the results, given in terms of a spectral eciency measure, are related to the trac capacity of the system since the eciency is de ned as a function of the maximum number of users supported per unit area and not the maximum Erlang trac supported per unit area.

[EE94] The analysis of reverse link trac capacity for CDMA cellular mobile networks developed in this chapter and in [EE94] shares many of the features of the above papers. In particular we employ the independent M=M=1 queue model for each cell, the service requirement is in terms of SIR, and each mobile is power controlled to a xed and equal power.


38

In the most general development [EE94] arbitrary network layouts, user distributions, and trac pro les are allowed, and lognormal shadowing is included in the propagation model. If however, a symmetric structure is imposed, the calculation of the service measure reduces to evaluating the probability of a compound Poisson sum exceeding a certain threshold. If the propagation model does not include shadowing, an analytic expression is available for the distribution of the random summands. The service measure is approximated using a standard Gaussian approximation and bounded with the Cherno bound and results are presented for one and two dimensional networks for various propagation environments and system bandwidths. The approach in [FRG94] is closest in spirit to this work but diers in several aspects. Firstly [FRG94] treats internal and external interference separately thereby requiring a numerical convolution at the last step. This is avoided in our approach where there is no distinction made. Secondly they give no analysis of, or justi cation for, the Gaussian approximation while we prove a central limit result for compound Poisson sums. Thirdly our analysis is strengthened with the use of the Cherno bound and an illustration of its asymptotic accuracy. Finally we present several results that explore how the service quality varies with the oered trac per cell, system bandwidth, and path loss exponent. Such results are not given in any of the above papers.

4.2 The M=G=1 Approximation In this section we discuss the assumptions leading to, and justi cation for, modelling the CDMA cellular network as a collection of independent M=G=1 queues. It is also shown that the gains from employing voice activity monitoring enter the analysis in a trivial multiplicative manner.

4.2.1 Basic Model The assumptions of the trac model we will use are as follows:

the call initiation processes in each cell are modelled as independent Poisson streams, all arriving calls are accepted into the network and remain in the network for the full call duration (no blocked or dropped calls),

call durations are generally distributed and independent of the arrival processes and other holding times,

mobility is not modelled and thus the mobile is associated with the cell of its call initiation for the duration of the call.


39

The rst and third points are standard assumptions from teletrac engineering that have been employed for several decades to model the stochastic nature of call arrivals to telephone exchanges and their circuit holding times. The second assumption is reasonable for systems operating with CDMA since there is no theoretical hard limit on the number of quasi-orthogonal codes available to assign to users. From a more practical point of view it is reasonable to assume that there are enough codes available so that the new call blocking probability is negligible for moderate oered trac. The nal assumption is a good approximation when the cell size is large compared to the distance a typical mobile will travel during a call. In particular the trend toward hand held mobiles drastically reduces the need to include mobility in trac models. The assumptions imply that each cell of our network behaves like an independent M=G=1 queue [Kle75]. This is one of the most basic queueing models and has a particularly simple form for the steady state distribution of the number of active calls. If the mean time between call arrivals is 1= seconds and the mean call holding time is 1= seconds then the trac to the system is A = = Erlangs. Let N be the rv representing the number of active calls in the system at steady state. Then N has the Poisson distribution n

P (N = n) = e?A An! : The independence of each cell in the network implies that the joint steady state distribution for the number of active calls in each cell is simply a product of Poisson distributions.

4.2.2 Inclusion of Voice Activity Eects Now let us suppose that once a mobile call is connected to the network the mobile user is ON with probability and OFF with probability 1 ? . This model results when voice activity monitoring is included and the subsequent suppression of transmission by a mobile after voice inactivity is detected. We are now interested not in the number of mobiles connected to a base station but in the the number of mobiles in a cell that are ON. Let this number be M . We have

P (M = m) =

1 X

n=0 1 X

P (M = mjN = n) P (N = n)

n! n=m m!(n ? m)! m = e? A ( mA!) =

m(1 ?

n

)n?m e?A An!

which is again Poisson distributed but with reduced trac load A. Thus the gains from voice activity detection enter the formulation in a simple multiplicative manner.


40

Before proceeding to the next section we make one nal point. In the rest of this chapter it is assumed that the trac oered to every cell of the network is equal. We emphasise that this equality applies to the parameters of a stochastic model and is distinctly dierent to assuming an equal static load in every cell. This along with the in nite, symmetrical, cellular layouts and uniform user distributions that we have assumed allows all calculations to be performed for one cell of the network only. The extension of this work to asymmetrical layouts, oered trac, and user distributions is straightforward from a theoretical point of view [EE94] and is not included here.

4.3 Modi ed Blocking Probability In this section we develop a simple expression for a QoS indicator which we call the blocking probability. Calculation of the blocking probability reduces to the evaluation of the probability that a compound Poisson rv exceeds a given threshold. The analysis of such an expression is left to the following section.

4.3.1 De nition In the absence of new call blocking we use as a performance measure the probability that a mobile initiating a call achieves an insucient SIR on connection. Although this measure is more closely related to outage probability than blocking probability we choose the latter name to re ect the trac modelling orientation. To calculate the blocking probability, B , we must determine the probability that an arbitrary call arriving anywhere in the cellular network receives a reverse link SIR that is insucient for acceptable QoS. If certain symmetries exist then B will be the same for mobiles arriving to any point in the network and we may just as well consider calls that connect to a particular BS. Associate with this target BS and its cell the index 1. Because of the standard power control assumptions, an arriving call would be received at BS1 with one unit signal power. We can thus easily translate the SIR requirement into a constraint on the total interference at BS1 at the call arrival time. Because the arrival process is Poissonian this is equivalent to the time average probability that the total received power at the target BS exceeds ?. In summary the expression for blocking probability is reduced to

B = P (I > ?)

(4.2)

where I is a rv representing the total power received at an arbitrary BS in the network at some time. ? is a measure of the size of the CDMA system and is related to the system


41

bandwidth (W Hz), the data rate (R bps), and the required bit energy to interference density ratio (Eb=I0 ) by [GJP91] ? = EW=R =I : b 0

4.3.2 Interference Suppose that there are M ? 1 cells apart from cell 1 that generate signi cant interference at the target BS2 with labels 2; : : :; M . Assume that the interference rvs for mobiles in cell i are iid with df Fi and the interference rvs from dierent cells are independent. Given Ni calls in cell i the total power received at BS1 is given by

I=

Ni M X X i=1 j =1

Iij

where Iij is the interference from the j th mobile in celli and the Ni are independent Poisson rvs with mean A. It is readily shown using characteristic functions that N X I =: Ij j =1

(4.3)

P

where N is a Poisson rv with parameter MA and Ij is an rv with df M ?1 M i=1 Fi being a : nite mixture of the original dfs. The symbol = indicates equality in distribution. In (4.3) the interfering cells are combined and the total trac into the conglomeration considered. Combining (4.3) and (4.2) we arrive at a simple expression for the blocking probability in the network 1 0N X (4.4) B = P @ Ij > ?A : j =1

We now present an example to illustrate and clarify the ideas of the last sections.

4.3.3 Example Consider the standard two dimensional layout of Figure 3.3 with A Erlangs of trac oerered to each cell. Assume the mobile locations within each cell are iid rvs uniformly distributed over each cell and that the propagation environment is governed by (3.1). The interference resulting at some target BS from a mobile randomly located in a cell of the network is characterised by the approximate df of (3.8). If only the target cell and the rst two surrounding rings of cells are taken to contribute signi cantly to the total interference 2

For example we might include only the rst two layers of cells in one or two dimensional systems.


42

P

at the target BS we have I = Nj=1 Ij where N is a Poisson rv with mean 19A and the Ij are a sequence of iid rvs with df p

F (z) = 19?1(u(z ? 1) + 6FI1;b (z) + 6FI 3;b(z) + 6FI2;b (z)):

(4.5)

In the above u(z ) is the unit step function and FIa;b (z ) are given in (3.8). To calculate the blocking probability as a function of the oered trac per cell we are faced with evaluating (4.4). Two methods of approximating this probability are described in the next section.

4.4 Blocking Probability Approximations In this section we consider techniques for approximating

1 0 NA X P @ Xj ?A j =1

where X1; X2; : : : are iid rvs and NA is a Poisson rv with mean A that is independent of the Xj . P A X and denote the mean and variance of S by E [S ] = Let us de ne SNA = Nj =1 j NA NA 2 2 S and var(SNA ) = S respectively. Then if E [X1] = X and var(X1) = X ,

S = AX S2 = A(X2 + 2X ):

4.4.1 Normal Approximation The Approximation The normal approximation is

SNA ? S =: N (0; 1) S

where N (0; 1) is a zero mean, unit variance, normal rv. We thus have

1 0N ? ? A X S A @ Xj ? = P (SNA ?) 0:5 ? erf P S

j =1

where

Zx 2 1 e?t =2 dt: erf(x) = p 2 0


43

Asymptotic Behaviour: Integral A To examine the asymptotic behaviour of this approximation as A ! 1 we rst assume that A takes non-negative integral values only. Since the sum of Poisson rvs is also Poisson, we have NA A X X SNA = Xj =: Yj j =1

j =1

where Y1 ; Y2; : : :; YA are iid rvs and

N1 X : Xj Y1 = j =1

E [Y1] = Y = X var(Y1 ) = Y2 = X2 + 2X : The sum with a random number of summands has been transformed into a standard deterministic sum of iid rvs for which we can apply the central limit theorem (CLT) in its simplest form. The CLT [Bil86] states that provided Y is nite and Y2 is positive and nite, SNA ?pAY = SNA ? S ) N (0; 1) as A ! 1: S Y A The symbol ) refers to convergence in distribution.

Asymptotic Behaviour: Real A We now consider the case when A is a positive real. As no appropriate result could be found in the literature we prove the required CLT here using characteristic functions. For convenience we use the following notation:

E [X1] = m1 E [X12] = m22 p A = so that S = Am1 and S2 = Am22 .

Theorem 4.1 If m1 is nite and m2 is positive and nite then SNA ? Am1 ) N (0; 1) as A ! 1: m2

Proof:

? Am1 , with Let ZA = SNAm 2

h

i

ZA (t) = E eitZA = e?im1 =m2 eA(X (t=m2)?1)


h

44

i

where X (s) = E eisX1 . This follows by conditioning and use of elementary properties of characteristic functions. Because m22 < 1 1 ? t2 ? t2 (t=m ) X (t=m2 ) = 1 + itm 2 m 2A 2Am2 2 2

2

where 2 (s) ! 0 as s ! 0. Thus t2

(t=m2) ZA (t) = e?t2 =2e 2m22 2 :

Fix t 2 (?1; +1) so that t2 2 (t=m2 ) 2 =2 2m2 ? t lim (t) = e lim e 2 A!1 ZA A!1 t2 2 (t) 2 =2 ? t 2m2 = e lim e 2 !0 = e?t2 =2

by continuity of the exponential at 0. Finally by the Continuity Theorem for characteristic functions ZA ) N (0; 1) as required.

4.4.2 Large Deviations Bound We now give an upper bound on the blocking probability using the Cherno bound. The asymptotic behaviour of this bound is discussed in the context of elementary large deviations theory. Consider rst the case when A takes on positive integer values and the Poisson sum can P be rewritten as the deterministic sum SNA =: Aj=1 Yj . The large deviation rate function is de ned by I (t) = sup [t ? MY ()] where is real and

h

i

MY () = log E eY1 = eMX () ? 1

is the log moment generating function (LMGF) of the Yj which is related as shown to the LMGF of the Xj , MX (). Provided MY () < 1 for all and that Y1 is not a bounded random variable in the sense that P (Y1 2 (a; b)) < 1 for all nite a and b, then from Cramer's Theorem [Buc90] 1 1 lim log P S = ?I ( ) (4.6) A!1 A

A

NA


45

for > Y . Moreover for all positive integral A 1 log P 1 S ?I ( ):

A

A

NA

The above bound is commonly called the Cherno bound and is directly applicable for any positive real value of A. To extend the limit result of (4.6) to the case when A is real is more involved but is readily accomplished either by modi ed use of Cramer's Theorem or by direct application of the more powerful Gartner-Ellis Theorem [LS92, DZ93]. Applying the above to our problem we have 1 log P (S ?) ?I ? NA

A

A

? = inf M ( ) ? Y A

MX () ? 1 ? ? = inf e A

and from Cramer's Theorem the bound becomes tight as A ! 1 with ?=A held constant.

4.5 Results In this section we use our previous results to examine the trac performance of one and two dimensional networks. After describing the network models used we compare calculated blocking probabilities by simulation with both the Cherno bound and Gaussian approximation for some representative cases. The variation in performance with both path loss exponent (PLE) and system size (?) is then investigated for one and two dimensional networks.

4.5.1 Network Models Two Dimensional The two dimensional model used in calculations is as in Section 4.3.3. In particular the df of the interferers is given by (4.5) with b = 0:53. It should be remembered that this df is implicitly dependent on the PLE.

One Dimension We consider the one dimensional equivalent of of the model in Section 4.3.3. The topology is now as in Figure 3.2 and the df of the interferers is given by

F (z) = 0:2(u(z ? 1) + 2FI1(z) + 2FI2 (z)) where FIa (z ) is de ned in (3.4).


46

Parameter Ranges In what follows we consider blocking probabilities in the range 0.01% to 10% with the oered trac limits altered to produce this range for each scenario considered. The two main parameters we have to vary are the PLE and ?. The PLE used lies in the set f2,3,4,5g with 4 being a typical value for existing macrocellular systems. ? takes values in f20,100,500g which might correspond to systems with Eb =I0 = 7 dB, R = 10 kbps, and W = 1 MHz, 5 MHz, and 25 MHz respectively. In all plots the ordinate represents the base 10 logarithm of blocking probability while the abcissa corresponds to the oered trac per cell divided by ?. The trac axis is thus normalised by the size of the system making capacity comparisons for dierent ? straightforward. Simulation points are accurate to within plus or minus 20% with 95% con dence.

4.5.2 Comparison of Bound and Approximation with Simulation In Figure 4.1 and Figure 4.2 we compare the Cherno bound and Gaussian approximation to simulation for the two dimensional network with PLE = 4 and ? = 100 and ? = 20 respectively. The following points are evident:

The bound overestimates blocking probability by about an order of magnitude in

both cases. This translates to under estimating trac capacity by about 10% in Figure 4.1 and 15% in Figure 4.2.

The accuracy of the approximation decreases as the oered trac and thus the

blocking probability decreases. The eect is less severe for the larger ? since in this case we are eectively summing a larger number of rvs and thus getting a better approximation to the tail of the sum.

The above points give some heuristic tips on when the Gaussian approximation is reasonable. Clearly for large values of ? and high blocking probabilities the approximation is excellent, however for low values of ? (< 20) and or low blocking probabilities (< 0:1%) the accuracy of the approximation may deteriorate rapidly. In the latter case the bound is a much safer and more robust technique.

4.5.3 Variation of Bound with System Parameters Figure 4.3 and Figure 4.4 show how the trac capacity varies with PLE and ? respectively for the two dimensional network. The one dimensional analogs appear in Figure 4.5 and


47

-0.5

Log Blocking Probability

-1

Chernoff Bound Gaussian Approx Simulation

-1.5 -2 -2.5 -3 -3.5 -4 -4.5 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 Normalised Traffic per Cell

? = 100, PLE = 4, Two Dimensions Figure 4.1: Blocking Probability vs Normalised Trac per Cell -0.5


-1

Chernoff Bound Gaussian Approx Simulation

-1.5 -2 -2.5 -3 -3.5 -4 -4.5 0.25

0.3

0.35 0.4 0.45 Normalised Traffic per Cell

0.5

?=20, PLE = 4, Two Dimensions

Figure 4.2: Blocking Probability vs Normalised Trac per Cell

0.55


48

-1


-1.5 -2 -2.5 -3 PLE = 2 PLE = 3 PLE = 4 PLE = 5

-3.5 -4 -4.5 0.25

0.3

0.35 0.4 0.45 0.5 0.55 Normalised Traffic per Cell

0.6

0.65

? = 100, Two Dimensions, Cherno Bound Figure 4.3: Variation of Blocking Probability with PLE -1


-1.5 -2 -2.5 -3 -3.5

Gamma = 20 Gamma = 100 Gamma = 500

-4 -4.5 0.25

0.3

0.35 0.4 0.45 0.5 0.55 Normalised Traffic per Cell

PLE = 4, Two Dimensions, Cherno Bound

Figure 4.4: Variation of Blocking Probability with ?

0.6

0.65


49

-1


-1.5 -2 -2.5 PLE = 2 PLE = 3 PLE = 4 PLE = 5

-3 -3.5 -4 0.45

0.5

0.55 0.6 0.65 Normalised Traffic per Cell

0.7

0.75

? = 100, One Dimension, Cherno Bound Figure 4.5: Variation of Blocking Probability with PLE -1


Gamma = 20 Gamma = 100 -1.5 Gamma = 500 -2 -2.5 -3 -3.5 -4 -4.5 0.2

0.3

0.4 0.5 0.6 0.7 Normalised Traffic per Cell

0.8

PLE = 4, One Dimension, Cherno Bound Figure 4.6: Variation of Blocking Probability with ?

0.9


50

Figure 4.6. In these plots the Cherno bound was used to obtain values for the blocking probability. We make the following points:

The capacity (for a xed blocking probability) is signi cantly reduced as the PLE decreases. The eect is less severe in the one dimensional network where each cell has only two cells adjacent.

The economy of scale for systems with large ? results in signi cant increases in

normalised trac capacity. This is important in comparing narrowband CDMA (low ?) to wideband CDMA (high ?).

4.6 Conclusion The analysis and results of this chapter provide a rst step toward understanding the trac capabilities of CDMA cellular mobile networks, an area which has been neglected in the literature to date. The key assumption is that each cell can meaningfully be modelled as an independent M=G=1 queue and we began the chapter with a thorough critique of the few papers where a similar assumption is used. After discussing the justi cation for, and consequences of the M=G=1 model, an expression for blocking probability was developed in terms of a compound Poisson random variable. Two techniques were then applied to approximate the blocking probability along along with corresponding asymptotic results. The numerical results provide an initial estimate of the trac capacity of CDMA networks and demonstrate the sensitivity to propagation parameters and system processing gain. The primary shortcoming of the preceding analysis is that it provides no information on how a network operator should control call admissions to the network so as to provide a more robust quality of service. This issue is addressed in the following chapter.

Chapter 5

Eective Interference and Trac Models with Product Form In this chapter we develop product form trac models for single and multiple cell CDMA networks with multiple classes of mobile subscriber. The key feature of this development is the speci cation of a exible call admission control procedure that details the numbers of mobiles of each class in each cell that the system operator should allow in order to maintain a certain quality of service. Familiar techniques from the analysis of statistical multiplexing at an ATM-based broadband ISDN link are used to give performance guarantees that overcome the variability in interference levels characteristic of CDMA cellular networks. The result is an admissible region bounded by a nite number of hyperplanes and a simple, ecient, call admission policy. The CDMA mobile network, operating within the admissible region described above, has a very similar form to a circuit-switched network operating with xed routing. This similarity allows the existing trac modelling techniques and network management strategies for general loss networks, to be applied to CDMA mobile cellular networks. In particular, with standard assumptions on the call arrival processes and holding times, the stationary state distribution has a product form on the truncated state space de ned by the call admission stategy.

5.1 Introduction The analysis of Chapter 4 was aimed at producing a fairly simple technique for calculating the trac capacity of CDMA cellular mobile networks and for investigating the sensitivity of this capacity to changes in the propagation environment and system parameters. The major assumption was that each cell of the network behaves as an independent M=G=1 51

CHAPTER 5. EFFECTIVE INTERFERENCE AND PRODUCT FORM

52

queue and thus has a stationary state distribution which is Poisson. Of course this assumption means that all calls are accepted into the network regardless of the state of the system, however in practice, the network operator would control access to the network in order to prevent the system from becoming overloaded. The M=G=1 models provide no information on whether or not a requesting mobile should be accepted into the network. It is to this problem that we turn in this chapter. Let us assume the network operator can accept or reject mobile requests based on the network state which consists of the number of currently active mobiles of each class in each cell of the network. The aim is to only allow network states for which the QoS is acceptable (signal to interference ratio above some minimum requirement) with a high probability. An initial scheme for a system with a single user class is described in Section 2.5. This approach assumes that there are an equal number of active mobiles in each cell and maximises this number subject to the QoS constraint1 . Once we have this number N , our call admission control algorithm simply rejects any call that arrives at a cell with N calls already in progress. We can then model each cell as an independent M=G=N=N loss system and calculate the new call blocking probability from the Erlang B formula. This approach to resource allocation is similar to xed channel assignment in conventional channelised systems and does not take advantage of the inherent exibility of CDMA networks. At times when neighbouring cells are lightly loaded, a cell may be able to hold more than N users and we would like to investigate the possible gains from a more

exible call admission control. Further the above approach appears to have limited scope for extensions to multiple classes of mobile call which we feel will be an integral part of future wireless networks. A rst attempt to model exible call admission schemes is presented in [Eve94a, Eve94b]. The author uses inter-cell interference factors to give a set of linear constraints that de ne the capacity region in terms of the number of users in each cell. These intercell interference factors are constants which are assumed to account for the interference from other cells, however as the author points out in [Eve94c], the other-cell interference demonstrates random behaviour strongly dependent on the location of mobiles, shadowing eects and voice activity. This issue is not addressed in [Eve94a, Eve94b] where the emphasis is on the trac modelling and performance of the call admission control algorithm using the exible capacity region. One possible approach to handle the randomness of the other-cell interference is to use While the calculation of N is by no means solved for general CDMA mobile networks, there is a body of literature centering on this static capacity problem [GJP91]. 1


53

the expected value. If there is much variation in the total interference however, the QoS may suer with signi cant periods of poor radio link quality. On the other hand, the peak interference value would provide a very robust interference factor, but the system would obtain no bene ts from statistical multiplexing, a position which is totally untenable for CDMA mobile. What we would like, is choose a value lying between the mean and peak values which can be used to de ne a capacity region where a certain QoS constraint will be met. A similar problem arises in the the admission of calls to a broadband ISDN network using ATM [Eve94c] where it is useful to assign an eective bandwidth to variable bit rate sources. It is suggested in [Eve94c] that the basic concept of eective bandwidth might be usefully applied to cellular CDMA systems, although no details are given. A similar remark appears in [MM94], where it is mentioned that eective bandwidth concepts may be used in CDMA networks to model the interference from voice speakers in cells at various distances, and to incorporate voice and data sources with bursty bit rate requirements. This remark appears to be motivated by the analysis of CDMA networks presented in [VV93, Han93], however it is by no means clear that the applications suggested above follow from these references. In [Han93, Section 9.7] it is shown how an eective bandwidth problem arises in a multiple receiver CDMA network with multiple classes of user and random resource requirements. This applies only for a macrodiversity model which dismisses the cellular concept and assumes all users are jointly decoded by all receivers in the network. The resulting capacity speci cation is in terms of the total number of users in the system and leads to an analysis very similar to the single cell case of Section 5.3. We emphasize that this analysis does not apply in the cellular context. In this chapter we investigate how eective bandwidth concepts can be utilised in the modelling of CDMA cellular mobile networks. These concepts allow us to associate an eective interference to each mobile dependent on its location relative to the target cell and to de ne a capacity region determined by a set of linear constraints. The resultant network model is of similar form to a circuit-switched network and the trac analysis for the latter can be directly applied to the CDMA cellular model. This work is also presented in summarised form in [EE95a, EE95b]. We begin the development in Section 5.2 with a look at the call admission problem for an unbuered resource shared by multiple classes of user with possibly variable resource requirements. The concept of eective bandwidth is explained and several simple methods for calculating these bandwidths are presented. The important point is that quite general single and multiple cell CDMA cellular networks can be analysed using the techniques examined in Section 5.2. The appropriate framework for modelling CDMA networks is


54

the subject of Section 5.3. Section 5.4 looks at the trac modelling of CDMA networks operating within the

exible capacity region developed in the preceding sections and illustrates the similarity to the analysis of xed circuit-switched networks. While the independence of the cells vanishes in the more dynamic form of operation, the stationary state distribution maintains product form provided some standard assumptions hold for the call arrival processes and holding times. The reduction of many CDMA network models to circuit-switched form is the most important contribution of this chapter, the analogy allowing several important operating strategies and analysis techniques from the circuit-switched literature to be directly applied to CDMA mobile systems. A selection of results is presented in Section 5.5 including eective interference values for a large number of propagation and system parameters. In Section 5.6 we conclude with a summary of the major insights and results gained in the chapter, as well as discussing possible directions for future work.

5.2 Eective Bandwidth Control of Call Admissions to a Shared Resource Consider several data sources multiplexed onto a single link of nite capacity, C bits per second. Requests to use the link arrive from each source as a point process in time and can either be accepted or denied. An accepted call is given access to the link for a random call holding time and a rejected request is lost. We are interested in the number of sources of varying type that should simultaneously be allowed access to the link, while maintaining a low link overload probability. The resource requirement for a particular source may be constant or variable during the call lifetime and there may be a single class of source or multiple classes. In all of the above cases we wish to associate an eective bandwidth with each call such that we allow any state in which the sum of the eective bandwidths of all connected calls is below the link capacity.

5.2.1 Calls with Constant Resource Requirement Single Class If every call requires a constant bit rate of , the link can carry N calls provided

N C


55

and the eective bandwidth of a call in this case is simply the constant resource requirement, .

Multiple Classes Now suppose there are J classes of call with constant resource requirements of 1 , : : :, J . The link can support Nj calls of class j , j = 1, 2, . . . , J , if J X

j =1

j Nj C

and the eective bandwidth of a call of class j is simply j .

5.2.2 Calls with Variable Resource Requirement If we allow the resource requirements of a call to vary throughout its lifetime the problem becomes much more interesting. How can we assign a bandwidth to a call with variable bit rate requirements such that the bandwidth is useful in controlling the overload probability? If the requirement from each source is bounded, then the peak rate is one possible bandwidth gure. This allows an admission control scheme that leads to zero overload probability, but may be very wasteful for bursty sources with a large dierence between peak and mean rates. Another possibility is just to use the mean rate which will lead to much better link throughput than if the peak rate was used. If however, there is much variability in the resource requirements, the overload probability may be higher than desired. In any case, the choice of mean rate provides no real control over overload probability. The ideal eective bandwidth would lie somewhere between the mean and peak rates and would lead to a capacity region where the probability of link overload was kept below a certain level. There has been a great deal of research on this problem especially in the light of its importance in ATM network operation [Hui88, DCL90, GAN91, Kel91a, Kel93, EM93, KWC93, MM94]. In what follows we describe two simple approaches for determining eective bandwidths of variable rate sources, the rst based on a Gaussian approximation and the second using the Cherno bound. The Gaussian approximation is mentioned in [Hui88], where it is not used due to robustness and accuracy problems for large deviations from the mean, and in [GAN91], where it is employed to dimension a collection of variable bit rate sources. The material on the Cherno bound approach is taken directly from [Kel91a, Section 2] and is based on the earlier treatment in [Hui88] which introduces the eective bandwidth concept for unbuered resources.


56

Gaussian Approximation: Single Class Suppose that there are N calls of a single class sharing the link and that the resource requirements of these calls are the independent and identically distributed (iid) random variables (rvs) X1, : : :, XN . We wish to guarantee that the probability of link overload remains below a QoS parameter, , which can be expressed as the constraint

P

N X i=1

!

Xi > C :

If = E [X1] is nite and 2 = var(X1) is nite and non-zero, then the central limit theorem (CLT) [Bil86] provides justi cation for approximating the sum of iid rvs by a Gaussian rv. Applying this approximation gives the QoS constraint

p P N (0; 1) > (C ? N)= N

where N (0; 1) is a zero mean, unit variance, normal rv. The above constraint is satis ed if and only if where is de ned implictly by

C ?pN N

p1

Z1

(5.1)

e?t2 =2 dt = :

(5.2) 2 Following on from (5.1), it is straightforward to show that the constraint can be written in the required form, N C , where the eective bandwidth of a call is given by 2 p = = 1 + 1 ? 1 + z (5.3)

z

with z = 4C= 2 2. A fully analytic form results if we use an approximation for the area under the tail of the complementary error function [AS65, GAN91] giving

p ?2 ln ? ln 2:

(5.4)

Assuming the Gaussian approximation is accurate, the above eective bandwidth de nes a simple capacity region (N C ) in which the constraint on overload probability is satis ed. We note that and is close to the mean when z >> 1 which corresponds low trac burstiness, a large capacity link and high overload probability. The advantages of the above approach include computational simplicity and the need to know only the mean and variance of the data streams. The main disadvantage is the uncertainty about the accuracy of the approximation. In particular when < 0:01 the accuracy may be questionable.


57

Gaussian Approximation: Multiple Classes We now extend the above approach from a single class to multiple classes. Let there be J classes of call and de ne the state of the link as the number of calls of each type accessing the link (N1; : : :; NJ ). Take Xji as the rv representing the load on the resource by the ith call of class j . All the rvs are independent and rvs in the same class are by de nition identically distributed. The probability of overload will be less than provided

1 0 J Nj X X Xji > C A : P@ j =1 i=1

Again invoking the normal approximation and proceeding as in the case of a single class, we are led to the constraint J X j =1

0J 11=2 X Nj j + @ Nj j2A C j =1

(5.5)

where j and j2 are the mean and variance respectively of calls of class j , and is as in (5.2) or (5.4). We note that (5.5) de nes a capacity region in terms of the state of the link, however this inequality is nonlinear in the state vector and we cannot associate a simple eective bandwidth with calls of each type unless we linearly approximate the region. Let RJ+ denote the positive orthant of J dimensional Euclidean space and call a state (N1; : : :; NJ ) 2 RJ+ admissible if it satis es (5.5). The complement of the admissible region is a convex set in RJ+ . This suggests the following linear approximations. Homogeneous Gaussian: an overbounding approximation by the hyperplane through the intersection of the admissible region boundary with the J Euclidean axes. The intersection point (0; : : :; 0; Nj; 0; : : :; 0), corresponds to the edge of the single class capacity region for class j calls. We can thus write Nj = C=oj, where oj is as in (5.3) with and replaced by j and j respectively. It follows that the overbounding linear constraint is given by J X

j =1

oj Nj C:

Heterogeneous Gaussian: an underbounding approximation by any tangent hyperplane to the boundary of the admissible region. If (N1; : : :; NJ) satis es (5.5) with equality, the tangent hyperplane at this point is given by J X j =1

2 0J 1?1=23 2 X (Nj ? Nj) 64j + j @ Nj j2A 75 = 0 2 j =1


58

and the underbounding linear constraint is J X j =1

where

uj Nj C

2 0J 1?1=23 2 X uj = 1 64j + 2 j @ Njj2 A 75 j =1

and

0

11=2

J X = 1 ? 2 C @ Njj2A : j =1

The rst approximation is simpler than the second, but has the disadvantage of overbounding the capacity region. The second approximation gives an underbound that is accurate near (N1; : : :; NJ), however it should be remembered that this is only true if the Gaussian approximation is exact. We move now to a more robust procedure based on bounding the overload probability.

Large Deviation Bound: Single Class Suppose once more, that there are N calls of a single class sharing the link with iid random resource requirements X1 , : : :, XN . We are again interested in the values of N for which

P

N X i=1

!

Xi > C

(5.6)

but rather than appealing to the central limit theorem we will use results from large deviations theory which are more robust for analysing tail probabilities. The main result we need is the Cherno bound, log P where

N X i=1

!

Xi C infs [NM (s) ? sC ]

h

M (s) = log E esX1

i

is the common logarithmic moment generating function (LMGF) of the rvs. We have already discussed this bound and its asymptotic behaviour in Section 4.4.2 suce to say that under some mild technical conditions the bound becomes tight as N ! 1 with C=N held constant [Buc90, Kel91a]. Using the Cherno bound it is clear that the constraint of (5.6) will de nitely be satis ed provided inf s [NM (s) ? sC ] log :


59

If the equality holds when N = N and the corresponding in mum is attained with s = s , then our admissible region has the form N C where the eective bandwidth of a call is given by (s ) = CsCM (5.7) + log :

Large Deviation Bound: Multiple Classes Now let there be J classes of call with Nj calls of class j using the link. Take Xji as the rv representing the load on the resource by the ith call of class j . All the rvs are independent and rvs in the same class are by de nition identically distributed with LMGF, Mj (s). Employing the Cherno bound, we know the overload constraint will be satis ed if

2J 3 X infs 4 Nj Mj (s) ? sC 5 log : j =1

(5.8)

Call a state (N1; : : :; NJ ) 2 RJ+ admissible if it satis es (5.8) and note the complement of the admissible region is a convex set in RJ+ , since it is de ned as the intersection of RJ+ with a collection of half spaces. We again consider two linear bounds on the admissible region. Homogeneous Cherno: a linear approximation to the admissible region boundary that provides an overbound on the capacity of the link. We choose the hyperplane passing through the points (0; : : :; 0; Nj; 0; : : :; 0); j = 1; : : :; J on the boundary of the original admissible region. However the Nj must be the maximum number of calls allowed on the link when only class j calls are present and these numbers can be calculated using the single class analysis of the preceding section. Heterogeneous Cherno: a tangent hyperplane at the point (N1; : : :; NJ) on the boundary can be used to give a linearization which underbounds the admissible region. If

3 2J X 4 NjMj (s) ? sC 5 = s; arg inf s j =1

then the probability of overload will be below if J X j =1

where

uj Nj C

CMj (s ) j = Cs + log

is the eective bandwidth of calls of class j .


5.3

60

Eective Interference Control of Call Admissions to CDMA Cellular Mobile Networks

In this section we show that many CDMA mobile networks can be analysed using eective bandwidth techniques with the link overload probability replaced by a QoS measure based on system state feasibility. Initially we study single cell systems, allowing mobiles to have various bit rates and error performance requirements. We then move to the multiple cell environment where other-cell interference, while complicating the analysis, is readily incorporated into the eective bandwidth framework. In the sequel, the system state refers to a particular realization of the random variables that in uence service quality. In the single cell model the system state consists of the bit rate and modem requirements of every active mobile, while in the multiple cell models the system state is augmented with the propagation losses from each mobile to each base station in the network. In contrast, the network state consists of a realization of the number of mobiles of various classes and types that are active in the network. In the single cell model the network state is the number of active mobiles of each possible class, while in the full multiple cell analysis, the network state is comprised of the number of mobiles of each class in every cell of the network. Much of the work on system state feasibility discussed in this chapter, also appears in [Han93, Chapter 8] where simple networks are considered under quite general conditions on received power levels. Here, the emphasis is on larger and more complex cellular systems operating with some restrictions on power levels.

5.3.1 Single Cell Models Consider a system comprised of a single base station to which N mobiles are connected. Suppose that at an arbitrary time instant, mobile i requires a bit rate of Ri and a bit energy to interference density ratio of (Eb =I0)i . Both of the above variables may be deterministic or random and may vary from mobile to mobile. For example the system might support several classes of constant and variable bit rate calls, and services with dierent bit error rate requirements2 . The vector of user requirements, or system state, ((Eb=I0 )1; : : :; (Eb=I0 )N ; R1; : : :; RN ) will be called feasible, if all the requirements can be met simultaneously. Diverse propagation channels and power control errors may also contribute to the variability in user demands. 2


61

Let Xi be the signal power received at the base station from mobile i, i = 1; : : :; N . Using a simple relationship between signal to interference ratio and bit energy to interference density ratio3 [GJP91], the system state is feasible if and only if there exist non negative values for the received signal powers such that P Xi Ri(Eb=I0)i ; i = 1; : : :; N (5.9) j 6=i Xj

W

where W is the spread spectrum system bandwidth. We also require that at least one of the Xi is non zero so that the left hand side of (5.9) is well de ned for all i. P P Writing ?i = Ri (Eb=I0 )i =W and making the approximation j 6=i Xj j Xj for all i, leads to an approximate condition for feasibility

PNXi X ?i ; i = 1; : : :; N: j =1 j

(5.10)

Allowed values of X1 ; : : :; XN satisfying the N inequalities of (5.10), exist if and only if N X i=1

?i 1:

(5.11)

This follows from noting that if (5.11) is satis ed, the assignments Xi = ?i ; i = 1; : : :; N are sucient, while summing (5.10) over i leads directly to (5.11) for any feasible system state. Based on (5.11) there exists a simple analogy between a single cell CDMA mobile system and the single link models discussed in Section 5.2. With a link capacity of one unit, mobile i uses up ?i units, and just as in Section 5.2, the resource requirements may come from several dierent classes, and be xed or random. We would like know the numbers of calls of each class that we should allow to receive service, while guaranteeing that the outage probability will remain below a nominal value. This is exactly the problem of controlling the number of sources connected to the link in Section 5.2 and thus the eective bandwidth concepts are directly applicable. In the mobile cellular environment, interference is the limiting factor and we therefore use the term eective interference to distinguish between the mobile environment and standard situations where eective bandwidths are applied. In what follows, we detail the use of eective interference in a single cell CDMA mobile system. Each case examined has a direct analogy with the equivalent case in Section 5.2. 3

We assume that background noise contributes negligibly to the total interference.


62

Single Class, Fixed Requirements If there is only one class of mobile service supported with the constant requirement ? = R(Eb =I0)=W , (5.11) becomes ?N 1 where ? is the eective interference of an active mobile.

Multiple Classes, Fixed Requirements If there are J classes of service supported with constant requirements ?1 ; : : :; ?J , then with Nj mobiles of class j , the vector of resource requirements will be feasible provided4 J X j =1

?j Nj 1:

The eective interference due to a mobile of type j is thus simply ?j .

Single Class, Variable Requirements In this case we consider N mobiles with resource requirements equal to the iid rvs, ?1 ; : : :; ?N , and we wish to guarantee that the feasibility condition of (5.11) is satis ed with probability (1 ? ). We are thus led to the constraint

P

N X i=1

!

?i > 1

(5.12)

which can be handled using the Gaussian approximation or Cherno bound as discussed in Section 5.2. This leads to the association of an eective interference, , to each call such that (5.12) is satis ed if N 1.

Multiple Classes, Variable Requirements Suppose once more that the resource requirements are independent rvs but allow each requirement to come from one of J distributions. If ?ji is a rv representing the resource requirement of the ith call from class j , then the condition on outage probability given Nj calls of class j , is 1 0

XX ?ji > 1A : P@ J Nj

j =1 i=1

(5.13)

This can be handled using the multiple class Gaussian approximation or large deviations bound leading to the assignment of an eective interference, j , to class j calls, in such a P way that (5.13) is satis ed if Jj=1 j Nj 1. 4

This result is equivalent to Lemma 8.1 in [Han93]


63

Example As a simple application of the above discussion we consider a single cell CDMA mobile network oering two voice services. The standard service has a data rate of 10 kb/s while the premium service operates at 20 kb/s. We assume that both classes require a constant Eb =I0 of 7dB and that the system bandwidth is 1 MHz. Voice activity detection facilities are present at the mobile so that the data rate generated by each mobile can be modelled as an ON/OFF source which we assume has 40% chance of being ON. We have two classes of call with variable resource requirements, ?1 (standard) and ?2 (premium), described by the distributions

P (?1 = 0:00) P (?1 = 0:05) P (?2 = 0:00) P (?2 = 0:10)

= = = =

0:6 0:4 0:6 0:4:

We wish to control the admission of calls to maintain the outage probability at less than 1%. With N1 standard calls, and N2 premium calls, we require

1 0 N1 N2 X X P @ ?1i + ?2i > 1A 0:01: i=1

i=1

Using the eective interference results, we can approximate or bound the set of allowable states by those satisfying the constraint 1 N1 + 2 N2 1. 1 and 2 are the eective interferences of the two call types. They allow us to operate the network as if the requirements of each mobile were xed and to take advantage of the statistical multiplexing of bursty data streams while at the same time guaranteeing that our QoS measure is met. We return to this example extended to a multiple set setting in Section 5.3.4. Before proceeding to multiple cell models some subtleties of the analogy between a single cell CDMA mobile system and a nite capacity link should be noted. A link capacity is generally a hard limit on what the link can handle and any overload results in loss of data. Controlling the number of calls accessing the link to limit the overload probability is thus a very critical operation. In a CDMA system however, the outage probability corresponds to a situation where all mobiles cannot simultaneously receive adequate signal to interference ratios. In a feature known as soft, or gradual, overload the system has the ability to share the resultant degradation in performance between all users. This implies a slight loss in performance which may be tolerable and therefore not lead to the dropping of any calls.


64

The dierence in severity between an overload and an outage is highlighted in the typical values for their respective desired maximum probabilities. While overload probabilities are usually kept below 10?4, outage probabilities of between 0.01 and 0.1 are common. This has implications in the accuracy of the relevant eective bandwidth/interference techniques. In particular, the Gaussian approximation would be rarely used for eective bandwidth calculations, however its accuracy is generally good in the outage probability range used in mobile. On the other hand, the Cherno bound will give conservative capacity regions for mobile applications and is more appropriate for bounding probabilities of large deviations from the mean.

5.3.2 Multiple Cell Models: System State Feasibility The multi-cellular networks we study involve each mobile communicating with exactly one base station at any given time, the assignment of users to base stations being xed and based on factors such as propagation path gains. Alternative approaches include the macrodiversity model of [Han93, Chapter 9] where all users are jointly decoded by all receivers in the network and cellular systems where the single base station assigned to a user may vary with the spatial loading of the network [Han95]. The feasibility problem in multi-receiver or multi-cellular CDMA networks is intimately related to power control algorithms and we refer the reader to [Han93, Han95, MM95, Yat95] for recent results in the area. For our speci ed multiple cell environment, a standard extension of the single cell analysis would address the following problem: Given the system state consisting of the number of mobiles in the system, their required bit rates and bit energy to interference densities, and their propagation gains to every base station in the network, is it possible to assign to each user a receive power level at its xed target base station in such a way as to satisfy the bit rates and modem requirements of every mobile? If so, then the given system state is feasible. Suppose the network consists of M cells with Nm users in cell m; m = 1; : : :; M . Let mobile i in cell m have a minimum SIR requirement in terms of bit rate, Rmi , and bit energy to interference density ratio, (Eb=I0)mi , given by ? = (Eb =I0)mi ; i = 1; : : :; N ; m = 1; : : :; M: mi

m W=Rmi Further suppose that mobile i in cell m is received at its target base station with a signal p at the base station of cell p. Then power of Pmi and causes interference of power Pmi Xmi


65

for feasibility of the system state, there must exist non-negative values for the mobiles received powers that satisfy

P PM PNmip P X m ?mi; i = 1; : : :; Nm; m = 1; : : :; M: p=1 l=1 pl pl

(5.14)

We also require that at least one of the Pmi is non-zero so that the above set of inequalities is well de ned. Due to the the coupling between the inequalities, this problem is not trivial even when the system state is deterministic and all bit rate and modem requirements are equal. An enormous amount of simpli cation results if we underbound the set of feasible states by making restrictions on the received power levels. In the simplest network we study this involves assuming that all mobiles are received at equal power by the base station to which they connect5 . This assumption, and other similar ones for more complex networks, leads to a separable set of inequalities allowing eective interference concepts to be applied to the other-cell interference and to the determination of admissible network states. We now turn to several special cases of the general feasibility problem.

Single Class, Fixed Requirements With a single class of user requiring a xed minimum SIR of ?, the feasibility conditions of (5.14) reduce to

P PM PNmip P X m ?; i = 1; : : :; Nm; m = 1; : : :; M: p=1 l=1 pl pl Since all users in a particular cell suer identical interference, there is no loss in generality in assuming Pmi = Pm ; i = 1; : : :; Nm; m = 1; : : :; M . This leads to the problem of the existence of non-negative P1 ; : : :; PM that satisfy

P PM PNmp P X m ?; m = 1; : : :; M: p=1 l=1 p pl

(5.15)

An almost identical problem is discussed in [Han93, Section 8.4] where it is shown that the power control problem has a solution provided M separable single cell constraints are satis ed along with a constraint on the Perron-Frobenius eigenvalue of a negative Metzler p are random, then this matrix is itself random and matrix. If the fading factors, Xmi guaranteeing that the eigenvalue constraint is satis ed with high probability is a dicult problem. As discussed above, the analysis is greatly simpli ed if we assume that all mobiles are received by their target base station at the same power. This leads to a set of constraints 5

This is a very common assumption in the literature on CDMA cellular mobile systems.


66

which provides a lower bound on the set of feasible system states, and it is shown in [Han93, Section 8.4] that this bound is fairly tight for an example two cell system. Writing P1 = P2 = : : : = PM = P in (5.15) gives the seperable inequalities

P PM PNp PX m ?; m = 1; : : :; M; p=1 l=1 pl

or, alternatively,

Np M X X p=1 l=1

?Xplm 1; m = 1; : : :; M:

(5.16)

Multiple Classes, Fixed Requirements In this section we wish to allow J service classes corresponding to dierent bit rates, Rj , and modem requirements, (Eb =I0)j , with the proviso that these values are deterministic. Let ?j = (Rj =W )=(Eb=I0 )j be the resultant minimum SIR requirement of a call of class j and assume that all calls of class j have their received power controlled to Pj . The last assumption is the obvious extension of the additional power control requirement used in the single class analysis. For convenience, write

Qm =

J N M X X Xpk p=1 k=1 l=1

m ; m = 1; : : :; M; Pk Xpkl

m is the interference at where Npk is the number of users of class k in cell p, and Pk Xpkl base station m from mobile l of class k in cell p. Qm is thus the total interference power received at the base station of cell m. The system state is feasible if there exists non negative values of P1 ; : : :; PJ with at least one Pj 6= 0 that satisfy

Pj =Qm ?j ; j = 1; : : :; J; m = 1; : : :; M: If a solution exists, then for m = 1; : : :; M ,

Qm =

)

J N M X Xpk X

m Pk Xpkl

p=1 k=1 l=1 M X J N X Xpk m max Q ?X h h p=1 k=1 l=1 k pkl M X J N X Xpk m ?k Xpkl 1: p=1 k=1 l=1

Conversely, if the implication of the previous line holds for m = 1; : : :; M , then a solution to the power control problem is obtained with Pj = ?j ; j = 1; : : :; J .


67

We have thus established that a system state is feasible if and only if M X J N X Xpk p=1 k=1 l=1

m 1; m = 1; : : :; M: ?k Xpkl

(5.17)

Single and Multiple Classes, Variable Requirements In the developments above, the minimum SIR requirement of a user was assumed xed for the duration of a call and all users belonging to the same class made equal SIR demands. If the SIR requirement of each user is variable during the call lifetime, then the instantaneous demands of each mobile will be dierent. Later we will model the instantaneous desired SIRs of mobiles by independent random variables with identical distributions for mobiles in the same class. For the moment however we will simply assume that we have a realization of the system state, consisting of the various fading factors and SIR requirements, and that we wish to determine if the state is feasible. It should be clear that it makes no sense to assume all mobiles within a class are controlled to a xed power. Rather, we will assume that received power is directly proportional to minimum SIR requirement for every user in the system:

Pmi = ?mi; i = 1; : : :; Nm; m = 1; : : :; M; where without loss of generality the constant of proportionality is taken to be one, and Pmi and ?mi are respectively the received power and minimum SIR requirement of mobile i in cell m. This is a fairly natural constraint that implies the mobile receive power increases as the corresponding SIR requirement increases. For example, if voice activity monitoring is used, then a user would have a minimum SIR requirement of zero during silent periods, and it is sensible to give this user a zero receive power level at these times. At a particular base station, the constraint ensures that the SIRs attained are in the same ratio as the requested SIRs, moving up and down together as the common interference power varies. The resulting power control algorithm is a simple, distributed scheme where each mobile requires only its current SIR requirement and the path-gain to its target base station to determine its transmit power. It should also be noted that there is no loss in generality in using these apparently stricter constraints in the analysis for xed user requirements presented above. The general feasibility inequalities of (5.14) become Np M X X p=1 l=1

?pl Xplm 1; m = 1; : : :; M;

(5.18)


68

after application of the ratio constraints. If we wish to group users into J classes then (5.18) can be rewritten J N M X Xpk X p=1 k=1 l=1

m 1; m = 1; : : :; M; ?pkl Xpkl

(5.19)

with the double subscript, pk, referring to class k mobiles in cell p.

5.3.3 Multiple Cell Models: Network State Admissibility The last section examined the system state feasibility problem. Remember that the system state consists of the the set of fading factors and SIR requirements for every mobile in the system, the state being feasible if an assignment of received power levels can be found that leads to the satisfaction of all SIR requirements. The results are encapsulated in the conditions of (5.16), (5.17), (5.18) and (5.19), all of which can be written in the common form M X J N X Xpk p=1 k=1 l=1

m 1; m = 1; : : :; M; Ypkl

(5.20)

m incorporate the fading factors and SIR needs. where now the Ypkl m are deterministic functions of the indices p, k and m Suppose initially that the Ypkl m = m ; l = 1; : : :; Npk . Then the feasibility conditions become and write Ypkl pk M X J X p=1 k=1

mpk Npk 1; m = 1; : : :; M:

(5.21)

Let the network state consist of the number of mobiles of each class in every cell of the network. We treat (5.21) as de ning an admissible region consisting of network states which guarantee system state feasibility. The admissible region is bounded by M hyperplanes in the positive orthant of M J dimensional Euclidean space. m as deterministic values then we would be done. However If we could treat the Ypkl there is typically a great deal of variability in the fading factors and SIR requirements which cannot be ignored6 . This variability results from factors such as the randomness in user locations within a cell, the shadowing from buildings and land features, and variable bit rate and modem requirements. m as random variables with p, k and We model this variability by treating the Ypkl m indexing dierent distributions. The feasibility constraints of (5.20) only apply to a realization of the random system state and instead we must consider the probability of 6

Unless the system bandwidth is so wide (small minimum SIRs) that the law of large numbers dominates


69

system state feasibility. If this probability is above 1 ? then the corresponding network state will be termed admissible. Our proposed problem then is to nd the set of network states for which

91 0M 8 pk J N M X = 1 ? ; Ypkl P@ : ; m=1 p=1 k=1 l=1

or equivalently,

9 8 M J Npk 1;A : m=1 :p=1 k=1 l=1

0M [ P@

A network state is thus admissible if the probability of every mobile simultaneously meeting its minimum SIR requirement is above 1 ? . We prefer to use a dierent de nition of network state admissibility however, that is easier to analyse and which is perhaps more sensible for CDMA cellular networks. The admissible region is de ned as the set of network states for which

1 0 M J Npk X X X m > 1A ; m = 1; : : :; M: Ypkl P@ p=1 k=1 l=1

(5.22)

Rather than requiring that the M inequalities of (5.20) are simultaneously satis ed with probability 1 ? , we use a weaker condition on the probabilities of separately satisfying the inequalities associated with each cell. Of course if the network state is admissible under the second de nition then a union bound gives

91 0M 8 pk J N M X 1 A M; P@ : Ypkl ; m=1 p=1 k=1 l=1

and by choosing = =M , admissibility under the second de nition implies admissibility under the rst de nition. For the remainder of this section we focus on the set of probability constraints of (5.22). m ; l = 1; : : :; Npk ; k = For xed m 2 f1; : : :; M g, we assume the random variables Ypkl 1; : : :; J; p = 1; : : :; M , are independent with distribution functions indexed by p and k. Each of the constraints can thus be handled using the eective bandwidth techniques for multiple classes discussed in Section 5.2. Whether the Gaussian approximation or large deviations bound is used7 , the result is an admissible region from the perspective of cell m of the form J M X X mpk Npk 1; p=1 k=1

where mpk is the eective interference at cell site m of a class k call in cell p. 7

Or indeed the underbounding or overbounding hyperplanes.


70

By repeating the above constraint at every cell in the network we arrive at our nal admissible region

8 9 J M X < = X S = :N 2 Z+M J j mpk Npk 1; m = 1; : : :; M ; p=1 k=1 n o = N 2 Z+M J j WN 1

(5.23)

where Z+ is the set of non-negative integers, the network state vector, N, is given by

N = (N11; : : :; N1J ; : : :; NM 1; : : :; NMJ )0; and W is the M by M J matrix of eective interference values. Before illustrating the above concepts with some examples, it is worth noting that the admissible region, S , has a form that is very familiar from the study of xed circuitswitched networks. If we treat each cell as a link with a capacity of one unit, then mpk is the resource required at link m by a call of class (p; k). The admissible region of (5.23) consists of all combinations of call classes that can be supported by the unit capacity links of the network. This analogy between CDMA networks and circuit-switched networks allows much of the extensive theory developed for the latter to be applied directly to the mobile environment. In particular the trac modelling of the next section is taken directly from the literature on loss networks.

5.3.4 Multiple Cell Models: Examples In this section we work through some practical examples of multi-cellular CDMA networks that can be analysed using eective interference ideas. A simple single class, xed SIR requirement system is studied rst and this basic model is then extended to include multiple classes of call with SIR requirements modelled by random variables.

Basic Model The simple model we consider here is similar to the one used in Chapter 48 and in [EE95a]. We consider a single class of call with constant bit rate and modem requirements leading to a xed minimum required SIR of ? for all mobiles. The propagation loss is due to distance only, as in (3.1), with shadowing and fading not modelled. Each mobile connects to its closest base station in a uniform, hexagonal, cellular layout, and is power controlled to be received at 1 unit of normalised signal power. Once more the forward and reverse links are orthogonal, and all our analysis centres on the reverse link. Background noise is not modelled. 8

With the exception of the trac model.


71

Type 1 Type 2 Type 3 Figure 5.1: Types of Interference Applying (5.22) for the special case of a single user class with xed requirements, for which the feasibility conditions are given in (5.16), leads to the network state admissibility constraints 1 0 M Np X X (5.24) ?Xplm > 1A ; m = 1; : : :; M; P@ that Xplm

p=1 l=1

where we recall is the interference produced at cell site m when mobile l in cell p is received at its home base station with one unit power. In our basic model this fading, or interference factor, is a deterministic function of the distances between mobile l and cell sites p and m. However if we assume the location of the mobile within cell p is a random variable, then the fading factor becomes a random variable. The only barrier to be overcome before the eective interference values can be calculated, is the determination of the distribution functions or moments of these fading factors. Because of symmetry and typically high path loss exponents in the mobile environment, we need only consider three distributions for the interference factors, as we now show. Consider the network from the perspective of cell m, our target cell. With reference to Figure 5.1, we break the cellular region over which mobiles contribute signi cant interference to the target base station, into three parts:

Type 1 mobiles are located within the target cell and cause one unit of interference at the target base station,

Type 2 mobiles are located in the rst layer of surrounding cells and contribute a variable amount of interference dependent upon their position,

Type 3 mobiles are located in the second layer of surrounding cells and also contribute


72

a variable amount of interference to the total9. Mobiles lying in cells outside the second layer and are assumed to contribute a negligible amount to the total other-cell interference. Suppose that the locations of mobiles within each cell are independent random variables with a uniform distribution over the cell. Then the distribution functions of the fading factors can be obtained directly from (3.8) and for xed m all interference factors are independent. With reference to (3.8) de ne

F1 (z) = FI0;b (z) = u(z) F2 (z) = FI1;b (z) p F3 (z) = 0:5 FI 3;b (z) + FI2;b (z) where the value of b is assumed xed and u(z ) is the unit step function. Ft (z ) is the distribution function of the fading factor for a mobile which is of Type t from the perspective of the target cell. Let nmt be the number of Type t mobiles seen by cell m and take Ztlm ; l = 1; : : :; nmt ; t = 1; 2; 3, to be a sequence of independent random variables with distribution Ft (z=?). The admissibility conditions of (5.24) can be rewritten

1 0 3 nmt X X Ztlm > 1A ; m = 1; : : :; M; P@ t=1 l=1

and eective bandwidth techniques can be applied to give the linear constraints 3 X

t=1

tnmt 1; m = 1; : : :; M

where t is the eective interference of a Type t call. These inequalities are readily transformed back to the network state domain by expressing the number of calls of each type in terms of the number of calls in each cell. The result is the set of inequalities M X p=1

mp Np 1; m = 1; : : :; M;

where mp = t if cell p is a Type t cell with respect to cell m. We remind the reader that if the number of mobiles in each cell satis es the above inequalities, then every cell has a probability greater than 1 ? of being able to satisfy the SIR requirements of its users. There are actually two Type 3 cells with slightly dierent distance from, and orientation to, the target cell, however we ignore these dierences here. 9


73

Before discussing some extensions from the simple model, it is noted that the above developments can of course be applied to one dimensional networks with appropriate change of interference distribution functions. Lognormal shadowing can be included in the propagation model for one or two dimensional networks as discussed in Section 3.4.3. The only change required in the above analysis to incorporate this eect is the use of (3.15) in place of (3.8) for the distribution function of the interference random variables. In the above multi-cellular environment we have assumed the cellular layout is symmetric thereby requiring only three call types to be de ned. If however the cells were of dierent sizes we would need to de ne a unique call type for each cell of the network from the perspective of each target cell. Apart from increasing the dimensionality of the problem, there is no diculty in extending all of the above models to handle asymmetric networks.

Multiple Classes Even with a single service class, the multiple cell environment leads to multiple types of call creating variable interference at the target base station. Here we extend the basic model to include J classes of user with xed minimum SIR requirement assuming that all mobiles within a class are controlled to the same power at their home base station. From (5.22) and (5.17) the probability constraints on network state admissibility appropriate for this multiple class problem are

1 0 M J Npk X X X m > 1A ; m = 1; : : :; M: ?k Xpkl P@ p=1 k=1 l=1

Once more we break the cellular network around a target cell into three region types. If cell m sees nmtk calls of Type t and class k, the feasibility constraints on SIR become

1 0 3 J nmtk X X X m > 1A ; m = 1; : : :; M; Ztkl P@ t=1 k=1 l=1

m has distribution function Ft (z=?k ) and for each m all random variables are where Ztkl independent. Eective bandwidth techniques for multiple classes with variable resource requirements are again applicable and lead to linear approximations or bounds for the boundary of the admissible region of the form J 3 X X

t=1 k=1

tk nmtk 1; m = 1: : : :; M:

In the terminology of cellular CDMA, we assign an eective interference to a call dependent on its type (location) and class (SIR requirement) which allows the interference to be


74

treated as if it were deterministic. Transforming back to the network state domain results in the nal speci cation of the admissible region boundary J M X X

where

mpk

p=1 k=1

mpk Npk 1; m = 1: : : :; M

= tk if cell p is a Type t cell with respect to cell m.

Variable SIR Requirements We now replace the deterministic minimum SIR requirements of the last section by random variables with users of class j having an SIR requirement with distribution function Gj (z ). Using the version of (5.22) with feasibility conditions from (5.19) we get the constraints

0 M J Npk 1 X X X m > 1A ; m = 1; : : :; M: P@ ?pkl Xpkl p=1 k=1 l=1

This is rewritten using the notion of cell types

1 0 3 J nmtk X X X m > 1A ; m = 1; : : :; M Ztkl P@ t=1 k=1 l=1

(5.25)

m is that of a product of random variables. The where now the distribution function of Ztkl remaining development is identical to the multiple class problem addressed above.

Example Consider an extension of the example from Section 5.3.1 to include interference from other cells. Remember that this example involved a standard and premium class of voice call both modelled as two state (ON/OFF) Markov sources. A standard hexagonal layout and propagation loss dependent only upon distance are assumed. For the moment we consider only a target cell. We identify three types of call, corresponding to calls within the target cell and in the rst and second surrounding layers, and two classes of call with diering bit rate statistics. Overall there are six dierent distribution functions that need to be de ned corresponding to standard and premium rate services for each of the call types:

H11(z) H12(z) H21(z) H22(z) H31(z) H32(z)

= = = = = =

0:6u(z ) + 0:4F1(20z ) 0:6u(z ) + 0:4F1(10z ) 0:6u(z ) + 0:4F2(20z ) 0:6u(z ) + 0:4F2(10z ) 0:6u(z ) + 0:4F3(20z ) 0:6u(z ) + 0:4F3(10z )

(Type 1, Standard) (Type 1, Premium) (Type 2, Standard) (Type 2, Premium) (Type 3, Standard) (Type 3, Premium)


75

2 3

7 1

4

6 5

Figure 5.2: Seven Cell Network of Example where u(z ) is the unit step function. m ; i = 1; : : :; ntk ; be iid random variables with common distribution function Let Ztkl Htk (z) with t 2 f1; 2; 3g and k 2 f1; 2g indexing independent sequences. The constraints of (5.25) are applicable and lead to an admissible region of sextuples satisfying 2 3 X X

t=1 k=1

tk nmtk 1; m = 1; : : :; M:

(5.26)

Suppose the network consists of seven cells as illustrated in Figure 5.2. Noting that there are two classes of call, the network state is the 14 dimensional vector

N = (N1;1; N1;2; : : :; N7;1; N7;2)0 and based on (5.26) and the simple transformation from the type domain to network state domain, the eective interference matrix W de ned in (5.23) is given by

2 66 11 66 21 66 21 6 W = 666 21 66 66 21 64 21

21

12 22 22 22 22 22 22

21 11 21 31 31 31 21

22 12 22 32 32 32 22

21 21 11 21 31 31 31

22 22 12 22 32 32 32

21 31 22 11 21 31 31

22 32 21 12 22 32 32

21 31 31 21 11 21 31

22 32 32 22 12 22 32

21 31 31 32 21 11 21

22 32 32 31 22 12 22

21 21 31 32 31 21 11

22 22 32 31 32 22 12

3 77 77 77 77 77 77 77 75


76

5.4 Trac Modelling and Control Issues Now consider the state of the network as the stochastic process

N(t) = (N11(t); : : :; N1J (t); : : :; NM 1(t); : : :; NMJ (t))0: Assume that calls of class j in cell m are initiated as a Poisson process with rate mj and as m and j vary over the sets f1; : : :; M g and f1; : : :; J g respectively, they index independent streams. A call request is blocked and cleared from the system if its acceptance would move the state out of the admissible region S , de ned in (5.23). If a call is accepted then it remains in the cell of its origin for a generally distributed holding time with mean 1=mj which is independent of other holding times and of the arrival processes. With these assumptions it is well known from the theory of circuit-switched networks [Kel86, Kel91b, Eve94a] that there exists a unique stationary distribution for the stochastic process N(t) with

8 Y M J N > < p0 Y (mj =mj ) mj ; N 2 S Nmj ! p(N) = P (N(t) = N) = > m=1 j=1 : 0; otherwise

where p0 is a normalization constant. The blocking probability for a class j mobile in cell m is simply

X

p(N) (5.27) N2Smj where Smj is the set of states in S that move out of S with the addition of one call of class j to cell m. Calculation of the blocking probabilities directly from (5.27) is infeasible for large networks, however the product-form solution allows a simple MonteCarlo acceptance-rejection technique to be employed [EM89]. Alternatively, we can use approximations from the calculation of blocking probabilities in general loss networks such as the Erlang xed point method [Kel91b]. We mention brie y that the in uence of limiting the number of users in any one cell is readily incorporated in the above model by adding extra linear constraints to those in (5.23). This allows the eect of base station hardware limits to be investigated as in [Eve94b]. It has already been noted that the above trac analysis falls directly from the theory of circuit-switched networks. Indeed the primary result of this chapter is the demonstration that quite general CDMA mobile networks can be modelled in this circuit-switched form. It is hoped that apart from the trac modelling above, this analogy will suggest sensible and ecient techniques for the operation and management of CDMA mobile networks. Bmj = p0


77

In particular, we note that our models of CDMA networks are the equivalent of circuitswitched networks with xed routing, equal link capacities, and quite a deal of symmetry in most cases. An important observation is that the control of the network operator is essentially limited to call admission control schemes which decide whether an incoming call should be accepted or rejected. Techniques such as trunk reservation may be useful for equalising blocking probabilities for dierent call classes, optimising network revenue, and implementing priority schemes. In any case, the analysis of such techniques as applied to cellular CDMA systems, can draw heavily from the well established theory theory for their circuit-switched counterparts.

5.5 Results In the following we present numerical results based on the examples of Section 5.3.4. The network considered is as in the basic model of that section except that the single class of voice user is now modelled as an ON/OFF source that requires an SIR of ? with probability and an SIR of 0 with probability 1 ? . will be known as the voice activity factor (VAF) since the above is a typical model of a system employing voice activity detection. Cell m thus sees three types of user with distribution functions

Ht(z) = (1 ? )u(z) + Ft (z=?); t = 1; 2; 3: Numerical eective interference values are presented in Tables 5.1-5.8 and Figures 5.35.6. All values have been normalised by dividing by the ON state SIR requirement ?. This allows more sensible comparison of how the eective interference values vary with system bandwidth or processing gain. The values shown are thus appropriate for use in the constraints 3 X t( ; ; ; ?) nmt 1=?; m = 1; : : :; M

nmt

t=1

where is once more the number of Type t users seen by cell m. As shown explicitly in these constraints, the eective interferences are functions of the PLE, VAF, quality of service factor and ON state SIR requirement which is essentially a measure of the size of the CDMA system. The values will also vary with the bound or approximation that is used in their calculation. In the results following we show how values vary with PLEs from 2-5, quality of service demands of 0.1%, 1% and 10%, and ON state SIR requirements of 0.1, 0.05, 0.01 and 0.002. These SIR requirements might correspond to a system with an Eb =I0 of 7dB and ON state processing gains of 50, 100, 500 and 2500 respectively. In most cases we use a VAF of 0.4 although this is varied in two of the tables.

CHAPTER 5. EFFECTIVE INTERFERENCE AND PRODUCT FORM Mobile Type Type 1 Type 2 Type 3

Heterogeneous Cherno 0.83 0.055 0.0016

Homogeneous Cherno 0.82 0.054 0.0013

Heterogeneous Gaussian 0.70 0.040 0.0015

Homogeneous Gaussian 0.70 0.040 0.0013

78

Mean 0.40 0.025 0.0012

1=? = 10, = 0:01, PLE = 4, VAF = 0:4 Table 5.1: Normalised Eective Interference Values for 3 Call Types Mobile Type Type 1 Type 2 Type 3





Mean 0.40 0.025 0.0012






Mean 0.40 0.025 0.0012






1=? = 500, = 0:01, PLE = 4, VAF = 0:4 Table 5.4: Normalised Eective Interference Values for 3 Call Types

Mean 0.40 0.025 0.0012


79

Tables 5.1-5.4 show the three eective interference values for xed system parameters calculated by the four methods outlined earlier. Each table corresponds to a dierent system size, or value of 1=?. We note that the Cherno bound gives more conservative values than the Gaussian approximation and that there is little dierence between the heterogeneous and homogeneous linear approximations. This is an indication that the boundary of the admissible region is close to linear especially for large values of 1=?. In most cases values are signi cantly greater than the corresponding mean although as expected this dierence becomes less pronounced 1=? increases and the economy of scale begins to bite. Figures 5.3 and 5.4 show the dramatic in uence of PLE on 2 and 3 respectively, for dierent quality of service factors. We note that for PLEs greater than 3.5 the contribution of Type 3 calls to the total interference becomes negligible. It is also evident that the Type 3 results cluster closer to the mean than their Type 2 counterparts and it may be reasonable simply to use mean values for Type 3 calls. Figures 5.5 and 5.6 show similar results except the Homogeneous Gaussian method is employed to calculate values and variation with system size rather than quality of service factors is shown. Tables 5.5 and 5.6 show the predictable variation of 1 with and ? for two values of the VAF. All values are calculated using the Heterogeneous Cherno approach. We note that for small systems and tight QoS constraints eective interference values are significantly greater than their corresponding means. This is especially evident in Table 5.6 where a lower VAF implies increased burstiness and extreme sensitivity to the system parameters. Type 1 calls are simply Bernoulli random variables and are essentially independent of PLE except for a very slight variation due to the use of an underbounding linear approximation. Finally, Tables 5.7 and 5.8 give reuse factors for dierent PLEs and ON state SIR requirements. The reuse factor is calculated from Reuse Factor = (1 + 62 + 123)=1 and is the ratio of the capacity of an isolated single cell to the per cell capacity of an equally loaded multi-cellular network. It can be compared directly to the reuse group size in FDMA/TDMA systems. In Table 5.7, the variation with system size does not show a clear trend due to the trade-os in statistical multiplexing gains of the inter-cell and intra-cell interference. Quite the opposite is true in Table 5.8 since a VAF of 1.0 leads to very little variation in 1 as ? is varied.


80

Normalised Effective Interference

0.12 0.11

alpha = 0.001 alpha = 0.01 alpha = 0.1 mean

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 2

2.5

3 3.5 4 Path Loss Exponent

4.5

5

1=? = 20, VAF = 0:4, Heterogeneous Cherno Figure 5.3: Normalised Eective Interference Values for Type 2 Mobile


0.025 alpha = 0.001 alpha = 0.01 alpha = 0.1 mean

0.02

0.015

0.01

0.005

0 2

2.5


4.5

1=? = 20, VAF = 0:4, Heterogeneous Cherno Figure 5.4: Normalised Eective Interference Values for Type 3 Mobile

5


81


0.11 1/Gamma = 10 1/Gamma = 20 1/Gamma = 100 1/Gamma = 500 mean

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 2

2.5


4.5

5

= 0:01, VAF = 0:4, Homogeneous Gaussian Figure 5.5: Normalised Eective Interference Values for Type 2 Mobile


0.022 0.02

1/Gamma = 10 1/Gamma = 20 1/Gamma = 100 1/Gamma = 500 mean

0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 2

2.5


4.5

= 0:01, VAF = 0:4, Homogeneous Gaussian Figure 5.6: Normalised Eective Interference Values for Type 3 Mobile

5

CHAPTER 5. EFFECTIVE INTERFERENCE AND PRODUCT FORM 1=? = 0:1% = 1:0% = 10% Mean 10 0.95 0.83 0.68 0.4 20 0.76 0.68 0.58 0.4 100 0.54 0.51 0.47 0.4 500 0.46 0.45 0.43 0.4 PLE = 4, VAF = 0:4, Heterogeneous Cherno Table 5.5: Normalised Eective Interference Values for Type 1 Mobile 1=? = 0:1% = 1:0% = 10% Mean 10 1.27 0.88 0.46 0.1 20 0.66 0.46 0.28 0.1 100 0.22 0.19 0.15 0.1 500 0.14 0.13 0.12 0.1 PLE = 4, VAF = 0:1, Heterogeneous Cherno Table 5.6: Normalised Eective Interference Values for Type 1 Mobile PLE 1=? = 10 1=? = 20 1=? = 100 1=? = 500 2.0 2.30 2.32 2.43 2.50 3.0 1.65 1.65 1.67 1.70 4.0 1.42 1.40 1.41 1.41 5.0 1.32 1.30 1.30 1.30

Mean 2.54 1.71 1.42 1.29

VAF = 0:4, = 0:01, Heterogeneous Cherno Table 5.7: Reuse Factors PLE 1=? = 10 1=? = 20 1=? = 100 1=? = 500 2.0 2.95 2.87 2.68 2.61 3.0 2.06 1.94 1.81 1.76 4.0 1.71 1.60 1.50 1.45 5.0 1.55 1.46 1.36 1.33 VAF = 1:0, = 0:01, Heterogeneous Cherno Table 5.8: Reuse Factors

Mean 2.54 1.71 1.42 1.29

82


83

5.6 Conclusion The key contribution of this chapter is the development of robust yet exible call admission control procedures for multi-cellular CDMA networks with multiple classes of user and possibly variable bit rate and modem requirements. This development was based on the notion of the feasibility of system states and the use of eective bandwidth concepts to handle the variability that characterises cellular CDMA networks. The resultant admissible region has a particularly nice form that is very familiar from the analysis of circuit-switched networks operating with xed routing. This allows a large body of existing teletrac theory and network management strategies to be applied directly to CDMA cellular networks. Numerical results demonstrated the sensitivity of network capacity to the propagation path loss exponent, burstiness of user demands, strictness of quality of service constraint and the eective size of the CDMA system. Results on trac performance have not been included in this chapter since the main thrust is on producing CDMA network models of the right form for application of standard teletrac theory. For a general discussion on trac issues refer to [Eve94b] while results employing the eective interference techniques of this chapter are presented in [EE95b]. Directions for future work include the incorporation of more general power control algorithms, allowance for imperfections in the power control process and the inclusion of user mobility and soft hando. Perhaps the most important area for research is the development of a deep understanding of the dynamic behaviour of CDMA cellular mobile networks. We believe the work presented in this chapter is a rst step in this direction.

Chapter 6

Conclusion In this nal chapter we brie y reiterate the main results and conclusions of this thesis. Remember that the original goal was to develop analytic techniques to assist in the design, performance analysis and general understanding of CDMA cellular mobile networks. We hope that this goal has been achieved. After a general introduction to cellular systems and spread spectrum communication in Chapter 2, methods of dealing with other-cell interference in CDMA cellular systems were discussed. Existing approaches were shown to be inadequate for our needs and so a novel characterisation of other-cell interference was presented. The end results were expressions for the distribution function of the interference from a mobile whose position within a cell was modelled as a random variable. For simple propagation models with no shadowing the distribution functions were obtained in analytic form, however numerical integration was required in the propagation model incorporating shadowing. These distribution functions were used continually in subsequent chapters to capture the variability due to uncertain mobile locations. The rst excursion into trac modelling came in Chapter 4 where each cell was modelled as an independent M=G=1 queue in an attempt to quantify the trac capability of CDMA cellular networks. The early sections of the chapter showed that the problem could be reduced to evaluating the probability that a compound Poisson random variable exceeds a given threshold. This probability was analysed using techniques from large deviations and central limit theory resulting in a bound and an approximation that become tight in their respective asymptotic regimes. Results indicated the sensitivity of trac capacity to the propagation environment and system parameters. Although the application of the M=G=1 model to CDMA cellular systems is not new, the treatment in this chapter was more detailed and advanced than any other attempts of which the author is aware. In Chapter 5 a more advanced trac model was developed where new calls were ac84

CHAPTER 6. CONCLUSION

85

cepted or rejected based on a stationary call admission control procedure. An eective interference is assigned to a user based on location and SIR requirements and these values form the basis for the determination of the capacity region used for call admission control. The calculation of eective interference values relies on techniques similar to those used for eective bandwidth calculation in ATM-based broadband networks. The resultant capacity region has a form very similar to a circuit-switched network and the trac modelling falls immediately from this analogy. The eective interference models we presented provide robust capacity speci cations for cellular networks operating with xed base station assignment and simple power control schemes. The major sources of variability modelled are random user locations within a cell and bursty SIR requirements for each mobile. Results were presented for a deterministic propagation model however lognormal shadowing is readily incorporated at the expense of some computational simplicity. A major result of this thesis is that variability in intra-cell interference will in many cases be the dominant eect for trac modelling. This may allow mean values to be employed for inter-cell interference, greatly simplifying the analysis and permitting models of increased complexity to be studied. There is currently a great deal of interest in more general models of cellular networks, particularly those operating with CDMA. Topics include exible base station assignment schemes, connection to multiple base stations and advanced power control algorithms. The use of such techniques may lead to signi cant capacity gains over the simple cellular system assumed here. It is also important to assess the performance of the dierent schemes when factors such as imperfect power control and user mobility are included. These areas will be the subject of future research.

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