Please note

The text in this file has been automatically extracted and may contain minor errors. For the original version please consult the paper copy

held in the Swinburne Library.

Traffic Modelling and Performance of Layered Cellular Networks with Overflow

Paul Fitzpatrick

School of Biophysical Science and Electrical Engineering

Swinbwne University of Technology

A thesis submitted for the degree of

Doctor of Philosophy

at

Swinburne University of Technology

March 1997

Abstract 4

This thesis reports on an investigation into the teletraffic modelling and performance of layered cellular networks that use overflow for new call and handover attempts in order to improve system performance It addresses the important problems how to model and analyse overflow in large cellular networks so that the overall impact across the network can be evaluated, the overflow policies that can be used with new calls and handovers to improve network performance and how these policies perform under a range of conditions

The general problem of modelling overflow in cellular networks is considered at length and from this it is concluded that the use of overflow in existing schemes such as directed retry, reuse partitioning and overlaid cells can also be described by the simple concepts of intra and inter layer overflow Using these concepts the author formulates the problem of overflow in layered networks as a subset of the general problem of overflow with multiple overflow routes and restricted overflow. This leads to the derivation of the mean and variance of the overflow traffic from a cell with multiple handover routes and restricted handover from the two-dimensional birth death model of the system

These expressions for the mean and variance are used in conjunction with the Equivalent Random Theory and splitting formulae in the development of the Splitting Formula Method or SF Method. The accuracy of three splitting formulae and a simple Poisson approximation are investigated and compared with simulation. From this we conclude

. that the splitting formulae of Akimaru and Takahashi and Wallstrom provide the best overall result. This method is used to solve the general problem of multiple user classes for an example two layer network. This leads to conclusions on the complex relationship between cell capacity, the proportion of uses that can access the microcells and the proportion of users that can overflow from the microcell to the macrocell.

The Modified Splitting Formula Method is an extension of the SF Method that includes handovers and overflow for handovers. A user perceived measure of the effect of new call and handover blocking described by the probability of call failure augments the Modified SF method. This is then used to analyse five new call overflow policies for a three layer network under conditions of extended handover, restrictions on handover for calls in cells at the edge of layers and spatial offset between the peak of the traffic demand and the cell layout. The results give new and valuable insight into the behaviour of layered cellular networks with overflow and the trade-offs in performance that exist when designing and operating these networks.

Acknowledgments

In the course of this PhD I have had the privilege s f the guidance and support of many people. All of them deserve acknowledgment for their efforts. In particular I would like to thank my supervisors Dr. C S Lee, Dr Bob Warfield and Dr. Jim Lambert for their commitment, encouragement and support.

Many thanks go to Taka Sakurai, my colleague at Telstra Research Laboratories for many years, who developed the simulation for the results in Chapter 3 and provided valuable feedback and discussion on many of the ideas developed in this thesis Thanks also go to my other colleagues at Telstra for their helpful discussion on the questions raised by this study and feedback on the problems of using layered cellular networks.

I am most thankfbl to Professor Les Berry for his assistance with identieing the splitting I

formulae.

I would like to gratehlly acknowledge Telstra for supporting this research project.

Finally, I wish to dedicate this thesis to my family In particular I dedicate it to my Mother and Father, Patricia and William for their support throughout my life. I also dedicate it to my partner Anne and our children Sian, Lucy and Sophie To them I express my love and appreciation for the continued patience, encouragement, tolerance and understanding that they gave me throughout the duration of this study

Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . Acknowledgments ............................................................................ 11 ... Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -111

1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 On the General Problems of OverdPow in Cellular Networks 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction 9

2.2 Overflow in Cellular Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 2.1 Overflow in Homogeneous Networks ................ ................... 11 2.2.2 Overflow in Hierarchical Layered Networks ............................. 13

2.3 Restrictions on Overflow ........................................................... 17 . . . . . . . . . . . . . . . . . . . . . 2.4 Overflow Policies for Hierarchical Layered Networks 20

2.4 1 On the Overflow Policy for New Calls . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.2 On the Overflow Policy for Handover Calls . . . . . . . . . . . . . . . . . . . 22

2 5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 The Analysis of Overflow in Microcellular Networks by Splitting Formula 24 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

. . . . . . . . . 3.2 A Review of Traffic Models for Cellular Networks with Overflow 25 3.3 Mean and Variance of Overflow Traffic for a Cell with Restricted

Overflow and Multiple Overflow Routes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 1 State Equations 27

. . . . . . . . . . . . . 3.3 2 Equations for Mean and Variance . . . . . . . . . . . . . . . 28 . . . . . . . . . . . . . . . . . 3 3 4 Comparison of Analytic Result with Simulation 32

3.4 Approximate Blocking for Individual Trafic Streams using the Splitting Formula Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 Comparative Performance of Splitting Formulae . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5.1 Graphical Results ................................. , . . . . . . . . . . . . . . . . . . . . . . . 38 3.5.2 Comparison of Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5.3 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6 Concluding Remarks and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Solution of the Problem of Serving Different User Classes using the SF Method 44 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 Model of a Network Serving Different User Classes ......................... 49

4.3 Network Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 I

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 New Call Blocking using the SF Method 52 ............................................................ 4.3.2 Network Capacity 53

................................................................ 4.4 Numerical Example 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Manhattan Network 54

4 4.2 Blocking Probability ...................................................... 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Network Capacity 60

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions 64

5 The Modified SF Method for the Teletraffic Modelling and Analysis of Hierarchical Multi-Layered Wireless Networks with Overflow 65

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction 65 . . . . . . . . . . . . . . 5.2 A Teletrafic Model of a Multi-Layered Wireless Network 66

....... 5 3 The Modified SF Method ................................................. 68 ............................ 5.3.1 Calculating Overflow Traffic into Each Cell 70

5.3.2 Proportioning New Call and Mandover Overflow Traffic Out of .................................................................... Each Cell 71

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Analysis 77 ............ 5.4 Performance Measures , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.4.1 Network Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4.2 Quality of Service Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 Numerical Example . . . . . . . . . . 84 5.5.1 Offered Traffic Model .............................. . . . . . . . . . . . . . . . . . . . 84

....................................................... 5.5.2 Handover Parameters 85 ............................................... 5.5.3 New Call Overflow Parameters 87 ................................................. 5.5.4 Handover Overflow Parameters 87

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Results and Discussion -87 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Concluding Remarks and Extensions 90

6 Performance of Layered Cellular Networks with Non-Uniform Teletraff~c Demand 92

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction 92 6.2 Review of Performance Issues for Hierarchical Networks . . . . . . . . . . . . . . . . . 93 6.3 Performance Criteria .................................................................. 95

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 1 Probability of Call Failure 96 ...................................... 6.3.2 Mean Number of Handovers per Call 97

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Network Model 98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Offered Teletraffic Model 99

6.4.2 Mobility Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. 100 ......... 6 5 Performance of Overflow Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

. . . . . . . . . 6 5.1 Impact of Intra Layer Cell Overlap on Overflow Policies 105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Performance with Restricted Handover 109

.................... 6 6 Performance of Overflow Policies with Spatial Variability 113 .......................................... 6.6.1 The Model for Spatial Variability 113

........................................... 6.6.2 Performance with Spatial Variability 114 . . . . . . . 6.6.3 Performance with Extended Overlap and Restricted Handover 119

..................................................................................... 6.7 Conclusions 121

125 7 Conclusion . . . . . . 7.1 Summary of this Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

. . . . . . . . . . . . . . 7.2 Recomnmendations for Further Research . . . . . . . . . . . . . . . . . 127

A List of the Author's Publications Resulting kom this Research 130 I

References 131

Please note

The text in this file has been automatically extracted and may contain minor errors. For the original version please consult the paper copy

held in the Swinburne Library.

Chapter 1

Introduction

Cellular mobile communications networks use a radio connection between the user and the network to provide untethered communications Each cell uses a base station (BS) to radiate the radio spectrum over a specific area. The cellular concept introduced by Macdonald in [Mac791 describes this process in detail and provides a basis for distributing the spectrum to each of the cells via frequency reuse. Frequency reuse allows the same spectrum to be reused in different cells resulting in an eEcient use of this spectrum. As the radio spectrum to provide this connection is a limited resource there is a need for network architectures that provide efficient use of this spectrum

The development of cellular mobile telephony into a popular cornrnunications medium has seen the demand for these services rise and hence a need for capacity beyond that originally envisaged for cellular networks While may techniques for increasing the capacity of cellular networks without the need for increasing the allocation of spectrum

' have been proposed, this thesis concentrates on the interaction of two of these techniques In particularly, the techniques under investigation are microcellular networks with their extension to layered network architectures and alternate routing or overflow.

The microcell Is a small cell that exists within the coverage of a conventional cell (usually called a macrocell or umbrella cell) and provides additional capacity when the macrocell. is congested. The development of the microcell concept and later its extension into a more general network architecture consisting of multiple layers of small cells was motivated by this need for increased capacity to meet the,expected demand for future mobile services. See for example [Ste89], [SN90], [JW91], [FGS92] and [RH94] for descriptions of the development of this architecture. The main virtue of the rnicrocell is its smaller size that allows for a closer frequency reuse between the same carrier frequencies and hence a higher efficiency of spectrum usage Microcells tend to naturally form layered network architectures with the macrocell because they can not completely duplicate the coverage provided by the macroceI1. This makes them ideal for the

1

application of alternate routing or overflow where calls can overflow from busy microcells to the macrocells.

Alternate routing or overflow is a well-known call control scheme that is used extensively in fixed circuit switched telephone networks where a second choice route between an origin and destination is provided for use when the first choice route is congested. Simply put, the calls access the first choice route and if a free channel exists then the call is carried. If there are no free channels then the call overflows to the second choice route. It then attempts to find a fiee circuit on the second choice route. If there are fiee channels on the second choice route then the call is carried, if not then the call is blocked

Overflow was applied to cellular systems long before the microcell concept was established. Then it was seen as a means for enhancing the capacity of existing cellular mobile communications networks without using more spectrum Such techniques as directed retry [Ekl86], reuse partitioning [Ha1831 and overlaid cells [Whi85] were proposed and anaiysed These techniques have provided substantial but finite capacity increases. When overflow is applied to the layered cellular network architecture the microcells behave as the first choice cells where ever possible, while the macrocell behaves as the overflow cell. The macrocell must also operate as a first choice cell for those calls that cannot access channels in a microcell. This provides the benefits of the closer frequency reuse and the capacity benefits of microcells with the call control benefits of an overflow system

1.1 Motivation

Future cellular networks built to provide Personal Communications Services are expected to use a multiple layer architecture to meet the capacity demands This multilayer architecture will build on the existing single and two layer architectures to use terrestrial cells with a range of cell sizes as well as satellites to cover the range of vehicular, pedestrian and office communication needs over the hll range of terrains. For example see [FGS92], [FGR92] and [HR94].

Planning and designing networks that incorporate overflow into this architecture will present a number of challenges. These challenges arise from the large number of design choices, the many constraints on cell locations and the complexity of the many overflow routes that arise in any given architecture. This will be hrther complicated because the conventional dimensi~ning rules that have been applied to single layer cellular networks

that are based on the Erlang loss system (see for example [Eve891 and [Eve94]) will no longer be applicable because of the nature of the overflow traffic and the increasingly complicated interactions between the traffic in neighbouring cells. I

The analytic approaches taken in investigating these networks and reported in the literature primarily focuses on the analysis of a small group of cells comprising a single macrocell with its underlying rnicrocells. See for example [FRG94], [HR94], [RH94], [LG95], [YLN95], [Fit96], [LG96] As it is difficult to tightly constrain cell boundaries in practice there will inevitably be cell overlap that extends beyond the boundaries defined by these simple groupings. An example of this is shown below in Figure 1. i for a network of three layers. The impact of overflow into neighbouring cells in the same or different layers that fall outside this simple grouping needs to be considered when attempting to understanding the overall effects of overflow on the performance of these networks. This in turn will require an analysis method that can deal with large networks.

Cells In Layers 1 & 2 Layer 2 coveragc overlap the boundary Layer 1 continues extends beyond Layer 1 coverag beyond Layer 2

Layer 3

Layer 2

Layer 1

Fgare I . d An example of a layered cellular network comprising three layers where smaller cells are not constrained to fall within the coverage area of larger cells.

Evaluating the teletraflic performance of these layered networks presents a dificuit problem to solve because unlike fixed telephony networks, new call and handover attempts that share a common cell can have many different overflow routes These overflow routes arise from the network architecture, the mobility of the users and variability in cell coverage. In addition, the impact of overflow on the performance of calls in progress through blocked handovers needs to be considered in the overall scheme of network performance.

The work presented in this thesis seeks to address these issues. The main motivations for the work presented in this thesis are :

to identifjr a range of individual problems that arise from the use of overflow in cellular networks with multiple overlapping cells that can be represented by a general problem and solve this problem,

to develop techniques for analysing large networks so that the impact of overflow on the whole network can be studied,

to propose a range of overflow policies that can be applied to cellular networks, and

e to analyse and investigate the pedorrnance that can be obtained using these overflow policies.

1.2 Overview

This thesis focuses on the telletraffic analysis and performance of layered cellular networks that use overflow In particular the author is interested to understand more about the impact of a range of overflow policies on the dimensioning and capacity of these networks. It has therefore been necessary to omit the radio propagation and transmission aspects fkom the modelling so as to concentrate on the teletraffic issues. While these radio issues are important in the overall context of network performance they remain a topic for future investigation.

As discussed, overflow was applied to cellular networks before the advent of the microcell concept. The techniques of reuse partitioning and overlaid cells inherently use layering and also use overflow to enhance capacity U7ith the introduction of networks employing microcells in a layered architecture with the macrocells overflow is also used and so the analysis of all these techniques will share a commonality. In Chapter 2 the general problem of overflow in cellular networks is examined with the aim of establishing a common approach to describing a range of problems that arise in these networks This Chapter examines overflow in both homogeneous and hierarchically layered networks using a graphical approach. Homogeneous networks describe those networks comprising a single layer of cells of unform size It also introduces the problem of restricted overflow in cellular networks. Finally a range of. overflow policies that can be used in a layered cellular network to control new call and handover traffic is proposed.

In Chapter 3 the Splitting Formula Method for analysing microcellular networks with overflow is developed. Here a microcellular network is simply a two layer network formed by layer of macrocells or umbrella cells overlaying a layer of microcelis. In order to develop this method the author derives an expression for the mean and variance of the

overflow traffic from a network with restricted overflow and multiple overflow routes I

This result is verified by comparing it with simulation The SF Method uses the Equivalent Random Theory introduced by Wilkinson and Riordan in [Wi156] and a I

splitting formula. The performances of three splitting formulae together with a simple Poisson approximation are examined and compared with simulations While this

1

approach gives the approximate performance its main advantage is that it can be used to analyse the performance of large networks.

The SF Method is then used in Chapter 4 to examine the general problem of multiple user classes in cellular networks. This problem is introduced in Chapter 2 where it is shown to represent a general problem that arises in cellular networks where users have restricted overflow. The solution of this problem provides important insight into a wide range of problems on the dimensioning of cellular networks where some users may have access to an alternate route and others only have access to a first choice cell

In Chapter 5 the Modified SF Method is developed by extending the SF Method from Chapter 3 to include handover and overflow of handover calls. A Quality of Service based performance measure is added to the Modified SF Method to enable the impact of new call and handover blocking to be incorporated into a single measure of user perceived performance.

In Chapter 6 the overflow policies proposed in Chapter 2 are investigated using the Modified SF Method. A set of performance criteria using the concept of a probability distribution of performance for the probability of call failure and the mean handover rate per call is developed This allows the large amount of data normally associated with a large network to be reduced to manageable proportions. The network investigated in this Chapter consists of three layers and is subject to a non-uniform distribution of new call tra& with the offered traffic distribution being spatially oEset from the network architecture. This allows the impact of diEerences-between the location of the cells and the traffic density to be investigated Further, the impact on network performance of changing the cell overlap and the handover between layers is investigated

Finally the conclusions are presented in Chapter 7.

% .3 Contribution

The original contributions that are presented in this thesis are listed below in the order in which they appear. The results of joint work are indicated.

Chapter 2

1. Define the general problem of overfl~w in multi-layered cellular networks to incorporate directed retry and other alternate routing policies and treat these with a unified approach.

2. Introduce the generalised problem of multiple user classes in layered wireless networks and show how this can be solved by modelling the network as a network with restricted overflow. This has been published in [Fit961

3. Show that the single layer and hierarchical cellular networks form a subset of the general problem of multi-layer networks with overflow. The hierarchical structure provides a set of constraints on the way calls can overflow between cells that may not exist in the non-hierarchical case.

4. Propose a range of overflow policies for a hierarchical layered cellular network. These have been published in [FLW97]

Chapter 3

1. Derive a general expression for the overflow variance fiom a common server group offered pure chance traffic with restricted overflow and multiple overflow routes This is important and usehl because it is used to solve the general problem of restricted overflow in layered wireless networks. This has been published in [FS95]

2. Compare analytic results fiom this expression with simulations to show the validity of the analytic results (joint work). This has been published in [FS95].

3 Develop the Splitting Formula Method for solving the general problem of restricted overflow in cellular networks based upon the Equivalent Random Theory and a splitting formula. One advantage of this approach is that it can be easily extended to large networks that would be intractable to solve using other analytic, numerical or simulation techniques. This work has been published in [FS95].

4. Investigate the comparative performance of three common splitting formulae with sirnulatioras to show that these give performance comparable with simulation and verifi their use in the Splitting Formula Method These results have been published in [FL96].

1. Analyse the general problem of multiple user classes with restricted overflow in a layered cellular network This result is important because it applies to a range of problems that arise when using overflow in cellular networks. This work has been published in [Fit96].

Chapter 5

1. Develop the Modified SF Method for analysing the general problem of overflow in multi-layer cellular networks based upon the SF Method with the inclusion of handover and overflow of handover calls. This is important and usehl because it is applicable to solving the network performance of large networks that would othenvise be intractable using other analytic, numerical or simulation techniques.

2. Incorporation into the Modified SF Method of a performance measure that accounts for both new call and handover blocking to give a single measure of performance for each cell.

3. Analysis of a range of overflow policies and compare the network performance for each of these when applied to an example hierarchical network

These are to be published in [FLW97].

Chapter 6

1. Introduce a new method for representing network performance based upon the concept of a probability distribution of performance. This is important and usehl because it allows the full network performance for large networks tc be evaluated when trading off the effects of various overflow, handover and network design strategies. This work is published in [FL97]. -

2. Evaluation of a hierarchically layered cellular network with a range of overflow policies and a non-uniform offered traffic distribution. The results from this evaluation are important because they give a clear picture of how using overflow improves network performance. The results from this a o r k appear in [FL97].

3 . Evaluation of a hierarchically layered cellular network employing a range of overflow policies under conditions of extended cell overlap and restricted handover policies The results from this evaluation are important because they show how extending cell overlap increases performance but does not result in a significant increase in

handover activity. They also show that an appropriately applied restrictive handover 1

policy can improve performance. The results from this work are published in [FL97].

4. Evaluation of a hierarchically layered cellular network employing a range of overflow policies under conditions of variability in the offered traffic model with extended cell overlap and restricted handover policies These results are important because they indicate the existence of an optimal relationship between the offered traffic distribution and the cell layout. This has application in the design of these networks.

Chapter 2

On the General Problems sf Overflow in Cellular Networks

2.1 Introduction

As discussed in Chapter 1 the use of overflow in cellular networks is well known and therefore its use in networks employing microcells or layered network architectures is a natural extension of the technique. Also, some commonality is expected between the problems arising from the use of overflow in each of these types of networks and in the analysis techniques that could be applied to solving these problems

The aim of this chapter is to present a generalised framework in which the problems arising from the use overflow in homogeneous and hierarchical cellular networks can be discussed A krther aim of this chapter is to show that a range of these problems constitute special cases of a more general problem called restricted overflow. The author introduces definitions for intra and inter layer overflow that allow overflow in cellular networks to be easily specified. Further, a range of overflow policies for new calls and handovers that can be applied to the class of cellular networks that employ a hierarchically layered architecture is proposed. The aim of these overflow policies is to improve the traffic capacity of these networks

Overflow in cellular networks is discussed in Section 2 2 . Graph theory [Mar711 is introduced as a means for describing the overflow routes in cellular networks. It is also used to identi@ the differences between homogenous and ,hierarchical and describe the different problems that can arise from the use of overflow in these networks. In Section 2.3 the problem of restricted overflow is presented and discussed. This is a unique feature of cellular networks and its causes are examined in this section Overflow policies for controlling both new calls and handovers are presented in Section 2.4. These policies provide a means of controlling the access of new calls and handovers to alternate

routes so that overall network performance objectives can be achieved. The conclusions are presented in Section 2.5.

2.2 Overflow in Cellular Networks

The use of overflow has been proposed as a means of improving the performance of cellular mobile communications systems Three of the earliest proposals that use overflow are the reuse partitioning scheme proposed by Halpern in [Ha183], the overlaid cell concept discussed by Whitehead in [Whig51 and the Directed Retry scheme proposed by Eklundh in [Ekl86]. These papers show that one of the underlying requirements to establish overflow in cellular networks is coverage of a given area by more than one cell This allows the mobile stations (MS) an alternate cell to which they can operate if the one offering the best signal (first server cell) is congested

It can be seen from these examples that the availability of an overflow route (or cell) is dependent on the existence of some sufficient level of signal strength that will allow the MS to communicate with an alternate base station. This can also be described in terms of first and second server cells or simply first and second servers. Define the first server for any 141s as that cell which provides the strongest signal strength greater than some minimum threshold to that MS. This minimum thresl~old is the power necessary to achieve a desired level of transmission quality between the base station (BS) and the MS Similarly, define a second server cell as a cell that provides a signal level less than the first server and greater than the minimum threshold to that MS. Many cells may exist that satis@ the conditions to be a second server. The usual practice is to rank the cells according to signal strength with the highest ranking given to the cell offering the highest signal strength. See for example p372 of [NIP921 and the application of this to the GSM system.

Therefore the first choice and overflow routes are provided by the first server and the second server cells. Further, the second sewer cell may change as the mobile moves within the area of the first server cell. The second server cell may or may not change when the MS changes first server cells. Also, there may be places where no second

- server cell exists. This presents a new set of problems compared with those encountered in the fixed network where calls between any origination and destination would use the same overflow route

The notation from graph theory (see for example [Mar71]) can be applied to represent the overflow paths in a cellular network. In particular directed graphs or digraphs are

used to represent cellular networks with overflow. Let each cell be represented by a '

vertex and the overfIow path between any two cells (in this case from cell a to cell b) be represented by an edge such that the initial vertex is the first server cell (in this example ceU a) and the terminal vertex is the second server cell (in this example cell b) Refer to Figure 2.1 for a simple example of overflow where the cells share a common overlap I

region and provide the second server to each other This is typical of directed retry between the two cells.

Figure 2.1 A representatzon of the overflow paths bemeen two cells using Graph n e o r y notation.

2.2.1 Overflow in Homogeneous Networks

Consider a network of n cells. In the most general sense overflow paths could exist for any MS in any ceii to all other cells in the network All possible overflow paths from one of those cells (cell j ) to the remaining n - 1 is depicted below in Figure 2.2 In practice however, the existence of any of these overflow paths will be constrained by the conditions for each of the n - 1 cells to be a second server to cell j for any particular MS

Figure 2.2 Representation ofthe overfow routes between thej-th cell each of the other $1 - 1 cells in a network of n cells nl total.

Taking the directed retry policy for overflow proposed in [Ekl86] for a homogeneous cellular network as an example we can see that the cell overlap that is typical for this type of network sets the overflow paths. This is shown below in Figure 2.3 for a cell and its six immediate neighbours.

Figure 2.3 Graph of the overflow routes for a cell and ~ f s slx immedzate neighbours. The solid lines show the notional cell boundaries and the dashed lanes show the

overlcrpplng coverage.

Clearly in a homogeneous network the overflow routes are provided by the natural overlap that occurs at cell boundaries. As this overlap between cells is shared, the overflow routes between the cells are bi directional In a homogeneous network large areas of cell overlap may not be desirable if they have the potential to degenerate network quality through increased interference (see [Ekl86]) Therefore there exists a trade off in the use of overflow as a balance must be struck between the capacity improvement and the potential to cause excessive interference When this balance is struck those users that fall outside the overlap area have no alternate route. This presents problems in terms of the fairness of access to overflow for all users. Clearly the performance experienced by the different classes of users will depend on their access to an overflow route. This is covered in greater detail in Section 2 3 under the topic of restriction on overflow

On approach to balancing the fairness aspect of overflow in homogeneous networks is the load sharing scheme proposed by Karlsson and Eklundh in [KES9]. This scheme uses handover to move calls in the overlap region of a busy cell to a neighbour cell This frees a channels for use by new call access attempts that fall outside the overlap region and have no overflow paths. Other network architectures that exploit overflow have been proposed. These fall into the category of hierarchical layered networks

2.2 2 Overflow in Hierarchical Layered Networks 3

Hierarchical layered cellular networks tend to occur naturally whenever there is a coverage area of one cell completely within the coverage area of another cell. They can also occur when the overflow routes between cells are controlled to be unidirectional I

The Reuse Partitioning scheme of Halpern [Ha1831 and the overlaid cells schemes described by Whitehead [Whig51 are early examples of layered networks The use of overflow in these schemes follows a pattern where one group of channels acts as a common pool for the users within the overlapping coverage area.

Figure 2.4 shows an example of a network that would arise from either reuse partitioning or overlaid cells. The ovefflow is in the direction from cell b to cell a only This is obvious as all calls falling within the overlapping coverage of cells a and b use cell b as their first server and then cell a as their second sewer. Where as calls to cell a that fall outside the overlap with cell b have only cell a as a first server and no second server The channels in cell a serve as a pool for use by users in cell b whenever it is congested This use of overflow is hierarchical with the overflow path being unidirectional.

Figure 2.4 A representatzon of the notional cell bounhr~es for Reuse Partitisn~rzg or overlaid cells and a graph representing the overflow between the two cells.

By combining the concepts of Directed Retry with Reuse Partitioning (or overlaid cells) the overflow paths in a network can be extended The overflow paths for two cells in such a network are shown below in Figure 2 5. Here cells a and c can share capacity in their region of overlap through directed retry. At the same time cells b and d use a and c respectively as overflow routes.

Figure 2.5 Overforv paths for a network of two cells using Dzrecfed Retry between neighbour cells with partially overfap and overglow between cells using Reuse

Partitioning.

Ovevfilow and MicroceIluPar Networks

The use of very small cells (rnicrocells) to provide additional capacity in the highest demand areas has been well documented in the literature. For example see [SN90], [JW91], [FGS92] and [HR94]. As part of these networks the microcells form another layer of cells under the macrocell or umbrella cell layer. Tt has also been proposed that the macrocell layer be used to provide alternate routes for calls that may be otherwise blocked when the rnicrocells become congested. These network architectures also naturally form a hierarchy of cells, again because the overflow is defined in a single direction only. That is. overflow is from the rnicrocells to the macrocell or from the lowest layers to the highest layers.

The overflow path can also be controlled through the cell selection algorithms In order to rnaxinuse the traffic carried by the network the aim has been to capture as much traffic as possible with the lowest layer cells (eg microcells). This is done because the lowest layer generally has the smallest cell size and highest reuse factor. To enable new calls to setup on these lowest layer cells the cell selection algorithms can be biased so that calls access these cells. In networks where cell selection is based solely on power levels then the lowest layers can be made dominant through judicious choice of power settings in the network design This leads to a network where the first server cell will be in the lowest layer of the network. If there are no free channels in the first server cell then the call can overflow either to a neighbour cell in the same layer or to a cell in the next layer, in the same way as calls overflow in the network using reuse partitioning.

Figure 2.6 Overflow paths between two microcells (b and c) and an overlayrng macrocell (a).

From Figure 2.6 we can see that overflow between microcells is possible where overlapping coverage between them exists. Further extensions of this are overlapping coverage at both the microcell and macrocell level (refer to Figure 2.7), and multiple overlap between the microcells and the macrocell (refer to Figure 2.8)

Figtcre 2.7 Overflow routes arisingfrom overlup bemeen macrocells, rnicrocells and macrocells and microcells.

Figure 2.5 Overflow routes arisrngfrsnz multiple overlap betwee11 macrocell and rnicrocells.

The grouping of cells into layers can be based on some common factors between the cells. One approach is to form layers by grouping cells according to the network technology used to provide the cells. For example in [HR94] the layers are provided by the satellite network and a terrestrial network of macrocells and microcells. This allows for each layer to be characterised by the features of the network that provides the layer, for example frequency band, transmission method (TDMA, CDMA etc), compatibility or

15

interoperability (eg data rates, signalling rates etc) However this does not account for I

layers formed within the one technology type Alternatively, some notion of grouping cells by size could be used to identifjr layers For example rnicrocells have become I

identified as a type of cell according to some typical size or range of sizes and possibly reuse pattern. The same notion can be applied to rnacrocells and picoceIls. Also, some I

combination of these two approaches could be adopted. However, comparing Figures 2.6 to 2 8 with Figure 2.2 the distinguishing feature is the unidirectional overflow between cells in different layers, while the common feature is the bi directional overflow between cells in the same layer. In particular we see overflow from the lower layers towards the higher layers is indicated by the direction of the arrow in the overflow path

The conditions for a hierarchical network can be determined from the preceding discussion as.

1. Cells that are connected by one or more bi directional overflow paths form a common layer.

2. Cells in the lowest layer have unidirectional overflow paths originating in that layer and terminating in other layers (higher layer).

3 . Cells in the highest layer have unidirectional overflow paths originating in other layers (lower layers) and terminating in that layer

4 Cells in intermediated layers will have unidirectional overflow routes from other cells in other layers that both originate and terminate in the intermediate layer. The paths from the lower layer cells will terminate in the intermediate layer and the paths to the higher layer cells will originate in the intermediate layer.

Therefore we conclude that hierarchical layered networks form a subset of the general network architecture that employs multiple overlapping cells. The hierarchical networks are distinguished by unidirectional overflow from the cells in the lowest layers toward cells in the higher layers and the general network with overlapping cells has bi directional overlap between cells. For example compare Figure 2.3 with Figure 2.8 for a network of six cells

Intra and Inter Layer Overflow

In general, overflow describes .the process where ar. unsatisfied call attempt is allowed one (or more) subsequent attempts to access a channel in other suitable cells before being blocked and cleared.

The overflow employed to control call access in hierarchical layered networks can therefore be classified into two broad groups The first is overflow between cells in the

same layer. This will be referred to as intra layer overflow. Intra layer overflow covers I

the familiar directed retry policy for new calls and could also be applied to handover calls. The second group is overflow between cells in different layers. This will be referred to as inter layer overflow Inter Iayer overflow describes the reuse partitioning scheme. Once again inter layer overflow could be applied to handover calls. 1

It can be seen that both intra layer and inter layer overflow can be described by the same mechanism. That is, when the call is unsuccessfbl in the first attempt at finding a free channel in the target cell it can immediately try to access a free channel in the next most favourable cell.

Let the number of layers in the network be L with the bottom most layer as 1 and the top layer as L. The number of cells in layer I is N, So for any two cells (1,n) and (k ,m) where I , k E (1, . . , L) and n E (1,2, , N , ) , m E (1,2, . , N,) overflow traffic from cell (1,n) to (k,m) for (k ;t I ) describes inter layer overflow. Overflow traffic from cell ( I , n) to (k ,m) for (k = I ) describes the intra layer overflow. It will be assumed that calls will connect to the preferred cell in the lowest layer before attempting to connect to the next preferred cell Therefore no overftsw would occur from celi ( I , n) to (k , rn) for ( k < I ) .

2.3 Restrictions on Overflow

The nature of mobility and the variability in cell overlap in cellular networks can provide a complex set of relations between the first choice and overflow route. In cellular

, networks two calls in the same cell may have completely different overflow routes. Some users may have no overflow route at all. Therefore there is not necessarily any uniformity in the overflow routes that can be applied to all the MS in a cell Restrictions on overflow describes this variation in the overflow routes (or second servers) that may exist in a cell and that manifestly controls the ability of a MS to access other cells and layers of the network. These restrictions have also been described as partial overflow in [FS95]. Therefore we see in layered networks that there is one stream of trafic offered to a common cell with the unsatisfied calls overflowing to a range of cells. The exact proportions of the overflow will be a hnction of the network topology, the cell overlap and the overflow policy.

These restrictions can arise because of the network layout and coverage They can also arise because of the technique used to establish the layers in the network Some of the causes for restricted overflow are discussed in [Fit961 and these will include.

1. Partial coverage

Partial coverage of an area results from poor signal level in that area. Figure 2 9 shows an example of a macrocell with underlaying microcells. The macrocell should provide coverage to the whole area, but because of shortcomings in the design of the cell or for other reasons there may be small areas of very poor signal strength. These low signal strength areas may fall within the coverage area of a microcell. For example, a microcell located on a street corner would provide good coverage inside the lower floors of a building where the macrocell coverage is very poor Coverage from the microcells may only be provided to the high traffic areas. This potentially leaves large areas with only macrocell coverage The degree of macrocell only coverage would depend on the traffic demand and the economics of providing broader microce~lular coverage.

Layer 1

Layer

only

coverage Layer 2

Layer 2 coverage only

Figure 2.9 A simple two layer network with partial coverage of some areas that can result in restricted over-ow.

2. Overlap discontinuity (at a layer boundaries)

Discontinuity in the overlap of cells at the edge of-a layer would result in users in some areas of a cell having an overflow path to another layer, while other users in the same cell would have a different path or no path at all. For example where one layer ends but other layers continue within the boundary of a higher layer. This is shown in Figure 2.10.

Cells in Layers 1 & 2 Layer 2 coverage overlap the boundary Layer 1 continues extends beyond beyond Layer 2

Figure 2.10 A Ewered network depicting overlap discontinuity at a layer boundary and partial overlap of a cell by cells in other layers

The overlap discontinuity will also result in the overflow paths for different users in a cell being diEerent. For example in Figure 2.10 it can be observed that users in the central cell of layer 2 have two distinct overflow routes into layer three depending on which part of that cell they occupy.

3. Techniques for supporting layered networks

The techniques used to establish a layered network can result in restrictions t o the overflow paths for users. Consider, for example a two layer network using overlaying macrocells to provide broad coverage and microcells to support high density traffic demand The network could support the two layers by

1. distinct frequency allocations within a single band (for example spectrum partitioning to provide the macrocell and microcell layers)

2. distinct frequency bands that are supported by single or multi band terminals, and

3 . distinct transmission modes within one or more frequency bands that are supported by single or multi band / multi mode terminals

Jn the network architecture for option 1 above, coverage defines the restrictions to overflow. This is the same problem as partial coverage which has already been described It should also be noted that in the case of classes of users defined by partial coverage that users may change classes as they move through the network. This will happen because some of the users will be changing location and hence changing the areas of the network that cover them It would therefore be expected that the numbers of users in any particular class would be dynamic and change with the mobility of the users.

Options 2 and 3 sees the restrictions to overflow defined by the type of terminal. Those users with a single band and or single model terminal are restricted to operate on those cells that support those terminals Users with multi band and or multi mode terminals can make use of all the cells that support both bands and or modes. Further, undt!r options 2 and 3, the layers using different frequency bands and or trans~nission modes may be provided by different service provider companies. This would allow users to choose to subscribe to access one or more layers in the network For example, a network that utilised a cellular network for the macrocell (high tier) provided by one operator and a PCS type network for the microcells (low tier) provided by a second operator Here, the overflow paths would be set by the extent of the user's subscription.

Each of these approaches allows users to be grouped into distinct classes. There are those who can access only one of the layers in the network. Then there are those who can access two of the layers, and so on, up to the class that can access all layers For a network of L layers the number of classes of users is equal to:

In all these cases the result is that, at worst, calls may not be able to overflow at all or they rnay only have one alternate route in the event of the first server cell being congested. So it is important to categorise users into these classes because each class potentially receives a different level of performance (in terms of the new call blocking probability) from the network. The difference in performance arises from the fact that

. even for users sharing a conlmon first server cell, they may not share a common second server cell (or overflow route). This aspect of overflow is studied in greater detail in Chapter 4 were new call blocking probabilities for the three classes of users in a network with two layers (L = 2) are investigated

2.4 Overflow Policies for Hierarchical Layered Networks

In this Section overflow policies that can be applied to controlling new calls and handovers in layered hierarchical cellular networks are proposed. These overflow policies use intra and inter layer overflow applied independently to new calls and handovers with the aim of increasing the overall network capacity while maintaining a desired level of performance for both the new calls and handovers A frame work for analysing networks with overflow that can be applied to these networks using these

1

overflow policies is presented in Chapter 5. A detailed analysis of the performance of the policies subject t o a number of network conditions is contained in Chapter 6

2.4.1 On the Overflow Policy for New Calls

One aim in using intra and inter layer overflow for routing unsuccess~l new calls is to overflow new calls to a cell that results in the best wanted signal strength and least interference in that layer It can be argued that when using overflow in a layered network preference should be given to inter layer overflow over intra layer overflow. This argument is based on the well-known problem of increased interference that occurs with directed retry when calls that overflow are too far from the second choice BS [EH86].

A problem of using overflow in multi-layered cells is then how to make a choice between routing the calls within the same layer or between layers. Restricting calls to only inter layer overflow would mean no overflow between cells in the same layer. This approach may be justified for the layers with the smallest cells on the basis of the additional interference contributed from calls in non-preferred cells. However, for layers where the cells are large, it may be desirable to use both intra layer overflow for the calls near to the cell boundary and then use inter layer overflow for the calls nearer the centre of the cell that would otherwise not satis@ any criteria for intra layer overflow. The impact of the exclusion of intra layer overflow for layer one is examined as part of the overflow policies.

Another aim of the overflow policy for new calls is to improve the probability that a new call will be accepted into a suitable cell but not at the expense of blocking a handover for a call in progress. The approach taken in this thesis is to allow the new calls and handsvers to compete on a first come first serve basis, but limit the number of times a new call may attempt to access a free channel compared with a handover. That is, for example, if a new call is unsuccessfbl at the first attempt it can overilow to an alternate cell. If it is unsuccesshl again at finding a free channel the call will be blocked and cleared.

The new call overflow policies studied in this thesis are:

1. No overflow 2 One ovedow attempt, with no intra layer overflow for layer one 3 One overflow attempt, with intra layer overflow in layer one 4. Unlimited overflow attempts, with no intra layer overflow for layer one 5 Unlimited overflow attempts, with intra layer overflow in layer one

2 1

It is important to note that policy 1 of no new call overflow does not in itself represent a viable policy but serves as a basis for comparison for policies 2 to 5.

2.4 2 On the Overflow Policy for Handover Calls

The goal with calls in progress that are subject to a handover is at all costs to prevent the loss of calls due to insufficient free radio channels. This can be achieved by treating handover calls differently from new calls The classic examples of this are to give priority to handovers over new calls when assigning traffic channels, to reserve channels specifically for handover calls, to provide some form of queuing for handovers so that the call is maintained until a free channel becomes available or to use some combination of these techniques.

The aim of alternate routing for handovers is similarly to ensure that handover calls have as many opportunities as possible to find a free channel in the network This will enable the call t o continue rather than being forced to terminate prematurely In this model, handovers are allowed to overflow as many times as necessary until either (1) they find a free channel and are carried or (2) they reach the top layer without finding a free channel and are blocked and cleared. As in a single layer network, handovers in a multilayer network normally try to handover to a neighbouring cell in the same layer first If this is unsuccessfbl because of insufficient radio channels then the call will attempt to overflow to a cell in the next highest layer Consequently, the ability of calls to handover to cells in higher layers provides an essential alternate route when one group of cells in a layer is congested.

2.5 Concluding Remarks

In this chapter it has been demonstrated that ovkrflow in conventional homogeneous single layer and hierarchical networks forms a subset of the general problem of overflow in networks with multiple overflow routes. The hierarchical structure provides a set of constraints on the directions that calls overflow between cells where the overflow is unidirectional between cells in different layers. These constraints do not exist in the non- hierarchical case where overflow paths in each direction (bi directional) between cells can be established whenever there is cell overlap. Compare for example Figures 2 1, 2 3 and 2 8.

Chapter 3

The Analysis of Overflow in Microcellular Networks by Splitting Formula

3.1 Introduction

In Chapter 2 the application of overflow to cellular mobile communications systems was discussed and from that we saw that each cell in the network will act as a common overflow route for unsatisfied call attempts from neighbouring cells. This is particularly evident in the networks employing a layer of microcells within the coverage of a macrocell. The question of how to determine the blocking probability for each of the traffic streams that overflow to this common route is important in evaluating the overall performance of any particular stream of new calls offered to a cell This is important when it is considered that restrictions to the overflow for different users will result in them experiencing different levels of performance.

In this Chapter the author describes a method for determining the approximate blocking probability for each traffic stream that makes up a composite traffic streams offered to a common overflow route. This method is based on the Equivalent Random Theory (ERT) and a splitting formula. Three splitting formulae are presented and their performance is compared to simulation and a simple Poisson model for overflow. The method provides an approximation to the blocking probability that can be applied to problems with large numbers of cells and channels per cell that would be intractable for

. other more exact analysis methods.

A review of the traffic models for cellular networks with overflow currently available in the literature is presented in Section 3.2. In Section 3 3 the mean and variance of the overflow traffic for a network with multiple overflow routes and restricted overflow is derived. The variance forms a necessary part of the blocking probabiiity analysis and this result is combined with ERT and the splitting formulae to produce the Splitting Formula Method for calculation of the approximate loss for each of the overflow streams This is

described in Section 3.4. A comparison of the results from a simulation study, the approximate analytic method using the splitting formula and a simpie Poisson approximation for a number of typical configurations of a network comprising two layers are given in Section 3.5. Section 3 6 contains the concluding remarks and discusses fixther extensions to the method.

3.2 A Review of Traffic Models for CeBl-~alar Networks with Overflow

The problem of calculating the blocking probability for traffic overflowing from networks that utilise common alternate routes has its origins in the analysis of overflow in fixed circuit switched telephony networks. Approaches such as the Equivalent Random Theory (ERT) by Wilkinson and Riordan [Wi156], the Intempted Poisson Process introduced by Kuczura in [Kuc73] and the Hayward Approximation reported by Fredericks in [FregO] have been developed to analyse the performance of circuit groups offered overflowing traffic. The literature shows that all of these approaches have been used in analysing the performance of cellular networks with overflow. The draw back of ERT and the Hayward -4pproximation is that they give only the overall lost traffic from the overflow route and not the lost traffic for the individual traffic streams overflowing to that route For example Frullone et a1 in [FRG94] analyse a two layer network using a model based on the Hayward Approximation. Here only the average blocking for all calls offered to the macrocell (including calls that overflow) is calculated.

The approach to analysing the directed retry policy used by Eklundh in [EM861 revolves around a model of a cell being offered three streams of traffic. Two of these are new calls in the form of pure chance traffic the can overflow to some other cell and those that cannot overflow because they have no second server The other is overflow traffic from neighbouring cells and defined by its mean and Gariance By grouping the overflow traffic with the new calls that have no overflow route the model can be simplified to two input streams. The overflow traffic and the new calls that cannot overflow are combined because if unsuccessfbl at finding a free channel they will both be blocked.

The combined stream can then be represented by an equivalent Poisson model using the ERT. The resulting model consists of two pure chance traffic streams, one offered directly to the channels in the cell (secondary group) and the other offered to a number of circuits (primary group) to produce the desired mean and variance of the combined pure chance and overflow stream and then offered to the channels in the cell (secondary group). This model is then used to produce a four dimensional state space

representation using the number of servers occupied in the primary and the secondary groups and the occupancy of two theoretically infinite overflow groups. The occupancy of the infinite groups captured the mean and variance of the lost traffic fiom the secondary group. The call congestion for the two traffic streams was calculated by applying a binomial transformation of the state equations and using Gauss-Siedel iteration with successive over-relaxation. While this approach was not considered in this thesis for application to layered cellular networks, it has been applied to the more general problem where the second group is offered two streams of overflow traffic. This is reported by Wilson in [Wi177]

The Interrupted Poisson Process (IPP) is a special case of the Markov Modulated Poisson Process (R.IIMPP) with two states for the Poisson process, on or off. This has been used by Hu and Rappaport in [HR94] to analyse the performance of a hierarchically layered cellular network. This network employed terrestrial based microcells and macrocell with an overlaying network based on a satellite system. The ove~flowing new calls and handovers were modelled using an IPP Under the assumptions of cells in each layer being statistically identical and behaving independently, a single cell in each iayer was analysed. While in this example the IPP model for each cell produces a state space of manageable dimensionality, it is difficult to apply this approach to the analysis of larger networks

The MbPP was used by Meier-Hellstern in [Meis91 to develop a methodology for analysing overflow systems. This method allows the blocking probability for each stream to be evaluated exactly. The analysis of large networks can be aided by suitable approximations. A similar approach is used by Lagrange and Godlewski in [kG95] and [LG96] in the analysis of the teletraffic performance of a hierarchically layered cellular network comprising two layers. Here the individual overflows from N microcells to a macrocell are modelled as WPs. The combination of the N IPPs results in a MMPP. On the assumption that all cells are identical, the size of the MMPP formulation can be reduced to manageable proportions. Once again this approach is difficult to apply to large networks because of the resulting dimensionality of the problem.

Jolley and Warfield [JW91] base their formulation of the problem on a flow model developed by McMillan in [McM91]. This relies on the assumption that all the cells in the network behave independently It allows the problem of large networks to be formulated, where using state based methods would result in large state spaces that are difficult to solve In [JW91] overflowing new calls are treated in the same way as new calls. This results in an underlying assumption of overflow being a Poisson process.

While this makes the formulation relatively easy even for large networks, it ignores the well-known characteristics of overflow traffic.

3.3 Mean and Variance of Overflow Traffic for a Cell with Restricted Overflow and Multiple Overflow Routes

In order to make use of the ERT for calculating call loss in an overflow group it is necessary to know both the mean and variance of the overflow traffic. The model in Figure 3.1 is used in this section to determine an expression for the mean and variance of the overflow traffic from a cell under the condition of restricted overflow and with multiple overflow routes. It consists of a primary group of x, channels and a secondary or overflow group of theoretically infinite capacity The primary group is offered pure chance traffic with an arrival rate A and a departure rate p Of the A Erlang of offered traffic, f can overflow to the secondary group and ( I - - cannot

Offered traffic Overflow Group xz channels

A = Alp Erl x , channels x2 4 00

Fzgure 3.1 A Two Layer Network as a Szmple Overflow Sysdem

Let the number of busy circuits in the primary group be denoted by i and the number of busy circuits in the overflow group be denoted by J. Then the statistical equilibrium probability that there are I calls in progress in the primary group and J calls in progress in the secondary group is denoted by P(i , j ) .

3.3.1 State Equations

The state transition rate diagram for the simple overflow system in Figure 3.1 is shown in Figure 3.2. From this state transition rate diagram the following state equations can be obtained:

Figure 3.2 State transition rate diagram for the szmple overflow system in Figure 3. I

( f ~ + x , + j ) ~ ( x , , j ) = AP(X, -l,j)+ f ~ ~ ( x , , j - l ) + ( j + l ) ~ ( x , , j + l ) (3.2)

f o r j 2 0 and

P( i , j )=O for i < O , i > x , , j < O .

Finally, the sum of all probabilities is equal to unity

3 .3 .2 Equations for Mean and Variance

The equations for the mean and variance of the overflow have been derived using the method described in [Sch76] and [Te174]. This is based on using equations (3. l), (3.2), (3.3) and factorial and conditional factorial moments. The factorial moments for two arguments are given by.

Further the conditional factorial moments are given by:

~ , ( j I i ) = x ( < ) r ! P ( i , j ) and F , ( j ) = C ~ , ( j ( i )

Summing (3 1) and (3.2) over i and j for i = 0, I, . ., x, and j = 0,1,. , x,, simplifjling and replacing x, with j gives.

Multiplying both sides of (3.4) by rl, summing over all j 2 r and applying the (3 identity

gives

To find a hr the r expression for the conditional moments multiply both sides of (3.1) by

rl and sum over all j r r, which leads to:

+ + = j = r J = r (i)

By simplifLing (3.7) and using ( 3 . 3 , the first two conditional factorial moments are found to be:

The mean overflow rn from the primary group is equal to the traffic carried on the infinite overflow group and is given by.

The variance of the traffic carried on the overflow group is equal to the overflow variance v and is given by:

V = & ( J ) + C ( J ) - ~ ~ ( ~ )

From equations (3.6), (3.8) and (3.9)

Substituting (3.12) in (3 10) leads to the following expression for the mean:

m = f ~ ( x , , A)

The overflow variance was found by substituting (3 12) and (3 13) into (3.11)

Remarks

It can be seen that the variance of the overflow traffic for full overflow can be found by substituting f = 1 in equation (3.15) It then simplifies to *

Figure 3.3 A server group cons~stzrlg of c channels wzlh n over-ow routes

This is the overflow variance of a network with full overflow to a single overflow route given by Equivalent Random Theory. See for example [Coo8P].

It should also be noted that the result for the mean given in equation (3.14) can be found fiom the one dimensiofial state transition rate for a M/M/mfm m-server loss system with two arrival streams of diserent rates. This is well documented in the literature. See for example pages 128 and 129 of [Coo8 11.

3.3.3 Variance with Multiple Ovedow Routes

The result derived in equation (3 15) can be used to find the variance of the overflow for a network with any arbitrary number of overflow routes to which a portion of the offered traffic may be routed.

Again let the total pure chance offered traffic to the network be A Erlangs (refer to Figure 3 3) Of this total traffic there are 72 portions of size f,A that can overflow to n different overflow routes and xn = l The traffic associated with each portion all

r = l

overflow to the same route. So for any portion i there is traffic cf;A that can overflow to the i-th route and (1 - J ) that cannot overflow to that route.

Equation (3.15) can now be used to calculate the variance (v,) of the overflow associated with the z-th overflow stream. This process can be repeated and the variance calculated for each of the remaining n - 1 overflow streams.

This approach can also be used to include the variance of the lost traffic that results fiom restricted overflow. Here the lost traffic simply becomes another overflow route with some portion of the offered traffic (f *) overflowing with mean m' and variance v'.

This gives the following relationship'

Note that because each of the n overflow streams are dependent that the sum of the variances of the individual overflow streams does not equal the variance if all the streams overflow to the same route (ie f = I), or

3.3.4 Comparison of Analytic Result with Simulation

A comparison of the analytic expression for the variance of the overflow is plotted in Figure 3 3 and 3 4. The graph shows the results using equation 3 23 and from simulation for 7 channels and 16 channels in the primary group andffO 1,0.5 and 0 9.

It can be seen from both these graphs that the result using equation 3.15 agree exactly with the simulation results, within the bounds of statistical variations.

In Figures 3.4 and 3.5 the graphs of the simulation and analytic results are contrasted against the values that result from using a simple proportional relationship between the overflow variance with partial and fill1 overflow. This is given below by

where v' is the overflow variance for full overflow (ie with f = 1) and is given in equation (3.16).

The approximate result is close for large f but not for lower values off and consistently overestimates the true variance. This is expected as the approximate result becomes exact for f = I .

0 2 4 6 8 10 12 14 16 18 Offered Traffic

Fzgure 3.1 Offered traf$c versus overflow variance for a przmaiy ,goup of 7 cha~nels.

Figtire 3.5 Offered traffic versus overflow variance for a primary groztp of 16 channels.

3 -4 Approximate Blocking for Individual Traffic Streams using the Splitting Formula Method

The Equivalent Random Theory introduced by Wilkinson and Xordan in [Wi156] provides a simple method for determining the traffic lost from a sever group where the traffic offered to that group is a mix of overflow traffic and pure chance traffic. The method presented in this section for determining the approximate blocking probability for the individual traffic streams overflowing to an overflow group combines the ERT with a splitting formula.

This result in itself is not new The use of splitting formulae for the analysis of alternate routing in fixed telephony networks has a long history. For example see Katz in [Kat67] and Wilson in [Wi171]. The important contribution in this section is the combination of the ERT, splitting formula and the overflow variance for a network with multiple overflow routes into a complete solution for the approximate blocking probability for cellular networks with restricted overflow. Layered cellular networks are particularly well suited to this approach because of the inherent multiple overflow routes and restrictions to overflow that arise from the very nature of mobility and discussed in more detail in Chapter 2.

In order to illustrate the operation of the Splitting Formula Method (SF Method) the author draws on a simple cellular network comprising a single macrocell with r p

underlaying microcells This is shown in Figure 3.6. Each of the rnicrocells has effectively two overflow routes. Either calls overflow to the macrocell or calls are lost. The call arrival process of fresh calls to the microcells and the macrocell is assumed to be Poisson and the call holding time is taken to be negative exponentially distributed. This gives rise to pure chance traffic to the coverage area comprising the macrocell and a group of rnicrocells. To simpli+ the analysis, handover and overflow between cells in the same layer have been ignored.

Referring to Figure 3.6, let the total traffic to the area covered by the macrocell and its underlaying rnicrocells be A . Define P as the proportion of the total traffic that can access the rnicrocells. DefineJ; as the fraction of offered traffic tq the i-th microcell that has an overflow route to the macrocell This accounts for the restricted overflow of traffic from the i-th microcell. The traffic to each microcell and the macrocell is given by.

n

x4 =PA and 4 = (1-P)A. r=l

Fzg~re 3.6 Model of a szmple two layer architecture comprzslng a single mncroceZ1 wwlfh n microcells. Each microcell has two overflow paths.

This assumes that the traffic offered to the microcells is divided equally between the microcells. That is the traffic density is uniform across all the microcells. There are c, traffic channels in the i-th microcell and c, channels in the macroceli. The traffic carried by the i-th microcell is given by x. The traffic lost at the i-th microcell because (1 - f;)A, Erlangs offered to the network cannot ovedow is given by mjf . The traffic that can ovedow to the macrocell is defined by its mean M, and variance y . The traffic offered directly to the macrocell is given by A,, the total traffic carried by the macrocell is Y, and the total traffic lost at the macrocell is rn.

The macrocell or overflow group in Figure 3.6 can be re-drawn in Figure 3 7 This shows the offered traflic to the macrocell represented by its mean and variance and the total loss from the macrocell composed of the sum of the individual losses due to each of the offered traffic streams. The total loss m from the macrocell can be found by applying ERT in conventional way, see for example Cooper [Coo81]. The individual mean and variances of the overflow traffic to the macrocell can be found using equations (3.14) and (3.15). In Figure 3.7 m, i E (0, . . , ? I ) is the traffic due to each of the i input traffic streams not carried by the macrocell.

Fzpre 3.7 A model of the rnacrocell or over-ow group

Let 3,' i ~ ( 1 , . . ,n) be the blocking probability for the A, (1 - f , ) calls offered to the i-th microcell. Since this microcell is offered only pure chance traffic 3,' is given by.

c, ! where ~ ( c , , A) = - is the Erlang loss firnction [Coo8 11

Let B, i E (I,. . , n) be the blocking probability for the f i A, calls offered to microcell i and Bo be the blocking probability for the calls offered to the macrocell. Then B, and Bo are given by:

and

In order to calculate B, and Bo values for m, i ~ ( 0 , . . ,n) need to be determined. The hnction of the splitting formula is to proportion the contribution of rn, from each stream of traffic offered to the rnacrocell to the total loss m The performances of three splitting formulae are investigated. These are given in equation (3.20) for the formula of Wallstrom, in equation (3.21) for that of O!sson both reported in [Pra67] and in Equation (3.22) for that of Akirnam and Takahashi which was reported in CAT831 .

rn and B=- M

Both the splitting formulae of Wallstrom and Akimaru and Takahashi rely on the mean blocking probability B of the overflow group. The splitting formula of Blsson relies only on the mean and variance of the individual stream offered to the overflow group

3.5 Comparative Performance of Splitting Formulae

In this section numerical examples are presented that compare the results for the blocking probability for the individual traffic streams offered to a network when using the SF Method described in Section 3 4 with the same results for a simulation study. In addition, a simple Poisson approximation to overflow is also compared with the SF

. Method and the simulation results. The performance of the three splitting formulae is compared for a range of network architectures. The examples presented in this section are based on the two layer network model shown in Figure 3.6 .

3.5 .1 Graphical Results

Offered Traffic (Erlang)

Figure 3.8 Blockingprobabili~ versrrs offered traffic to the microcellfor NelWork 2 w z t h n = 2 , ~ = 7 , ~ , = 7 , ~ = 0 6 7 , f = l .

0 1 2 3 4

Offered Traffic (Erlang)

Figure 3.9 Blocking Probability versus Offered Traffic to the macrocell for Network 2 wlthn=2, c = 7 , c, = 7 , /3=0.67, f -1.

Wallstrom Olsson

Poisson

4

2 3 4

Offered Traffic (Erlang)

Figue 3.10 Blockzngprobabilzty versus offered traffic to the microcell for Network 3 with n = 2 , c = 7 : c O = 7 f = 0 . 5 , P = 0 . 5 .

+ slm Wallstrom

.- - - - - Olsson n A&T

0 b ~ s s o n \

5 10 15

Offered Traffic (Erlang)

Figure 3.11 Blocking probability verszrs offered traffic to the macrocell for Network 3 withn=2, c = 7 , c O = 7 f = O 5,p=O 5. .

2 3

Offered Traffic (Erlang)

Figure 3.12 Blocking probability versus offered traffic to the microcell for Network 4 wz thn=2, c = 7 , c, =14, P = 0 . 5 , f =0.5.

+ slm

Wallstrorn - - - - Olsson - - . - - - - A&T

5 10 15

Offered Traffic (Erlang)

Figure 3.43 Blockirigprobabilify versus offered traffic to the nzacrocell for Nehvork 4 w~th n = 2 , c = 7 , c, = 14, P = 0 . 5 , f =0.5 .

I

0 2 4 6 8 10

Offered Traffic (Erlang)

Figure 3.14 Blocking probability versus offered trafJic to a microcell for Network 6 w i t h n = 7 , c = 7 , c, =14 ,P=0 .9 , f =0.5.

0 2 4 6 8 10

Offered Traffic (Erlang)

Figure 3.15 Blochngprobability versus offered traffic to the macrocell for Network 6 w i t h n = 7 , c = 7 , c O = 1 4 , P = O . 9 , f =0.5.

0 1 2 3 4 5 6

Offered Traffic (Erlang)

Figure 3.16 Blocbng probabrlity verszis offered trafJic to nzzcrocelI number 1 for Network 9 with n = 7 , c = 7, c, = 7, p = 0 9 , f = 0.5 for mzcrocells n = 1,. . ,3 and

f = 0.9 for microcells n = 4, . ,7.

O 4 I + slm

Wallstrorn 2 035 - .----- Olsson E 2 0 3 t A&T

Offered Traffic (Erlang) I Fgure 3.17 Blocking probability versus offered traffic to microcell number 6 for Network9with n=7, c = 7 , c o = 7 , P = 0 9, f =0.5 formicrocellsn=1,..,3 and

f = 0.9 f i r rn~crocells n = 4,. ,7.

0 2 4 6 8 10

Offered Traffic (Erlang)

Figure 3. I 8 Blockingprobah~lzty versus offered traf3c to macrocell for Nettvork 9 wrth n = 7 , c = 7 , c o = 7 , p = 0 . 9 , f =0.5 formicrocellsn=l,.., 3and f =0.9 for

Comments

From Figures 3.8 to 3.18 it can be seen that the results for the blocking probability for the traffic offered to the microcells using the splitting formulae tend give a wide variation that straddles the simulation results. The results using the formulae of Wallstrom and Akimaru and Takahashi tend to over estimate the simulation values The results using the Olsson formula tend to under estimate the simulation result. The interesting outcome is that for large P all methods including the Poisson approximation give quite accurate results. Refer to Figures 3 .14 and 3.16 where P= 0.9 While for moderate values of P they give what appears to be just adequate results when compared with simulation. Refer to Figures 3 8, 3.10 and 3.12.

By contrast the blocking probability for the traffic offered directly to the macrocell calculated using the splitting formulae appears to agree well with the simulation results under most conditions. These results will be discussed in more detail in Section 3 4 .3

3.5.2 Comparison of Numerical Results

The splitting formulae are compared in detail by calculating the ratio of the blocking probability for the i-th cell obtained using the splitting formulae to the mean value determined by simulation (ijt""). Therefore let x, = B , / B ; = ~ The mean and variance of the xr values over a range of koffered traffic values are then determined for each network according to the following

1-1 mean = - k

k 2

k.. ]=I -(&) var =

k(k - 2)

where x, is the j-th sample value of x, at some value of offered traffic and there are a

total of k points taken for each study. All the results within each study are evaluated over the same range of offered traffics.

Table 3.1 Results comparing the ratio of blockingprobabibties for traffic offered directly to the Macrocell from a slmulatr'on study to that calculated with the splitting

formula.

Macrocell

Wallstrorn Olsson Akimaru and Poisson Takahashi

Mean Var Mean Var Mean Var Mean Var

Network 1: n = 2 , c = 7 , c, =7,/3=0.5, f = l

0.980189 0.000928 0 991347 0.001079 0 982751 0.000943 0.99341 9 0.001 147

Network2: n = 2 , c = 7 , c, = 7 , P= 0.67, f = 1

Network4: n = 2 , c = 7 , co =14,P=O5, f =O5 I

0.974757 0.00407 0 991854 0.004576 0.980368 0.004223 0 982388 0.004532

Network 5: n = 7 , c = 7 , c, = 14, P=O 1, f -0.5

Network 8: n = 7, c = 7, co = 14, P= 0.9, f = l

0999541

Network9: n = 7 , c = 7 , co = 7 , P = 0 . 9 , f =0 .5 for cells 1,2,3, f = O 9 for cells 4,5,6 & 7

1.06E-07 1.74E-07 0.999426 0.99991'7 4.17E-09 1.15E-08 1.000028

Table 3.2 Results comparzng the ratio of bkockingprobabzlities for traffic offered to a MicrocelIfiom a simulation study to that calculated with the splitting formulae.

Microcell

3.5 .3 Discussion and Conclusions

For the range of networks examined it can be observed that the blocking probabilities for the traffic streams offered to the macrocells are all close in comparison with the simulation result. In particular it can be noted that the results for the Poisson approximation are in some instances closer to the simulation result than the splitting formulae.

In contrast, the results for the blocking probability for the traffic offered to the microcells show that the Poisson approximation consistently and significantly under-estimates the blocking probability compared with the simulation The splitting formulae produce results that are quite close to the simulation results with the formula of Akimaru and Takahashi giving the best outcome relative to the simulation. This is closely shadowed by the Walistrom formula The formula of Olsson seems to give a mixed performance, sometimes only marginally better than the Poisson approximation.

On the basis of the work presented here the splitting formulae of Akimaru and Takahashi [AT831 in equation (3.22) and Wallstrom in equation (3 20) appear to give the best result when compared to a simulation study. This reinforces the use of a two moment model using these splitting formulae for calculating the blocking probability in layered networks. It can also be noted that where only the blocking probability for traffic offered directly to the macrocell network is needed, a Poisson model may be sufficient.

3.6 Concluding Remarks and Extensions

In this Chapter a method for the analysis of overflow in layered cellular networks using splitting formulae has been presented. In applying this method to layered networks, the assumptions of no overflow from higher layers to. lower layers and no overflow between layers have been made Also, handover traffic is ignored. This method was developed to examine the general problem of blocking probability and dimensioning for the problem of serving different user classes in layered cellular networks and is applied to that problem in Chapter 4. It can therefore be applied to network architectures where the overflow paths can be described by Figure 3.19 below. or example the overlaid cell or reuse partitioning schemes described in Chapter 2.

Figure 3.19 A general model of the ovei-jlow paths f i r a network of one common over-ow cell with n o v e r -w i n g cells.

A fhrther extension the SF Method is to include handover between layers and overflow of handovers. This is undertaken in Chapter 5.

Chapter 4

Solution of the Problem of Serving Different User Classes using the SF Method

4.1 Introduction

In Chapter 2 the problem of different user classes arising in layered cellular networks because of restrictions in the ability of calls to overflow was described In this Chapter an analysis of this general problem applied to a two layer cellular network is presented The analysis method uses the SF Method described in Chapter 3 . The aim in analysing this problem it not only to gain insight into the design and dimensioning of macrocell and rnicrocell networks but it is also aimed at gaining a greater insight into the common problems of restricted overflow in cellular networks.

In section 4.2 the network model for this problem is developed and it is shown how this problem can be modelled as a network with restricted overflow. The network performance parameters are discussed in section 4.3. A numerical example based on the well-known Manhattan Model for street microcells is presented in section 4.4 This example examines the performance experienced by users in each of these classes and examines the impact of the proportion of each user type on the blocking probability experienced by the three classes. It also examines the impact of the proportion of the three user classes on cell capacity. The concluding remarks are given in section 4.5. It is assumed throughout that a fixed frequency allocation scheme is used in both the macrocell and microcell network

4.2 Model of a Network Serving Different User Classes

The network consists of a layer of macrocells that provide contiguous coverage. Microcells are implemented to support high density traffic demands. Calls arrive in the catchment area of a macrocell according to a Poisson process. The catchment area of

the macrocell is defined as that area within the boundary of the macrocell It includes all 1

microcells that fall within this boundary. Further, it is assumed that all rnicrocells within the macrocell boundary do not extend outside that boundary. Independence between cells is also assumed. This allows a grouping of one macrocell and its underlaying rnicrocells to be considered as a distinct entity.

Calls from class I users will attempt to connect to the microcell and if unsuccessfbl they will be blocked and not return Calls from class I1 users will attempt to connect to a macrocell and again if they are unsuccessfUl, they will be blocked and not return. Calls from class I11 users will first attempt to connect to a microcell, if unsuccessffil they will overflow to the macrocell. If, afier trying to connect to a macrocell, they are still unsuccesshl then they will be blocked and not return. All calls are assumed to have the same negative exponential holding time. Call arrivals from neighbour cells due to handovers are not considered in this analysis. Also, directed retry of new calls between cells is ignored.

Let the total traffic offered to the catchment area of a macrocell be A Erlangs. Define P as the proportion of this total offered traffic that is offered to the rnicrocells within the catchment area of that macrocell. Thus /? gives the proportion of the total offered traffic due to class I1 and I11 users. It is assumed that this traffic is uniformly spread anlongst the n microcells within the catchment area of the macrocell.

Further, define as the proportion of the offered traffic from the i-th microcell that can overflow to the macrocell. Then the following equations describe the relationship between the total traffic offered to the macrocell catchment area and the contribution from each user class.

For a network where the offered traffic to the rnicrocelils is uniformly distributed between all the rnicrocells then the following equations apply:

Microcells

Figure 3.1 Nebvork Model for a Two Layer Network Supporting Three User Classes

, ( ~ - L ) P A An = and A: = - f#A.

n n

where A', A: and A: are the offered traffics from the class I users to the macrocell and the class I1 and I11 users to the n-th microcell respectively In this study it is assumed that each microcell experiences the same proportion of class PI1 calls. Therefore f ; =....=fn = f.

The network described above can be modelled as a network with restricted overflow and this is shown in Figure 4.1. The network in Figure 4 1 consists of a single macrocell with n underlaying microcells There are c, traffic channels in the i-th microcell and c, channels in the macrocell. The traEc carried by the i-th microcell is given by Y,, the traffic Iost fiom class I1 users at the microcell is given by 'm,. The traffic fiom class 111 users that overflows from the i-th microcell t o the macrocell is non-Poisson and is defined by its mean m, and variance v, The total traffic from class I1 and 111 users carried by the inacrocell is given by Y, and the total traffic lost from class I and I11 users at the rnacrocell is given by m. The traffic offered from ciass I users to the macrocell can also be defined in terms of its mean and variance, where m, = v, = A' because this is pure chance traffic.

4.3 Network Performance I

The network performance can be viewed from two perspectives. These are the new call blocking probability and the network capacity. These quantities are described in the following sections.

4.3.1 New Call Blocking using the SF Method

Define the blocking probability as the probability that a call access attempt will not find a free channel and hence be lost. For class P calls the blocking probability is B', for class TI calls offered to the i-th microcell it is B," and for class I11 calls offered to the i-th microcell it is B,"'.

The blocking probability for calls fiom class I1 users is given exactly by

where ~ ( n , A) is the Erlang Loss Function

Applying the Splitting Formula Method developed in Chapter 3 the approximate blocking probability for each of the other user classes can be determined. Using the splitting formula derived by Akimaru and Takahashi [AT831 the blocking probabilities are given by.

since z, = 1 and

,, m m { l + ( z l - l ) g s ( ~ , z ) B, = A for z = (I,. . . . , n) ,

A;"'M I+(z - I )~ , (M,z )

(M + ~z)c, V v n n where gs (m, z) = , z = - , z =A, y=C v,, M = x m l

M(c , , -1+~+32) M ' m, Z=O 1=0

From equation (3.14) and (3 15) the mean and variance of the overflow traffic from the i-th microcell are given by:

It should also be noted that with equal traffic offered to each microcell the blocking probabilities for each user class in each cell will be BY = B: =. = B: = B" and

III - 111 B, - B ~ = .... = ~ r f l l = ~ ~ .

4 3.2 Network Capacity

One of the primary aims of using microcells is to provide additional capacity. The number of users in each of the three call classes is defined by the quantities P and j; The network capacity can be defined as the total offered traffic A that the catchment area covered by the macrocell and microcells will support for a given blocking probability or Grade of Service (GOS) value to the users in that area. The GOS experienced by each call class will differ since each is subject to a different part of the overall network and the class 111 calls have the benefit of having an alternative route available to them.

The network capacity can then be defined in two ways. Firstly, it is the total offered traffic to the catchment area of the macrocell such that on average all users experience a GQS that is not greater than the target value. Remember that higher or larger GQS equates with worse performance. Alternatively, it is the total offered traffic to the

, catchment area of the macrocell such that on average any user in any call class experience a GOS thzt is not greater than the target value.

The average blocking probability for calls initiated to this area is given by

Therefore in the first instance the network capacity is the total offered traffic A to the catchment area subject to the constraint that

B, = GOS (4.7)

In the second instance the network capacity can be expressed as the total traffic A offered to the catchment area of the macrocell and the microcells such that

m a x ( ~ ' , B" ,B"' ) = GOS . (4 8) 1

Either of equations (4.7) or (4.8) can be used to represent the performance constraints ,

on the maximum cell capacity. The graphs in section 4.4 are given in terms of as it ,

would be expected that on average any user attempting to make a call would experience a blocking probability of 0 02.

4.4 Numerical Example

In this section the variation in the blocking probability for each user class with variations in p and f is examined. The impact of /? and f on the network capacity for a single macrocell with an underlaying network of microcells is also investigated. All the analysis is based on the application of the SF Method presented in Chapter 3 It should be noted that in a11 the graphs presented in this section the value of p = is never actually achieved and this point represents a point that can exist in theory but may only be approached in practice. This happens because for P= there are no class I calls offered to the macrocell and therefore there can be no loss for these calls. However, if just one class I call is made then it will experience some finite blocking probability. Therefore the graphs for call blocking and network capacity do not include a point for ,8= but are calculated ,

for a maximum value of p= 0.999999.

4.4.1 Manhattan Network

The examples presented in this section are based on the "Manhattan Model" described by Gudmundson in [Gud92]. The network architecture used is shown below in Figure 4.2 It consists of a grid of streets with blocks spaced at an interval of d metres The streets are overlayed with a rnacrocellular network such the one macrocell will cover an area of 0.5 krn? The microcell base stations are located at the street intersections.

6 Microcell Base Station locations

Figure 4.2 Example of a Manhattan layour for microcells coverrng 25 street mtersectrons. The shadzngs indicate the notional microcell coverage and are not

zntended to zndicate a frequency plannrng strategy.

Two example networks were studied and these are based on a spacing between intersecting streets of d equals 100 metres and 200 metres. This results in 100 and 50 microcells per square kilometre respectively. As-the macrocell is taken as covering an area of 0.5 square kilometre in each scenario, the total number sf microcells (n) within the catchment area of the macrocell is 50 and 25 respectively. Throughout all the examples the microcells have equal numbers of channels, c, = c, =. . . = c, = c

4.4.2 Blocking Probability

The blochng probability experienced by each of the three user classes depends primarily on the level of offered traffic. For a given total offered traffic the blocking also depends on the values of p andS, the number of microcells (n) and the number of channels in the

macroceli and microcells. The variation of blocking probability for a range of values of p and f is presented.

Variation with p and f

The typical variation of blocking probability with P and f is shown in Figures 4.3 to 4.5 for a 25 rnicrocell network and Figures 4.6 t o 4.8 for a 50 microcell network The general trends in blocking for the three user classes are summarised in Table 4.1. From the graphs it can be seen that the blocking probability for Class 111 users reaches a maximum before decreasing when P is increased. This results because the macrocell is being offered less traffic as P increases and so the overflow suffers less blocking The occurrence of this maximum depends on the total offered traffic and the macrocell capacity.

Table 4.1 The effect on the blockmngprobability4br each user class oj chungzng P and f.

The blocking probability for class II users continues to rise with increasing /? because of the resulting increase in offered traffic to the microcells The blocking probability for class I users fdlls at high P because of the reduction in the offered class I calls to the macrocell. However, this is effected by the level of overflow to the macrocell from the class 111 calls ( A ) . This will vary depending on the capacity in the microcells (number of channels c, and number of microcells n ) and the total offered traffic.

Class I f=O 9 . . Class I f=O 1 ---- Class II

Class 111 f =O 9 Class Ill f=O 1

Beta

Figure 4.3 Blocking Probability versus Beta for A = 50 E, n = 25, c = 7, c, = 7.

Figure 4.4 Blocking Probability versus Beta for A = 100 E, n = 25, c = 7, c, = 7.

I 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1

......... ... '...... L . . . . . . . . . . . . , v 7 - ' . . . . . . . . . - . . . I - * ' * - - -

1 OOE-01 1

1 00E-02 --

1 OOE03 -- n 0 0 e 1.00504 --

1 OOE05

1.OOE-06

1 ODE07

. -- . , --a _----

1

/ @

J - //

Class I f=O 9 ,

/' Class I f=O 1 #' *

4 Class II P f - - Class Ill f=O 9

z I

8 Class Ill f=O I

-- ' / ' >

I /

-

I - - - - - - - Class l f a 9 - - - - - - Class If* I

Class I I Class Ill f=0.9 Class Ill f=O 1

Beta

Figure 4.5 Blochng Probabrlity versus Beta for A = 100 E, n = 25, c = 7, co = 2 1

1 ' L . . - - - - - L - - - - - - - * . - - - - - - . - . . - - . - * - . - - - I - - . -_____ - - - - - - - - - - - _ - - -.

- '\

- --- aass 111 f a 9 rn C - Y Class I l l f a 1 g 1E-06 - _ _ _ .-.I .. . - * - . _ . - . - - m - -

1 E-09

1 E-I 2

Fzgure 4.6 Blochng Probability versus Beta fov A = 25 E, n = 5 8, c = 7, co = 7

- - - - aass I, f = ~ 1 \ - --- Class II

Class Ill f =O 9 Class Ill f=O 1

I Q 1 00E-09

Beta

Fzgure 4.7 Blocking Probabilzty versus Beta for A = 100 E, n = 50, c = 7, c, = 7.

0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0.8 0 9 1 - -:. ._.

0.001 _--- -

I-

/ -- .- / -f , / m C , - Class I f4.9 X o 1E-06 -- sass I f=0.1 0

8- z 2 +' Class II

Class Ill f=O 9 1E-09 --

Beta

Frgure 4.8 Blocking Probability versus Beta for A = 150 E, n = 50, c = 7, c, = 2 1

4.4.3 Network Capacity

As explained in Section 4.3, the evaluation of network capacity can be based on the GOS performance averaged over the three user classes (equation 4.7). In the examples presented in this section a target GOS of 0.02 is used. That is a blocking probability of 2%. A network consisting of a single macrocell with 25 and 50 microcells is considered

For both the networks of 25 and 50 microcells we can see that, as expected, the network capacity increases as the proportion of class I1 and I11 users (ie P ) increases. This occurs because a greater proportion sf calls are making use of the capacity of the microcell (or low tier) network

However, for some cases when f<l the network capacity reaches a maximum and then begins to fall as ,L? increases. This is seen in Figures 4 10 and 4 12 This occurs whenever the hnction for the average blocking ( B ~ , , ) has a minimum. In turn this will

happen when the blocking experienced by the class I1 begins to rise relative to the falling blocking for the class I and I11 users As a result of P increasing and more calls (both classes 11 and 111) being offered to the microcells the blocking of the class I1 users rises relative to the falling class I and I11 blocking. This is particularly evident when there is a relative abundance of macrocell capacity. Therefore, in these instances, being able to increase p offers no increase in cell capacity. However this can be reversed if there are sufficient network resources in the microcells to satis@ more of the class I1 call demand (for example larger c and or n) or the values of can be increased. The class I1 calls

are most effected because they have no alternative for call connection, whereas the class , I11 calls can overflow t o the macrocell

This peak in the cell capacity can be moved by changing the relative capacity of the macrocells and microcell This can be seen in the case of Figures 4.10 and 4.12 where there is an additional number of microcells (50 compared with 25). lit can also be seen for Figures 4.1 1,4.12 and 4 13 where reducing the macrocell capacity and increasing the microcell capacity respectively effectively moves the peak to the right of the graph. When microcell capacity is added to the cell (by increasing c , ) there is the additional effect of an overall increase in cell capacity.

For p in the range p < 0 S we observe that adding microcell capacity (either as c, or n ) has little impact on cell capacity. As expected, the major impact on capacity for low P is achieved by increasing the rnacrocell capacity. Therefore it is concluded that as a network matures and the microcell network expands with more users capable of accessing microcells then the operator must be able to balance the resources in the rnacrocell and microcells to match the given value of /3. It is also possible that ,O and f

60

values will vary from cell to cell and potentially with time of day as the users move ,

around the network.

Figure 4.9 Capacr fy A versus P fir n = 25, c = 7, c, = 7

F~gure 4.10 Capacity versus P for n = 25, c = 7, c, = 2 1

Beta

Figure 4.11 Capacity 2 versus /3 for n = 5 0, c = 7, c, = 7

0.1 0 2 03 0 4 05 0.6 07 08 09 1 Beta

Figure 4.12 Capacrty 2 versus P for n = 50, c = 7, c, = 2 1

0 1 02 03 0 4 0 5 06 0 7 0 8 0 9 1 Beta

Figure 4.13 Capacity 2 versus P for n = 50, c = 14, c, = 2 1

4.5 Conclusions

It is clear from the example networks studied in sections 4.4.2 and 4 4.3 that the maximum network capacity is achieved when both and f are approaching one. This may be difficult to achieve or control in practice, especially if both the quantities P and f are predominantly influenced by factors outside the control of the network operators. Examples of these types of factors are where customers can choose their handset types or user mobility. The opportunity exists for the network operator to take control by adopting a strategy for designing the network, influencing customer choice or offering subscriptions that will maximise both P and f .

It has been seen that in some of the cases where f<l, there is a peaking in the network capacity before P I . Once again it may be difficult for operators to achieve network conditions that allow this capacity to be achieved. However, it should also be noted that the total network capacity decreases rapidly from its maximum value as P decreases. It is possible in a network for the values of P and f in each cell to vary from time to time throughout the day. This would then result in a corresponding variation in the network capacity. This problem may lead to temporary instances of congestion.

One way to address this problem is to use techniques that equalise the blocking probability for each of the user classes. This would reduce the maximum capacity at the expense of giving it some uniformity for the range of and f values Equalisation of performance based on network solutions would use techniques such as channel reservation or state-dependent call acceptance. Equalisation based on charging would use different rates for different performance (for example, lower performance means a cheaper rate). This is a subject for hrther analysis

In this chapter the author has shown that the general problem of a layered ~ellular network that is required to serve different user classes can be modelled as layer network with restricted overflow. The restrictions arise from the fact that the users can be classified in terms of their ability to access different cells in the network. These results give an insight into the problem of ogtimising a network for maximum capacity while achieving a given Grade of Service.

Please note

The text in this file has been automatically extracted and may contain minor errors. For the original version please consult the paper copy

held in the Swinburne Library.

Chapter 5

The Modified SF Method for the Teletraffic Modelling and Analysis of Hierarchical Multi- Layered Wireless Networks with Overflow

5.1 Introduction

In Chapter 2 a range of overflow policies for new calls and handovers was proposed. Also a number of restrictions on overflow that can occur in cellular were described In this Chapter an analysis method for estimating the network performance of a hierarchically layered cellular network utilising these overflow policies and operating under these restrictions is derived.

The aim of this Chapter is to develop a model that allows for any conditions of cell overlap that can be used to provide an alternate route for blocked new call or handover attempts. The basic features of layering incorporated into the model presented here are.

e handover of calis between overlapping cells in the same layer

handover between cells in overlapping layers where layer coverage corntnences or ceases

the overflow of unsuccessfbl new calls and handovers

the capability to model unlimited overflow of handovers and new calls between any layers

the capability to model restricted overflow of new calls and handover calls

A hrther aim of the model is that it is capable of analysing large networks. This requirement arises from the need to account for the interaction between cells in the operation of the overflow policies The ftndamental aim of alternate routing is that calls

overflow from the congested cells to the less congested cells Therefore it is to be expected that the performance for any user will depend on the congestion in the first choice cell and all the other cells that the call may overflow to ,

In Section 5 2, a traffic model for hierarchical multi-layered networks is developed based on the flow model approach of [McM91] This extends that work to multiple hierarckal layers. This model is further developed in Section 5.3 to the Modified SF Method. This is achieved through the addition of overflow traffic arising from intra and inter layer new call and handover overflow and incorporation of the SF Method. The SF Method developed in Chapter 3 with some additional approximations is used to determine the approximate loss and overflow from the individual offered traffic streams to each overflow route This forn~ulation allows the treatment of multiple overflow routes with restricted and unrestricted call overflow between any overlapping cells in the network. It also allows large networks to be treated and can deal with any arrangement of cell overlaps In Section 5 4 the network performance framework is presented This is followed in Section 5 5 by the analysis and results of an example network. Further remarks and extensions to the work presented in this Chapter are discussed in Section 5 6.

5.2 A Teletraffic Model of a Multi-Layered Wireless Network

Starting with the approach of McMillan In [McNI91] and [McM93], a traffic model for layered wireless networks can be constructed under the assumptions that the call arrivals

' to each cell are independent and that the cell blocking probabilities are also independent. Commence by letting the number of layers in the network be L with the bottom most layer as 1 and the top layer as L. The number of cells in layer 1 is N, and c,, is the number channels in cell n in layer I or cell (1, n), where n ~ ( 1 ~ 2 , ., N,J and 1 E (1,2, , L) . The allocation of c, ,, channels to cell (1, n) is based on a fixed channel assignment.

Let the arrival rate of new calls to cell (1,n) be ill,n and the arrival rate of handovers t o cell (1,n) be el,,. The total arrival rate of new calls to the network is then given by A. = zL zNi il,,. Both new calls and handovers are mbdelled has having a Poisson

1=1 n=l

arrival process. The modelling of handovers as having a poisson arrival process has been investigated by Guerin [GuP187], where it was shown that this is a reasonably valid assumption. Ca!ls depart all cells due to conversations completing at mean rate (per call) ,u. Calls will also depart cells because they handover to cells in the same or other layers. Those calls in cells on the edge of a coverage area will need to handover to a cell in the

6 6

next higher layer when they leave the coverage area of a layer or they will be able to I

handover to a cell in the next lower layer when they enter the coverage area of a lower layer These handover calls will depart cell (l,n) for cell (k,rn) at mean rate (per call) I

y . All call departure rates are assumed to be parameters of negative exponential

distributions

The mean departure rate from (1,n) is given by

A l , n Layer I , Cell n

Figure 5.1 A traffic model for n single cell in n multi-layered network

The approach of McMillan in [McM91] can be used to yield the following expressions for the offered new call traffic ( A , . ~ ) and handover traffic for cell (Z, n)

where the mean arrival rate of handover calls (el,,). into cell (1,n) is.

Y,,, is the traffic carried by cell (k, m) and

where Bk ,,, is the call blocking probability or average call congestion in cell (k,m)

Similarly, the traffic carried (y , , ) and the traffic lost (m,,) in cell ( 2 , n) are given by I

m1.n = B,.~(A,," + elan) ( 5 7 ) 1

The blocking probability for any cell (I, n) B,,, is given by.

4 . n = ~(cl , . , A,, + @I,, ) ( 5 . 8 )

where ~ ( c , A) is the Erlang loss function for a network of c channels offered A Eriangs.

As in [McM91] Equations (5.1) to ( 5 8) can be solved iteratively using a fixed point analysis to give the call blocking probability, carried and lost traffic for each cell

5.3 The Modified SF Method

The approach taken in this Section is to extend the model presented in Section 5 2 through the addition of overflow traffic and include the technique exploited in Chapter 3 of using a splitting formula to calculate the approximate loss for individual traffic streams This will be formulated to include the overflow policies for new calls and handover that were described in Chapter 2.

In this section the traffics offered to cell (Z,n) in the form of overflowing new calls and

handovers from neighbouring cells and cells in other layers are modelled by their mean and variance. Define the total mean overflow into cell (Z,n) due to overflowing new calls as "M,, and the variance as "y,, . Furthermore, define the total mean overflow into

0 cell ( ~ , n ) due to overflowing handovers as MI,, and the variance asel/;,, Refer to

Figure 5 2. It is proposed to develop a set of equations that allow the calculation bf lost traffic from each cell using the ERT and that can be solved iteratively

0 l ,n ayer I , Cell q,n AW,n7 A Y , n ( 2 , n) qJz V I , ~

Figure 5.2 A traffic model for a cell in a layered network with over-ow

68

A Assume for the moment that the values for A ~ , f , , Tl;,n, @M/,, a n d o ~ , , are known The 1

method for calculating these values is addressed in Section 5.3.1 The total offered traffic to cell (L,n) now comprises traffic from Poisson sources (new calls and I

handovers) and non-Poisson sources (inter and intra layer overflow). Now, define the quantity (m,,) as the traffic not carried in cell (Z,n) and (v,,n) as the variance This is

necessary since the traffic not carried in a cell is not necessarily all lost as there exist alternative routes for it via ovedow So, (m,,) and the traffic carried in cell (1,n) (x,,) can be found by applying Equivalent Random Theory (ERT). Using the standard approach (see for example [CooSl]) an equivalent pure chance offered traffic A;:, and an equivalent number of channels c,:, can be determined that yields the same values for the mean offered traffic to cell (Z,n) and yvl1 the variance of the offered traffic to the cell (refer to Figure 5 3) . The total mean and variance of the traffic offered to cell (1,n) is

given by.

F~gzrre 5.3 The baszc Eqzcivalent Random Model- for the tl-aff~c ofleered to cell ( I , n)

Once again, applying ERT yields the mean traffic not carried by the cell m,, and its variance v,., '

It is assumed that the new calls, handovers, overflowing new calls and overflowing handovers arriving to cell (1,n) behave as independent traffic sources and that the blocking probabilities are independent.

For each cell (1,n) the values of A,:, and cl:, can be found using the standard

approximate solutions given by Rapp and found in [Coo8l] and reproduced here as

A,:, = y,, + 3 4 , (z , .~ - 1) where Z,,n = is the variance to mean ratio or

peakedness of the offered trafic and

The carried trafic in cell (1,n) is now given by

Because calls that do find all channels busy in cell (1,n) have a number of possible alternate routes based on the overflow policy, the actual number of calls lost from the system needs to be determined for any particular overflow policy.

5.3 1 Calculating Overflow Traffic into Each Cell

It can be seen from Figure 5.2 and the description surrounding it that the mean A4, rl and variance V;, of the overflow traffic offered to cell (1,n) will comprise traffic that is

overflowing from the neighbour cells in the same and other layers (for example cell (k ,m) for k ~ ( 1 , . . , L) and rn E (1, . , N , ) ) and that this will be dependent on how the values of mk ", and v,.,,, are split between lost new calls and handovers and overflowing

new calls and handovers

Define "mi; as the mean new call traffic that overflows from cell (k,rrz) to cell (l,n), A I n v; ,~, as the variance of the new call traffic that overflows from cell (k, rn) to cell (l,n),

as the mean handover trafic that overflows from cell (k, m) to cell (1,n) and " v ~ ~ ~ as the variance of the handover trafic that overflows from cell (k,m) to cell (1, n) Then

equations for each of the mean and variance of the overflowing new calls and handovers offered to cell (1,n) from all other cells can be written

Once again, it is assumed that the overflowing new call and handover traffic to any cell (I,n) from all the other ( ( k , rn) for k E (1, . , L) and m E (1, . ., Nk) ) cells behave as

independent trafic sources.

Now for any two cells (l,n) and (k, m) the proportion of ca!ls that overflow from cell (k , m) to cell ( I , n) needs to be determined so that equations (5.14) to (5 17) can be applied.

5.3.2 Proportioning New Call and Handover Overflow Traffic Out of Each Cell

The Splitting Formula Method described in Chapter 3 is used to determine approximately the proportions of calls that overflow from one cell to another. On the basis of the comparison of splitting formulae presented in Chapter 3, the splitting formula introduced by Akimaru and Takahashi [AT831 will be used in the remaining analysis presented in this Chapter.

Three ovedow models are considered below based on the call control options described in Chapter 2 The first covered is the general overflow model that has no limits on the number of overflow attempts for new calls and handovers other than overflow is either between cells in the same layer or to cells in the next highest layer The second model arises when new calls are allowed only one overtlaw attempt and the final model'refers to a special case when there is no intra layer overflow of new calls allowed in the lowest layer

The General Overflow Model for Unlimited Overflow Attempts

The general overflow model accounts for unlimited overflow attempts of both new calls and handovers This condition exists under the policies 4 and 5 for new calls described in Chapter 2 For a general overflow model the diagram of Figure 5 2 can be redrawn to take account of there being no distinction between the treatment of new calls that are freshly offered and those that have overflowed. Similarly there is no distinction between

the treatment of handovers that have overflowed and those that are making their first attempt. This is shown in Figure 5.4

A l,n +I'M n A l,n + A ~ l,n Layer E , Cell n

Fzgure 5.4 The trafJic model for a cell where no restrictions apply to new calls and ha~zdovers

Applying a suitable splitting formula the mean of the trafic not carried in cell (I,n) (mi,,, v,,,) can be split into the mean new call traffic that can overflow ("rn,,,) and the

mean handover traffic that can overflow ('mi,,) This assumes that all calls offered to

the cell that are allowed to overflow actually have another cell to which they are capable of overflowing. The situation of calls having no alternate cell that they can overflow to is dealt with later in this section

Applying the splitting formula of Akimaru and Takahashi [AT831 given in equation (3 22) the following equations can be determined for cell (Z,n)

and

A m/,n= m,n+'m/,n .

Next the variance of the traffic not carried by cell (Z,n) ( v , , ~ ) must be split over the two I

overflow streams (new calls and handovers). A search of the literature reveals that there are currently no similar splitting formulae that can be applied to splitting the overflow variance when the circuit group is offered overflow traffic. Wilson in [Wi177] investigated the moments of the traffic overflowing from a common circuit group offered rnultiple streams of overflow trafic found no analytic or empirical expressions for the individual overflow variances. Wilson [Wi177] and later Fitzpatrick and Sakurai [FS95] develop an exact analytic solution for the individual overflow stream variances when the common circuit group is offered traffic from a number of independent pure chance traffic sources In that case, the splitting of the traffic streams results in dependent traffic streams (see also Wallstrom and Reneby [WR79]). It can be seen that for these dependent overflow traffics the sum of the individual variances is greater than the total variance. That is v , , c vrn where is the set of all streams over which the overflow

9 I

variance is split.

It may be expected that the individual overflow streams from a circuit group offered overflow traffic streams will also be dependent as suggested by Wilson [Wi177] The approach taken here is to split the variance in the same proportions as the means. This does not account for the dependence between the individual split streams, nor does it account for the offered traffic streams being a mix of pure chance and rougher than pure chance trafic. Taking this approach it is expected that the estimated variances will be smaller that the real variances on the basis that splitting the variance in the same proportions that the means are split will result in v , ~ = ~ p ~ ~ n compared with v,, < zpv:, for the exact result with pure chance offered traffics and dependent

overflow traffic streams

Splitting the variance in this fashion the two values of variance are given by

As described in the introduction, one of the properties of a layered network is that cells in one layer will not uniformly overlay cells in a lower layer. In most cases calls in a cell (or group of cells) will overflow to a common cell in the next layer However, there may be some calls that do not conform to this arrangement because of the propagation conditions. Similarly for inter layer overflow it is expected that only. calls at the cell

boundaries would satisfy conditions that allow inter layer overflow All calls in the cell 4

that satisfied these conditions may still have different cells as neighbours. Therefore the next step is to fhrther proportion the overflowing new call and handover traffic from cell . (1,n) into those parts that overflow to the other cells (k, m)in the network.

Define the fraction of new calls that can overflow from cell (1,n) to any cell (k,rn) as A k m f;; for k s (1,. . . , L) and m E (1, , ) . This fraction of traffic will be largely

dependent on the portion or degree of overlapping cell coverage and also the traffic distribution in the cell Also define as the fraction of new call traffic that has no

alternate route, that is it cannot overflow to a cell in any layer because there is no alternative cell option This condition can arise because of coverage anomalies or because of restrictions applied by handset type when the network uses multiple modes or frequency bands to support the layered architecture or where the customer has subscribed to only a subset of all the layers in the network. Therefore

and clearly ^ ~ f ; f = 0 in all cases

Similarly, define the fraction of handovers that can overflow from cell ( I , n) to (k, rn) as O k.m for k ~ ( 1 , . ,L ) and m ~ ( 1 , . ., N ~ ) , and also define 'A,: as the fraction of

handover traffic that has no alternate route. Then

and = 0

The splitting of the overflow is then performed on the basis of a linear combination, based on the values. This yields the following relationships:

0 8 The new call and handover ( rn,,,) traffics that are lost from the cell because of

the restrictions on overflow can be found from

L N k L N k e # also Aml,n=Am:n + 7 "m:: and @ml,.,= m,,. + 'nl;:

The total traffic lost from cell (1,n) is then

One New Call Overflow Attempt

In the call control options 2 and 3 presented in Chapter 2 it was proposed that all new calls that have overflowed once and are unsuccessful at finding a free channel would be lost from the network. Freshly offered new calls and all handover calls can overflow to another cell, where that alternative route exists. This restriction on the overflow of unsuccessfhl overflowed new call attempts can be modelled by rearranging Figure 5 2 to group all the handover traffic together This is shown in Figure 5.5 below.

A

Layer I , Cell n

F ~ p r e 5.5 The rnodrfied trayfic niodel for a cell wzth overfo~v that combines all haizdover traffic.

Now the mean traffic not carried in cell (l ,n) ( m ) can be split into the mean 0 I! overflowing new call traffic lost from the cell ( rn,,,), as well as the mean new call traffic

that can overflow ("rn,.) and the mean handover traffic that can overflow (ernl,,z). Here

distinction is made between overflowing new calls that will not be given another connection opportunity and new calls that have no alternate route in terms of an overflow cell although it will be seen that they both contribute to the total traffic lost from the cell. This leads to new equations for the mean overflowing new calls that are lost from the cell and the mean new call traffic that can overflow. The mean handover traffic that can overflow is still given by equation (5.19).

and

The splitting of the overflow is then performed on the same basis as in the previous Section with equations (5 21) and (5.22) applying Also, the equations that describe the mean and the variance of the new call trafic that overflows from cell (1,n) to cell (k ,m) as A m ~ ; m , A v ~ ~ and the mean and the variance of the handover traffic that overflows from

O k cell (1,n) to cell (k ,m) as r n l , ; : ) ' , ' ~ ~ ~ are still given by equations (5.25) and (5 26) The new call (%;)and handover (Om:,) traffics that are lost from the cell because of the

lack of a inter or intra layer overflow cell are still given by equations (5.27) and (5 28).

The total traffic lost from cell (1,n) is

Layer One as a Special Case - No Intra Layer Overflow

New call overflow policies 2 and 4 in Chapter 2 do not allow intra layer overflow in the lowest layer under the argument that this will avoid the problems of increased interference that is associated with calls in a second server cell. If this is done the lowest layer (layer one) will be offered only pure chance trafic in the form of new calls and handovers This allows the mean and variance of the overflowing traffic from layer one to be calculated using the exact expressions derived in [FS95]. Layer one can therefore be treated as a special case. This results in the following equations for the mean and variance of the new call and handover overflow traffic from cell (1, n) to (k, m) :

5.3.3 Analysis

The equations (5 9) to (5.37) for each cell and the relevant overflow policy can now be combined with ERT and solved iteratively until a fixed point is reached Starting with approximate values of the mean and variance of the total offered traffic to each cell based on the new call traffic values for each cell (E,n), values for B,,,, Y;,, and m,, can be

calculated. The handover and overflow traffic is determined according to the handover rates (per call) and the fractions of overflow between each cell. These are combined to produce a new estimate of the handover traffic, overflowing new call traffic and overflowing handover traffic to produce a new estimate of the total offered traffic mean and variance. These new values are used to recalculate the values for each of the B,,Iz, Ir;,, and nz,, values in the network. The process is repeated until all values converge and

a fixed point is reached.

This application is simply the fixed point method used by McMillan in [McM93]. The equations for the blocking probability represent a continuous mapping of the closed space [O, 11' where X = zL AT, into itself From Brouwer's Fixed Point Theorem as

I = I

given by Simmons in [Sim63] a fixed point exists. This iterative solution has not been rigorously tested for a unique solution under all conditions, but in practice it was found that very small and large new call arrival rates, that is A. --+ 0 and A -+ oo resulted in steady state blocking probabilities in each cell of B,,, -P 0 and B,,, -+ 1 respectively. It

was also found in practice that for a range of starting values for il the results converged

to a fixed point with a number of different arbitrary starting values for all the input variables. However this is no proof that multiple fixed points do not exist.

5 -4 Performance Measures

The performance experienced by callers in a layered network can be viewed from a number of perspectives. Traditionally, performance is seen from the network operators' perspective. Here it is measured in terms of the loss and blocking probabilities of traffic offered to each cell This is usually done on a cell by cell basis as it allows the network operator t o then check the performance of each cell against targets and to direct remedial action when a fault occurs.

An alternative paradigm for evaluating the performance of the system is in terms of how a user sees the performance that they receive from the network Such measures represent a Quality of Service (QoS) measure that takes into account the performance of the network as a whole rather than looking at separate aspects of the performance of individual cells. For example, new call and handover blocking represent two aspects of how a network performs. However, a customer may be more concerned with firstly getting access to the network and then, once access is established, maintaining the connection until they are ready to finish the call. The new call blocking is still a measure of the success (or lack of success) in gaining access to the system, but the handover blocking probability on its own does not indicate the probability of the call maintaining connection until terminated by the user since a typical mobile call may experience inultiple handovers.

In this section two approaches for evaluating performance in a layered network are presented. The first is based on the conventional approach to what the author terrqs here "Network Performance" The second is a quality of service based model that is also used in Chapter 6 as the primary means of comparing network performance The benefits of using the QoS approach are firstly that it gives a view of performance as the users might see it and secondly it can reduce the amount of data to be reported (in this case by one half, where there is new call and handover blocking).

5.4.1 Network Performance

From the cell perspective, the loss and blocking probability for new calls and handovers to each cell may be determined Here, the new calls lost from a cell would include fresh new calls with no overflow path as well as direct retry and overflow calls that fail in that

cell. While it is possible to distinguish lost overflow calls (@m:,) From lost new calls

with no overflow path, this hardly makes sense since the overflowing calls are not distinguishable by their cell of origin. This gives the new call blocking probability for cell (1, n) as:

Similarly for handover loss, all the handovers that fail (both overflowing and non- overflowing handovers) are grouped together to give the overall handover blocking probability for each cell ( I , n)

Under conditions of fbll overflow for new calls and handovers, that is A~.:=@&t = 0 , no

calls will be lost except when restrictions such as those described in Section 3 are placed on overflowing new calls. In this instance, only the overflowing new calls blocked will be counted in the individual cell blocking probabilities. No handovers would be lost until they reached the top layer in the network and, on meeting congestion, had no hrther alternate routes This makes little sense of using individual cell blocking probabilities as a measure of network performance since under some overflow policies no calls are lost from cells in some layers

From the network perspective the blocking probability for new calls ("B) and handovers ('B) are given by the ratio of the total lost calls (new calls or handovers) to total offered

calls (new calls or handovers) These blocking probabilities are given below as

and

5.4.2 Quality s f Service Measures

A suitable QoS measure that can be applied to the multilayer network was developed using as its basis the approach taken by Jolley and Warfield in [JW91]. This approach uses a Markov chain model of call progress with the exact number of states depending on the overflow options used. The steady state conditions of the network are used to determine the one step state transition probabilities for the Markov chain. The Markov chain models of QoS for the policies using unlimited and limited new call overflow from Chapter 2 are described in detail in the following sections.

General Overflow

For the general overflow option that allows unlimited overflow of new calls the Markov chain model for the progress of a call through the network comprises S = 3xL N, + 2

1=1

states. These states can be divided into four subsets Begin by defining the state space as S and containing the four subsets AS,eS7nS,yS with each of the subsets defined as.

A "S = {"s,,, , ."s .., s ,,,L ) contains the states for n ~ ( 1 , . . . . N,) and 1 E (1,. . . L ) where a new call is attempting to access a channel in cell ( 1 , ~ ) .

O s = { @ e 0 s,,, , . . s,,, ., s, N L } contains the states 's,., for n ~ ( 1 , . . . , N,) and 1 E (1, . , L ) where the call in progress is attempting to handover into cell ( l ,n) .

n "S = {ns,,, , . " s ,,,, , . . s,.,,,~ } contains the states "s,., for n E (1,. . . , N, ) and I E (1, . , L ) where the call is in progress in cell (1,n).

Y S = {*s,,'sz} contains two absorption states State 's, absorbs calls that complete

normally, while state 's, absorbs calls that are blocked for any reason

Let ~ ( t ) , t 2 0 describe the state of the Markov chain at time t Define the onestep

transition probability as

and 6t refers to some time increment

Then, using equation (5 42) the one-step transition probabilities for each of the states in the state space can be determined from the appropriate offered traffics, lost and overflowing traEcs, steady state blocking probabilities, handover rates and other network values. These are described below:

1. New call attempting to access a channel in cell (1,n)

A k .m

P( A s l . n . A ~ k , m ) = n 1 . n ,Vk 2 and ~ ( " s ~ , ~ , " s ~ , ~ ~ ) = 0,'dk c l

' / , n + A M I , n

P ("s , ,~ ,"s , .~) = 0,Vk < I accounts for new calls not being allowed to overflow to cells in

a lower layer

2. Handover attempt into cell ( 1 , n )

B k.m

p( @ s ~ , . , " ~ k . ~ , ) = m , n , 'dk > i and ~ ( ' s ~ , , , ~ s , ~ ~ ) = 0, V k 5 Z (5.47)

@ 1 , , , + @ ~ 1 , ~

P ( @ S ~ , ~ , B S , . , ) = 0,Vk < l accounts for overflowing handover attempts being treated in

the same manner as overflowing new calls

3 Call in progress in cell ( I , ? ? )

4. Absorption states I

All calls must eventually enter one of the two absorption states That is, a call will either I

successhlly terminate or it will prematurely terminate because of failure to find a free channel. When a call enters one of these states it remains permanently in that state. I

5. Probability of Call Success and Failure

Next the probability of call success or failure can be determined from the one-step transition probabilities of the Markov chain Begin by defining the following probabilities-

call in progress in cell (1, n) wccesshlly completes} = P, ( "s,,,)

~ { a call in progress in cell (1,n) prematurely terminates) = P, (%,,,)

new call attempting to access a channel in cell (1,n)is accepted into the network and completes

call attempting to handover into cell ( l ,n) completes in state ys, = PC ( " ~ 1 , ~ )

new call attempting to access a channel in cell ( I , n) fails or is accepted and terminates prematurely

So it is now possible to write P,("s,.,), P, ("s,,,), pC("si,,) and ~ ~ ( ' s , , , ) in terms of the

one-step transition probabilities as:

As shown in Theorem 5 of [?3PS72], for the set of transient states comprising subsets A S ,"S and ' S that are finite, a unique solution to equations (5.52) to (5.56) exists.

Limited New Call Overflow

With restricted overflow applying to new calls an additional subset of states @S is introduced to describe a new call that has overflowed once and is attempting to access a channel With the addition of this subset of states the one-step transition probabilities for state subsets ' S and nS remain as given in equations (5.46) to (5.48) and (5 49) to (5.5 1) respectively. For state subsets ' S and 'S the one-step transition probabilities become:

1. New call attempt states

The new call attempting to access a channel in cell ( / , n )

A k n~ *',' , Y k > l and P ( ' S , . ~ , ~ S ,c.,) = 0,Yk < l

2. Overflowed new call attempting to seize a channel in cell ( I ,* )

3. Probability of Call Success and Failure

The probabilities PC(%,,,), P~(@s, . ,~) and P, ,,) for the network where overflow of

new calls is restricted remains as given in equations (5.52) to (5.54). Because of the altered overflow conditions P, ( "s,, ) becomes:

5.5 Numerical Example

The aim of the example presented in this Section is to examine the impact of various intra layer and inter layer overflow policies for new calls and handover calls on the probability of call failure F',(I,n) where P,(I,n) = ~,("s,,,) The scenarios considered in these examples are based on the network architecture shown in Figure 5.6. This network consists of three layers. The layers are aligned on the central cell of each layer. The lowest layer (layer one) contains nineteen cells and both the middle (layer two) and top (layer three) layers contain seven cells each The cells in each layer of Figure 5.6 are marked accordingly, with the central cell of each layer being cell 1 for that layer. Each cell in the network contains 7 channels therefore c,., = 7

5 - 5 1 OEered Traffic Model

The total arrival rate of new calls to the network is A calls per second This new call traffic is offered to each cell in the network according to following model The area covered by the network is divided into Q blocks, each the same size as a cell in layer 1. The arrival rate to the network from each block is then proportioned as K r 2 , where r is the radial distance of the centre of the block from the centre of cell (1,l). This gives the arrival rate of new calls to each block as

where q ~ (1 ,2 , . ,Q) and rq is the radial distance of the centre of the q-th block from

the centre of cell (1,l)

The total arrival rate of new calls to each cell is. then given by the sum of the offered traffic from each of the blocks that fall within the coverage area of the cell Where a block is covered by more than one cell, the traffic from that block is divided amongst the cells according to the proportion of area coverage by each cell. At the edge of the network some blocks fall partially outside the coverage of the network The total traffic from these blocks are included in the cell that partially covers them.

/' 0 \ " \

3 Laver 3 '

Figure 5.6 Three layer network model for nurner~cal example

5.5.2 Handover Parameters

The departure rates (per call) y:: due to call handovers for each cell in the network

need to be specified This could be done by developing a mobility model for each cell that relates departure rate to mobility. Some examples of mobility models based on user velocity are contained in [I-IR86], [Gu687], [EWS89] and [Nan93]. Use is made of the works of Guerin in [Gue87], Nanda in man931 and Rappaport in [Rap931 to provide the example mobility model that yields handover departure rates.

Firstly refering to Guerin and Equation (5.1), the following relationships hold .for the mean departure rate due to handover (17) or the average time spent in a cell (v')

where h,., is the average number of handoffs per call in cell (l,n) and ,u-' is the call

holding time. This result is confirmed by Nanda in [Nan93]. Also Nanda shows that the "handoff rate" or "mean number of handoffs per call" h increases "as the square root of the increase in the cells per unit area". However, he also states that this can break down

for small cell sizes. It is assumed that for the cell sizes in this example that a square root ' relationship holds.

Define the number of cells per unit area in layer I as K~ and the average number of handoffs per call in layer k as h, , the the following expression can writen :

substituting (5 65) into ( 5 66) gives

where ( r l , ) is the mean departure rate for calls from cells in layer 1 This assumes that all cells in the same layer exhibit the same mean departure rate The relationship between mean departure rate (77,) for any cell in layer Z and the departure rates y:;: from cell ( l ,n) to each of the ( k , m ) neighbouring cells is governed by the number of neighbouring cells and the assumptions on how the mean rate to each is proportioned In this example it is assumed that the total mean rate of calls leaving a cell is divided equally between all the neighbouring cells While this assumption is made in this example, the method presented in Section 5 4 can take account of any proportioning of the total handover rate between the neighbouring cells

Examination of Figure 5 6 shows that the K values for each layer can be determined in terms of the cell density in the lowest layer (layer 1). This results in K, = 9 ~ , , K, = 1 6 ~ ~ giving 77, = 4 q 3 and 7, = 7, /3. Starting with some knowledge of the value of v3 it is possible to determine both 77, and v2.

Rappaport in [Rap931 presents an examination of the likely range of dwell times (or average time spent in a cell) arising from a range of mobility characteristics. For the various cell sizes and "platform types" it is possible to draw some likely values for q, that can then be used to determine 7, and 17, For the examples presented here the value of 7;' = 400 sec has been used

Further constraints on the handover process apply where handovers need to be made between the various layers. Particularly, this covers the cells at the edge of layer 1 and 2. For these cells, calls are allowed to handover from layer 2 to layer 1 as well as from layer 1 to layer 2. This accounts for the movement of users between layers at a layer edge. It also applies for the boundary of layer 2 with Iayer 3 For these edge cells symmetrical handover rates are assumed This means for example that for cells (1,s) and ( 2 , ~ )

2.2 - 1.8 y,,, - y2.?. The exact handover rate is that for the lower layer. This means for example i

that y:: = y;;: are derived from q,. Handover rates for the cells at the edge of a layer

are only given where there are overlapping cells in the next highest layer Therefore , there is no wrap around effect at the edge of the layer.

5.5.3 New Call Overflow Parameters

Throughout the examples in this section the intra layer cell overlap is taken as 5% per neighbour. That is a total overlap of 30% with six neighbours The inter layer cell overlap is taken from Figure 5 6

5.5.4 Handover Overflow Parameters

The handover overflow values (qf;',m) are based on the inter layer cell overlap

boundaries from Figure 5.6 These values remain unaffected by the changes to the new call intra layer overflow policy. Because the aim is to provide the best opportunity for handover calls to find a new channel, handover overflow is in the upward direction only This results in 'J",."' = 0 for k 2 I Note that there are no handover overflow values for

layer three as all new calls and handovers that are not satisfied in this layer are lost from the network This combined with the previous result gives Oh!;"' = 0 for k G (1, . , L)

5.5.5 Results and Discussion

The probability of call failure given that a call is initiated to cell (1,n) ~ , ( l , n ) was evaluated for each of the cells in the network. The results for ~ ~ ( 4 1 ) versus total new

call arrival rate A are presented in Figure 5.7 for each of the five new call overflow policies described in Chapter 2 This shows that the strategy of unlimited new call overflow for all layers (Policy 5 ) gives the largest improvement in performance for cell (1,l). The strategy of unlimited new call overflow with no intra layer overflow in layer

one (Policy 4) provides allnost the same level in performance as Strategy 5

0 0 2 0 4 06 0 8 1 1 2 1 4 1 6 1 8

Total New Call Arrival Rate (callslsec) I Figzcre 5.7 P, (1,l) versus Total New Call Arrival Rate for each of the Bve new call

overflow policres: 1. No Overflow, 2. O~re overflow attempt, with no ~~ztra layer over-ow for lajler orze, 3. Orze oveflow attempt, tvrtkt intra layer overflow In layer one, 3.

Unlzmited overflow altempts, with no irztra layer oveiflow for layer one, 5. Urdimited over-ow attempts, ~uith intra layer overflow in layer one.

The graph in Figure 5 7 only gives the result for the central cell. In order to observe the effects of overflow on the surrounding cells ~ , ( l , n ) for all of the cells in the network is given in Table 5 1, for a value of a total new call arrival rate to the network of A = 0 8 calls per sec Note that no values are given for ~,(2,1) and ~,(3 ,1) This is because of the assumptions that (1) all new call attempts go to the lowest iayer first, that is A2,, = ,I3,, = 0 and (2) complete overflow, that is ''~,~~='f~~~ = 0.

The overflow polices for new calls and handovers evaluated under this network architecture and offered traffic model shows that the unlimited new call overflow policy (Policy 5) offers the greatest impact on eclualising the probability of call failure over all the cells in the network. Also, most of the effect achieved through unlimited overflow can be achieved with unlimited overflow and no intra iayer overflow in layer one (Policy 4) The main advantage of adopting such a policy as limiting intra layer overflow in the lowest layer would be to limit the detrimental effects of cochannel interference that may arise when intra layer overflow is used in small cell environments. The calculation of interference arising fiom inter layer overflow is not considered in detail in this thesis and needs hrther study. The policies that limit the number of times a new call can overflow to one provide only a small level of equalisation in the network performance.

Table 5.1 ~ , . ( l , n) for all the cells in Layer One, T+vo and Three. A = 0.8 calls per sec

I Laver 1 I

I Laver 2 I Cells2to 7 1 0.00150 1 0 01156 1 0.01309 1 0.03224 1 0.03074

Laver 3 Cells 2 to 7 1 0 00275 1 0 01192 1 0.0.1354 1 0.03482 1 0.03215

The performance improvements for cell (1,l) are gained at the expense of the rest of the network. From Table 5.1 it can be seen that the reduction in ~,(1,1) is accompanied by an increase in ~ ~ ( 1 , 1 2 ) for all the other cells. This results from the overflow of

unsuccessful calls fi-om heavily loaded cells and the corresponding increase in offered traffic t o the cells with the lowest new call traffic Also, by progressively allowing more calls to ovefflow between cells there is a greater opportunity for a call to find a free channel. This leads to the probability of call failure becoming more uniform across all the cells in the network Figure 5 7 shows policies 4 and 5 give approximately the same

performance. This indicates that if the additional interference generated by intra layer overflow in layer one is significant and it can be stopped by ceasing intra layer overflow, with the result of only a small increase in the probability of call failure and hence little impact on the capacity of the network.

5.6 Concluding Remarks and Extensions

In this Chapter the author has developed an analytic method that allows the evaluation of the approximate performance of the overflow policies proposed in Chapter 2. Its application t o hierarchical networks represents a special case of the general problem of overflow in networks with overlapping cell coverage. The hierarchical network provides one particular set of constraints on overflow between cells In the extreme case overflow between any cells is possible provided that an overflow route is available by virtue of the overlap between cells and the existence of conditions that allow overflow Further study of this network will provide valuable insight into the impact of overflow on the overall performance experienced by all users It will also assist in providing insight into the factors effecting the design of such networks

Such an extension of the analysis contained in this Chapter is to consider the handover performance in terms o f the mean number of handovers per call. Another interesting area is to investigate the changes in the performance of the network when variations are made in the location of the network relative to the traffic peak. This is valuabie because in many cases it is not possible for a network designer to obtain the desired cell location and a compromise must be made. This problem is examined in Chapter 6

Other open questions that remain t o be addressed relate t o the performance s f the overflow policies under different network architectures that may be hierarchical or non- hierarchical in nature and under conditions of restricted overflow. Questions remain about the performance of the overflow policies under different offered trafic models and mobility models that define different handover rates. The optimum provisioning of resources (eg spectrum, infrastructure) to each of the cells so that performance and capacity are maximised for a given network architecture and network cost is open to further investigation. These areas remain to be addressed at a later time.

The main contributions of this chapter are a framework for studying overflow that enables overflow polices for a range of network architectures, offered trafic models and mobility models to be investigated. The analysis and example studied are framed in the context of hierarchical layers but need not be restricted to this. The analysis is easily

extended to cover the general concept of cell overlay without the need t o define any sub grouping by layers. This provides an extremely flexible framework for investigating overflow The approximate analytic approach provides a simple formulation o f the problem that takes account of the relationship between cells and the impact of overflow on neighbouring cells by allowing large networks to the investigated.

Chapter 6

Performance of Layered Cellular Networks with Non-Uniform Teletraffic Demand

6.1 Introduction

In Chapter 5 the author presented a method for anaiysing layered cellular networks that use overflow. The overflow policies and the network to which they were applied provide a number of interesting problems on the impact of these policies on network performance Further, it is important to understand more clearly how these policies may assist the designer of layered networks given their inherent action of spreading the offered traffic across the network.

The aim of this Chapter is to study the performance of the network presented in Chapter 5 under these overflow policies for control of new calls and handovers when is subject to a number of variations in its architecture Specifically, the performance of these overflow policies is studied for different degrees of intra layer cell overlap and with constraints on the handover policies In addition the impacts of variations in the spatial relationship between the cell layout and the offered traffic are also investigated

In section 6 2 the performance issues for layered cellular networks are discussed and the key performance issues highlighted. This sets the path for the analysis that follows later in the chapter. Expressions for the performance criteria are defined in section 6.3 In section 6.4 the network model under investigation is described This covers the model for the offered traffic distribution and the mobility model that drives the handover rates and other handover behaviours A study of the impact dn network performance of the overflow policies is presented in section 6 5 . In section 6.6 a study of the impact of spatial variability in the relationship between the offered traffic model and the network is presented. The impact of cell overlap on the performance of the overflow policies is also discussed in section 6 6 Finally section 6 7 contains the conclusions and hr ther discussion.

6.2 Review of Performance Issues for Hierarchical Networks

Considerable investigations into the teletrafic performance of layered networks have been reported in the literature A representative sample can be seen in [EWS89], [Jo190], [JW91], [FGS92], [FGR92], [FL92], [Nan93], [FR694], [HR94], [PTT94], [RH94], [CR95], [LG95], [YLN95], [BMM96] [Fit96], [L696], [GHK96]. The focus or aims of the studies reported in these papers can be broadly classified into the following problem areas

1. Resource Allocation

The problem of resource allocation between the layers in a network is important to the efficient use of scarce radio spectrutn. The optimum resource allocation problem has been studied in [FGR92] and [FRG94] This problem focuses on the optimum allocation of resources between the layers in a trow layered network to achieve maximum capacity for a given level of network performance These studies relied on the new call blocking as the criterion for network performance in determining the capacity of the network

2. Mobility:

These studies cover such problems as handover rates for macrocells and microcells and formulating mobility models that can be applied to layered networks. The primary work on mobility models for layered networks comes form Wan931. This paper reports a number of key conclusions on the relationship between handoff rate (mean number of handovers per call) and cell shapes, cell densities and user densities The main performance issue in this paper is the mean number of handovers per call.

3 . Performance Characteristics of Layered Networks.

This category covers those papers that report on the performance characteristics, capacity and dimensionins calculations for layered networks in such terms as the new call blocking, handover blocking and forced termination probabilities. Examples can be seen in [EWS89], [FGS92], [FL92], [LG95], [YL.N95], [Fit961 and [GlX96] These papers are generally exploring the properties of layered networks (for example two layer networks of microcells and macrocells) but also contribute new analytical methods. The main result is to show the capacity benefits that layered cellular networks can provide and the dimensioning algorithms for these networks.

4. Performance of New Call and Handover Control Strategies:

These studies primarily describe the performance of particular new call and handover control policies that can be implemented within the framework of a layered architecture Examples o f these papers can be found in [JW90], [JW91], [HR94], [PTT94], [RH94], [CR94], [BMM96] and [LG96]. The performance is typically measured in term of the new call and handover blocking (forced termination) probabilities and some measure of handover activity (for example mean number of handovers per call) The main contribution of these papers is to propose new analysis methods or call control options that use the layered network architecture

In particular, the paper of [BMM96] investigates the performance of a two layer network with the impact of handover from the top lay (macrocells) to the bottom layer (rnicrocells) as its main focus The analysis is restricted t o a single macrocell with two microcells and uses the same analysis method as [HR86] This analysis is similar to that presented in Section 6 5 2 and concludes that allowing handover down from the macrocell t o the microcell results in reduced new call blocking and handover failure probability but with an increase In handover activity. This concludes with a similar result to that presented in Section 6.5.2 however the measure of call and handover failure used are different.

Other works have been concerned about the impact that using small cells will have on handover. For some examples refer to [Chi90], [JW91], [RM92] and [Rap93]. The need for fast reliable handover algorithms in a small cell environment to avoid excessive call dropping has been proposed. The impact of handover performance on dropped calls in progress arising from this increased number of handovers has been highlighted as a major area of concern

While these papers highlight the additional handovers because of smaller cells, additional handsver activity will also arise from handovers between layers Handovers between layers are most likely to occur at the boundary of a layer Calls moving into the coverage area of lower layer can take advantage of the capacity of this layer and handover t o a cell in that layer. Similarly, for calls leaving the coverage area of a layer there is a need t o handover to the next higher layer so that the call continues.

In summary, there are a range of issues surrounding the design and performance and optirnisation of multi-layer wireless networks that remain open questions. This Chapter is concerned with investigating the performance of the overflow policies proposed in Chapter 5. This is undertaken for a range of conditions described below.

Spatial variation between the offered traffic and the cell layout

The aim of this is to investigate how the performance varies when the cell plan is not coincident with the offered traffic An important factor in designing cellular networks is the speed at which the cells can be implemented. Delays in developing new sites can lead to congested cells and customer dissatisfaction Therefore, site availability is crucial to speed of cell implementation In a trade off between speed of implementation and cell selection it may not always be possible achieve the optimum cell location An import design consideration is how the network performance is effected if the cell layout varies from that planned.

* different levels of intra layer cell overlap

Exploitation of cell overlap is a key feature of ceilular network. The aim of this is to investigate what benefits in terms of capacity (or reduced call failure) can be achieved by increasing intra layer cell overlap. This must be balanced against any increases in handover activity that may result for the increased intra layer cell overlap

e restricted inter layer handover

The literature has confirmed that an increase in handover activity is expected for the smaller cell networks that would form on integral part of a layered network This section aims to study the level of additional handover activity that may occur due to inter layer handover occurring for calls entering or leaving at the edge of layers

6.3 Performance Criteria

In order to evaluate the relative merits of the overflow policies and the impact of variability in the offered traffic it is necessary to use some performance criteria. The impact of any call control policies on probability of blocking for new calls and calls in progress is an important measure of performance. Also, the literature tells of the importance of the average number of handovers per call to the reliability of the system

One problem that arises when dealing with large networks is how to take account of the variations in performance between cells. This is particularly evident when the traffics offered to each cell are not equal. This problem of accounting for variation in performance between cells will be exacerbated when irregularities due to the overflow patterns and non-reciprocal handover rates are incorporated into the modelling. It is possible to consider the worst case performance and design for this to be equal to the

performance goal or design specification on the assumption that all other users will experience better petforinance However, it still may be important to know how many users are experiencing this worst performance and how better is the best performance Other questions concerning the cost to provide this level of performance must also be considered.

To give a broader view of network performance without being overwhelmed with the results for each cell the author introduces the concept of a probability distribution of performance The aim of using this approach is to present the maximum information on the overall network performance in the simplest possible form. The approach to determining the distribution of performance for each of the criteria is similar but each is described separately below

6.3 1 Probability of Call Failure

Consider a call that initiates at some location placed randomly on the plane that is described by the network coverage This call will fall within the coverage of a cell with some probability The call will then experience some level of performance given that the access attempt was made to that cell. This level of performance is ~,(l ,n) , the probability of call failure given an access was made to cell (I, n)

Define X as the random variable that denotes the probability of failure for any call attempt and takes the value x = ~'(l, n ) for any call attempt to cell ( I , n). Then the

probability density of the performance experienced by call access attempts in the network can be written as

where 5 is the set of all cells where x 5 P,(I,II) 5 x + a5c

This result can be used to calculate the expected call failure probability E [ X ] and the variance of the call failure probability V a r [ ~ ] using the standard approach (see for example Section 5 3 [Pap841 ). These quantities can then be used as a measure of overall network performance The ideal network perforlnance would exhibit the desired goal value or design specification for E [ X ] with ideally ~ar[X] = 0 or in the practical case as small as possible

6.3.2 Mean Number of Handovers per Call

In assessing the mean number of handovers per call it should be noted that no particular , handover algorithm is considered, nor is the propagation environment taken into account Neighbouring cells in the same layer or other layers are assumed t o be given by the network architecture Nevertheless this serves to give a good indication of the impact on the mean number of handovers per call of the range of call control policies considered in this study.

Once a call enters a cell for whatever reason it will remain in that cell until it either needs to handover or it completes successfUUy For calls in progress the mean time spent in the cell is given by r7;1, where

Define Z as a random variable that denotes the mean departure rate from a cell for any call in progress and takes the value z = q1, for any call in progress in cell ( 2 , ~ ) . Then

the probability density of the mean departure rate for calls in progress in the network can be written as

where Tyn is the carried traffic in cell (1,n) and 5 is the set of all cells where z I ql,n 5 z + d z

The relationship between the mean number of handovers per call h and the mean time spent in a cell is given by [Gue87] as 77= Cz x p The mean call departure rate for the network is given and assumed to be a constant. Therefore the distribution for the mean number of handovers is given by

E [h] = p-' E [z]

and ~ a r [ h ] = p-"ar[~]

J

Mean Number of Handovers for n Single Layer Network with Homogenoras Cell Size

As the comparison point for the multilayer network the single layer network is chosen It should be observed that it may be impossible to design a single layer network t o support the same traffic density that is supported by a multi-layer network Additionally, the single layer network would need t o use cells with dimensions that approach the lower limit for conventional macrocell radius

The mean handovers for a single layer network of homogenous cell size can be simply determined from v= h x p since both r1 and p will be given and constant for all cells in the network

A simple graph of the hnc t ion h = p-' x 77 will allow the performance of the multi- layered network to be compared with the ranze of performance that could be expected from a uniform network with homogenous cell size

I 1 ""t ; 351

0 100 200 300 400 500 6lm j Mean time in cell (sec) 1

F~prr-e 6.1 Meat? trzrmher- of haildoversper call vers2r.s nlearl depnrfzlr.~. rnfe,pom n cell with .a meart caN hold trnte p-' = 180 seconds.

6.4 Network Model

The network is based on that used in Figure 5 6 of Chapter 5. The same assumptions apply in the following examples It is important to note that there is no division o f the traffic between cells according to mobility, so all calls assume a common mobility model with mean call hold time p-l and a mean cell dwell time or mean time that a call remains

in any cell being determined by the layer in which the call is operating This is outlined in 4

Section 6 4 2 on the mobility model

6.4 1 Offered Teletrafic Model

The teletraffic model for the arrival of new calls is based on the inodel presented in Chapter 5 . This allows for a non-uniform distribution of new call arrivals across the cells in the network An extension of this model is made to allow the position of the traf5c peak to vary relative to the cells in the network This allows investigation of the performance of the network under conditions of spatial variation between the location of the traffic peak and the network To achieve this, the coordinate system is drawn from the work of Macdonald in [Mac791 Using this coordinate system, the distance between any two points ( 4 , j, ) and (i2, j , ) is given by.

Using this relationship the new call arrival rate from each of the Q blocks in the network (refer to Figure 6 2) can be evaluated from equation (5 63) where r can be calculated from equation (6 6) above in the following way Define the point (i, , j, ) = (0,O) as the

centre of the cellular network in Figure 6 2 and also the centre of the Q blocks that the network area is divided into Define the point (i2, j,) as the centre of the offered traffic distribution Then the new call arrival rate to each block q ~(1,2, , Q) is calculated

using equation (5 63) The new call arrival rate to each cell is then calculated as the sum of the new call arrival rate from all the blocks q that fall within the area of the cell

Note that the position of each of the offered traffic blocks remains fixed in relation to the cell layout, however the new call arrival rate from each of the blocks can be altered by changing the location of the centre of the traffic peak (i?, j,) relative to the centre'of the network (0,O)

Flg21i-e 6.2 Qflered ti-uflic nnlodel of Q = 123 blocks 1~1th the ceflfre o f the trc!fl~c d e m n ~ ~ d alrgrleu'sl-11th the ceiztr-e ofthe cell i~etwork i, = i, - = 0, j, = j, = 0 .

6 4 2 Mobility Model

The values for the mean departure rate due to handover yfn"' between any two cells are

determined using the mobility model presented in Section 5 5 2 of Chapter 5 This allows the total rate at which calls leave a cell t o related to:

1 . assumptions about the mean cell dwell time for any layer as one simple starting assumption,

2 the number of neighbouring cells in the same layer that is defined by the network architecture, and

3 the degree of overlap of the cell by higher layer cells which is also defined by the network architecture

In terms of the analysis, this means that as calls move between layers the mean time that they spent in any cell is determined by the relative cell density Therefore calls carried in the lowest layer should need to handover on average more often than those in the higher layers The results presented throughout this chapter are based on an average time spent in a cell for layer three of r/3' = 400 seconds

Where the network is operated with no handsver down, this is taken t o mean that calls in higher layers ( two and three) cannot handover t o cells in lower layers Therefore yk;I"' = 0 for all k < I

6.5 Performance of Overflow Policies

In this section the impact of the overflow policies on the probability of call failure and the mean number of handovers per call is reported In particular, this covers the impact of different levels of intra layer cell overlap and the impact o f using handover t o control the flow of calls into the lower layers where calls traverse a layer boundary

The first results, presented in Figures 6 3 t o 6 6, report on the network performance of the overflow policies 1 to 5 described in Chapter 2

Probability of Call Failure

0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8

Total New Call Arrival Rate (callslsec) / Flg.1rr.e 6.3 Expectedprohahrl~ty of call farlz~re versus total new call alprnlal rare. Tolal

rP1tiZr layer cell o~~erlap of 0.3 .

0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2

Total New Call Arrival Rate (calislsec)

Figztre 6.1 Vcrriatrce of thc probahzlity of cnll$rrlzrre verszrs riew call ar.rt\lal rale. Total ultra lnyer cell overlap of 0.3 .

Mean Number of Wandovers per Call

New Call Arrival Rate (callslsec)

Fip4r.e 6.5 Expected mean rurnlher of harldover-s per call verszrs i l e ~ ~ cnll ai.rr\)al rate. Total rnfl-a layer cell overlap of0.3 .

0 4 ' ----t------+ ; --t----i------

0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8

Total New Call Arrival Rate (calislsec)

F1g1r.e 6.6 Vat-lance of the ntecrrl ~nrmher of handovers per call \7ers~rs total rleui ccdl arrival rate. Total ir~tra layer. cell o\jerlap of 0.3 .

Comments

It can be seen froin Figure 6.3 that overflow policies 4 and 5 give the best result in terms of the expected probability of call failure Figure 6 4 shows the reduction in the variance of the probability of call failure that policies 2 to 5 provide by overflow spreading the traffic load more evenly over all the cells in the network. The peak in the curves for the VarCX] is expected because as the new call arrival rate increases it is expected that all cells in the network will tend toward being fully loaded This will occur regardless of the overflow policy, but will be different for each of the overflow policies As all the cells in the network become hlly loaded the probability of call failure for all cell will tend toward unity with the result that the variance will tend toward zero.

In terrns of the handover performance, Figures 6 5 and 6 6 show that policies 2 to 5 tend to decrease the expected mean number of handovers per call over the network compared with the no overflow policy The best performance is provided by policies 4 and 5 In particular policy 4 clearly gives the lowest result for the expected mean number of handovers per call This is due t o forcing all new calls in layer I that need to overflow into layer 2 where the cells are larger and the mean number of handovers per call is less

I t can be noted that while overflow results in an overall decrease in the expected mean handover rate per call, it results in an increase in the variance of the mean handover rate per call This is expected as overflow works to place unsuccessfUl calls into alternate cells that may be in a higher layer This will result in a greater spread of calls across the different layers in the network

* The mean number of handovers per call over the network is expected to be highest at low network loading This occurs because most of the calls will be carried by the smallest cells and hence tend t o have a shorter mean time in the cell As the new call arrival rate increases there are a number of factors at work that shape the results For policy 1, with no new call overflow the calls carried by the smallest cells will be finite and limited by the number of channels in these cells As the new call arrival rate increases the number of calls lost from these cells will also increase At the same time the number of calls carried by the larger cells will be increasing and so there will be a large proportion of carried traffic in large cell with a consequent long time in the cell and hence smaller mean number of handovers per call for those cells For policy 1 this results in a flattening of the curve in Figure 6 5 as all the carried traffic in each cell tends to saturate to a maximum value At the same time the variance of the mean number of handovers per cell rises as more calls are carried in the large cells to a near constant value when the network carried traffic saturates. It is interesting to observe that the mean and variance

curves in Figures 6 5 and 6.6 have complementary shapes. No explanation for this phenomena is offered at this stage

6.5.1 Impact of Intra Layer Cell Overlap on Ovefflow Policies

In this section the impact of the intra layer cell overlap on the network performance for a range o f overflow policies is reported. The intra layer cell overlap is that overlap between cells in the same layer. This is the important factor that effects the degree of intra cell overflow compared with inter cell overflow for both new call and handovers In changing the intra layer overlap there is a flow on change in the fraction of calls that will overflow via inter layer overflow. This arises because in each cell the users are considered to be uniformly spread over the area of the cell. Therefore by increasing the intra layer overlap more calls will have an alternate route within the same layer This will result in fewer calls overflowing via alternate routes in the next highest layer

Two degrees of intra cell layer overlaps are used to produce the results These are 5% overlap with each of the six neighbouring cells in the same layer (0 3 total) and 10% overlap with each of the six neighbouring cells in the same layer (0 6 total) For each of these intra layer overlaps the remaining overflow routes are to the next layer Calls in the top layere of the network that are not satisfied by intra layer overflow are lost New call attempts will have additional constraints depending on the overflow policy Some additional constraints will be imposed on handovers and results covering this work are reported in Section 6 5.2

Probability of Call Failure

0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8

I Total New Call Arrival Rate (callslsec) I

Figr4r.e 6. 7 Ex~~ecledpr~~l~ahi l i ty of call far1zri.e vel;~rls tolal new call cn-I-ival /-ale. Total ir~ti-a luyer. cell overlap of 0.6 .

I Total New Call Arrival Rate (caUslsec)

Figure 6.8 Varr~nlce of [he probabllity of callfail~li-e versus folal tlew call ai-rrval rzzte. Total rizti-a layer cell overlap of 0.6 .

Mean Number of Handovers per Call

1 5 ' +---I--- +------a P-i

0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8

Total New Call Arrival Rate (callslsec)

Figzrre 6.9 Expected it7eai1 haildoverl~ per. call ver.szrs lotaltzeli) call aw~val mte. Toin/ mtra Iayer cell overlap of 0.6 .

Total New Call Arrival Rate (callslsec)

figure 6. I 0 Vcn-rance of the rpienrl nzrn7bc.r of handovers per call verszrs total n e ~ ) call arrival m fe . Total rnfra layer cell overlap of 0.6 .

The impact of increasing the intra layer cell overlap from a total value of 0 3 to 0 6 is shown in Figures 6 7 to 6 10 It can be seen in Figure 6 7 that the call failure probability is significantly reduced for policies 2 to 5 with 60% intra cell overlap compared with the same policies and 30% overlap. The best resuIt is again for policies 4 and 5 Also, significant reductions in the variance of the call failure probability can be seen in Figure 6 8. Both Figures show the same trends as were observed for an intra layer call overlay of 30%. The reduction in the call failure probability is expected since the increased intra cell overlap provides more calls with an overflow path within the same layer and this in turn will allow more calls in total to overflow through both the intra and inter layer routes.

Again, policy 4 gives the best result in terms of the expected mean number of handovers per call because of unsatisfied new call attempts froin layer 1 overflowing to layer 2 For new call arrival rates less than around 1 call per sec the results for the expected mean number of handovers per call are marginally greater for a cell overlap of 0.6 compared with a cell overlap of 0 3 (compare Figures 6 5 and 6 9) In paricular this can be seen in detail for policy 5 in Figure 6.9 This is expected because the increased intra layer cell overlap results in more calls staying in the lower layers with a consequent increase in the mean number of handovers per call due to the shorter time spent in these cells.

At new call arrival rates greater than around 1 call per sec we see the convergence of the expected mean number of handovers per call toward the same values for both intra layer cell overlaps of 0 6 and 0 3 This result is not expected if based on the argument that the increased intra layer cell overlap results in more calls staying in the lower layer cells or fewer calls needing to overflow up to a higher layer and therefore contributing to a higher mean number of handover per call for those calls This argument is the same one used to explain the decrease in call failure probability due to more calls being able to overflow into intra layer cells and results in more carried traffic in the lower layer cells

However, this position fails to take account of the effect of the increased carried traffic per cell on the flow of calls between intra layer cells due to handover As the number of calls moving from one cell to another is dependent on the carried traffic (see equation (5 4)) then we can see that a greater flow of handovers between cells will result from a higher carried traffic. At high load, few free channels will be availble and this in turn will provrde more opportunities for handovers to be blocked and lead to overflow to a cell in the next higher layer This would act to force more calls into the higher layers and hence reduce the expected mean handovers per call Therefore it is concluded that these two mechanisms conter each other with the overall result that at high level of offered traffic

108

the degree of cell overlap has neligable impact on the mean number of handovers per call.

We note that the curves for the expectation and variance mean retain their complementary shapes

6 5 2 Performance with Restricted Handover

The performance of the network can also be influence by the control of handover between cells in different layers In particular the handover of calls from higher layers to lower layers at a layer edge Here it would be advantageous to use handover to direct calls into the lowest layer in an attempt to offer more traffic to the channels in these cells However, this is likely to result in an increase in the handover activity It is not clear to what extent this will increase the handover activity It is also not clear how much the call failure probability will be reduced Further it is not clear what impact the new call overflow policies (2 to 5) will have on the handover activity compared with the policy of no new call overflow

Let the handover between cells in the same layer be called intra layer handover and similarly the handover between cells in different layers be inter layer handover As discussed in Section 5 5 2 of Chapter 5 inter layer handover can be used at the boundaries of layers where coverage by cell in that layer commences or ceases In this Section the same approach is used to control inter layer overflow that was used in Section 5 5 2 That is, handover up to a higher layer is always allowed, but handover clown to a lower layer can be either allowed or disallowed

The aim of this section is to investigate the impact of allowing and disallowing this inter layer handover in the downward direction on the performance and to contrast it with the performance achieved when using overflow policy -5.

Probability of Call Failure

7 lJnrestnsted HO, OL=O 4

1 New Call Arrival Rate (callslsec)

Frgrre 6.1 I Expectedprohnbrlrty of calf fazlzlre versus total new, call arr-ival rate for oveijlow policy 5 (zalrestrrcted new call overflo~/). TWO 111tra cell 017er.l~rp coildrt~o~rs

are lrsed nrzd ilo htmduver dowrl oil the cells at the edge of layers one ar~d fwo.

New Call Arrival Rate (callslsec) I Frglrrae 6.12 Val-rance of the probability of callfnrlwe ~~eistrs total new call ari.11lc71 rdcrte

for overflow polrcy 5 (2rrwestricted rlew calf overflo~v). Tbvo rwtrn cell overlap cowdrtror~s are used nrld 170 handover dowrl on the cells a[ the edge of layers oile ntrd

two.

Mean Number of Handovers per Call

1 8

1 6 \ --*. - '3.

"Y 1 4 t *% U1 - '-3

;h 1 3 %L 1

Unrestr~cted HO,OL=O 3 -Z

Restr~cted HO,OL=O 3 --L-

--C <L=2rL- -------I-

Unrestricted HO,OL=O 6

Restr~cted HO.OL=O 6 1 ! ____l---.___t___ i

0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2

Total New Call Arrival Rate (callslsec)

Frgire 6.13 Expected n?ear~ handovers per call ver.vlrs total new call art-~val I-ate.for overfZo14~ pollcy 5 (zirli-estr-rcted rrew call overj7ow). TM~O rrltra cell over.lcrp cond~trons

are zrsed arzd 1 7 0 hancloser do~vr~ on the cells at the edge cf layers oite nrrd two.

Unrestricted HO.OL=Q 3 1 - - - - Restricted HO,OL=O 3

- . .-- . . Unrestricted HO,OL=O 6 1 I / - - - . - Restricted HO,OL=O 6 1

Total Mew Call Arrival Rate (callslsec)

F~gzire 6.14 Vmiarzce of the meat? nzin~ber of ha~zdoversper call ver..s?rs total riew call avrrval raze for overflow polzcy 5 (iit~restricted rlcw call overjlowt). FYvo lrjtra cell

overlcrp corzdrtion.~ are 1rsc.d and no hcrndover dowr~ (zi? the cells at Ihe edge oflayer=r orre and two.

Comments

It can be seen from Figure 6 11 that for overflow policy 5 and both 0 3 and 0 6 intra layer cell overlap that the expected call failure probability improves marginally when inter layer handover downward at the layer boundaries is not allowed This is more noticeable in both Figures 6 11 and 6.22 for higher new call arrival rates The improvement is due to an overall balance of call movement in an upward direction. This overall movement of calls upward is due to handovers down for calls at layer edges being disallowed which keeps some calls in higher layers than would otherwise occur Further, this allows more new calls t o access cells in the lower layers and generally frees channels in the lower layers t o accept intra layer overflow attempts for both new calls and handovers

In terms of the expectation and variance of the mean number of handovers per call we observe from Figures 6 13 and 6.14 that the use of inter layer handover at the layer boundaries has a significant impact on the handover activity For both values of irltra layer cell overlap the expected mean number of handover per cell is significantly reduced by restricting the inter layer handover However, this benefit is offset by a corresponding increase in the variance Again, this reduction in the expected mean number of handovers per call is due to an overall balance of call movement in an upward direction into the higher layers where the ceils are larger and the handover activity reduced

Therefore it is concluded that the use of handover in this way provides a significant benefit in terms of a reduction in the average handover activity with no penalty in terms of increasing the expected call failure probability

6.6 Performance of Overflow Policies with Spatial Variability

In this section the impact of spatial variability in the offered traffic model is investigated for a range of overflow policies Again the performance is measured in terms of the criteria described in Section 6 4

A total new call arrival rate of /Z = 0 8 calls per second was used in all the investigations reported in this section This value was chosen on the basis of an E [ X ] of

approximately ten percent when the offered traffic peak and the network centre are located coincidently This is an expected probability of call failure of approximately one in ten While this value may be h i ~ h from a network design perspective, where a figure of one per cent or less may be require to meet customer expectations, it represents a level o f performance that can serve as a reference point for comparing the performance of the proposed overflow policies

6 6 1 The Model for Spatial Variability

The model for spatial variability is based on Figure 6.2 To generate the new call arrival rate from each of the Q blocks the centre of the traffic peak is moved relative to the blocks The traffic from each block is given by equation ( 5 63) and is restated here.

where P, is the radial distance of the centre of the q-th block to the centre of the traffic peak The distance cl can be found using equation (6 6) with the centre the Q blocks iocated coincidently with the centre of the network at (1,l) This is given as

where (i,, j , ) are the coordinates of the q-th block and the location of the traffic peak is

varied by changing its position relative to the centre of the network by some value (6

In the examples presented in this chapter the traffic peak is moved along a trajectory described by the relationship 6, = 6, over the interval 6, = 6, = 0 to 6, = 6, = 1 5 R,

where R is the radius of the smallest cell in the network (this is also the radius of each of the Q traffic blocks)

figure 6.15 Model of spatla1 var-ltrhll~ty hefht~ec.11 I/?e h-gfflc peak and the network

6.6.2 Performance with Spatial Variability

In this section results are presented for the performance of the network under overflow policies 1 to 5 when subjected to the spatial variability in the offered traffic model described in the previous Section The results are presented as a function of the variation 8, and 6, and the distance is measured in terms of the cell radius R

Probability of Call Failure

1 Distance (cell radii)

F~gzii-e 6.16 Expected call fazlzire prohahllity versus ofjsef hctwee~~ the offc.1-ed tt.affrc drstribzitron a d the cell layofit. The aistance w grverl rn ternls of i~~ziltrples of the cell

rndiris.

Distance (cell radii) I Figure 6. i 7 Variance of the cnllfnilzii-e probab~lity vei-.slrs o#.set he f~~eei? the offered

traffic d~.sh.rbzrt~on nird rhe cell Iayolrt. The drsrmlce is grven rn tern~s of rnzrltiples of the cell mdrlis.

Mean Number of Handovers per Call

I I Distance (cell radii) 1

F1grr.e 6.18 Expected nteall nzm~ber of haridover. per. call \ler.siw oflset betweew the offered trnfflc u'rstrrh~rtroi~ aazd the cell Zayozrt. 7he distar~ce 1s g~: i \ )e~~ la7 terns of

m~iltzples qf fhe cell rnu'l~rs.

Distance (cell radii)

Figure 6. I 9 Variance of the nlean i~zmtbel. ofhaizdo\~ersper call ver-szrs offset betweell the offL.red traffic d1strib2rlio1? and the cell Inyo2rt. The d~st~nzce 1s gz\)ei~ 1a1 terms of

nt zrltiples of the cell radl zts.

Comments

From the graph of expected call failure probability versus distance (Figure 6.16) we can see for overflow policy 1 (no overflow) that the expected call failure probability rises as the distance the along the trajectory 6, = 6, of the traffic peak relative to the network

centre increases For the same change in distance, overflow policies 2 to 5 result in the expected call failure probability remaining constant until at total distance of between 1 5 and 2 cell radii In this region a minimum in the expected call farlure probability is reached before increasing more rapidly as the total distance increases

This can be explained in terms of the change in cell overlap as the offered traffic peak moves toward the junction of cells (2, l), (2,2) and (2,3). As this point is approached the overflow traffic that would normally flow to cell (2,l) is now split among the three layer 2 cells This resuIts in a broader distribution of the overflow traffic and hence an overall reduction in the call failure probability As the traffic peak moves hrther along the 6, = 6, trajectory the limits t o both the extension of layer 1 and the overall network

size (refer to Figure 5.6) and the number o f channels per cell area in the upper layers begin to impact significantly on the results and the call fa~lure probability begins to rise

Two important conclusions that can clearly b e seen from these results are that

1 the impact of overflow is t o smooth out the increase in the call failure probability compared with overflow not used, and

2 a minimum in the call failure probability is reached as the traffic peak is approaching a point where the network provides a greater opportunity for overflow traffic t o be distributed over a number of cells in the next higher layer

The variance of the call failure probability, Figure 6.17, exhibits a minimum at a marsinally larger distance than the expectation However, for distances in the range 0 t o 1 5 cell radii the variance remains approximately constant This indicated that while the expected call failure probability is remaining constant so this is also true for all the cells in the network.

When no new call overflow is used the handover activity in,Figure 6 18 shows a marginai decrease up to a distance of approximately 1 5 cell radii Beyond this value the expected mean number of handovers per call decreases more rapidly. This is due to there being more calls in the higher layer cells as the traffic peak moves toward the edge of layer I The variance also exhibits a reasonably flat behaviour up to a distance of I cell radius and then increases as distance is increased. This is consistent with the changes in cell occupancy that will result from the movement of the traffic peak

117

4

Finally. it can be seen that overflow smooths the variations in the call failure probability that result from movement of the traffic peak There is clearly a benefit to be gained by locating the traffic peak close to an area where the overflow can be shared among a group of cells. Also, doing this results in a decrease in the hand activity due to more calls being carried in higher layer cells The movement of the traffic peak appears to have no major impact on the handover activity other than that caused by more calls being offered to the higher layer cells that arises from the traEc peak moving toward the boundary of the lowest layer Inclusion of new call overflow will improve both the call failure probability and the mean number of handovers per call relative to no new call overflow The mean probability of call failure is reasonably insensitive to the location of the traffic peak relative to the network, even when no overflow is used However, the use of overflow significantly reduces the mean probability of call failure regardless of the location of the traffic peak relative to the network Further reductions in the mean probability of call failure can be achieved by locating an area of maximum overlap coincidently with a traffic peak The additional benefit of this is to reduce the mean number of handovers per call

6 6 3 Performance with Extended Overlap and Restricted Handover

0 0 5 1 1 5 2 2 5 3

Distance (cell radii)

Flgrre 6.20 Expecfed call farlzrre pi-ohahilrty vet-szis offset befween the offered traffic drstrihzrtron arid the cell Inyouf. The distm~ce IS grven 1n terrz7.s of mtrltrples of the cell

/-crdrlrs.

0 0 5 1 1 5 2 2 5

Distance (cell radi~) ,

Frgire 6.21 Varratlce of the call farlure prohahrlr~ lw-srrs ofjset hetween the offered traffic distrlbzit~or~ ar~d the cell Iayo~it. The disrance IS give^ rrl terms of ntlillzples ofthe

cell radi~rs.

Mean Number of Handovers per Call

! I Distance (cell radii)

Flgtre 6.22 Expected mean ~limber of handovers pel- call versrrs offsel befioeerl the oflered trcrffrc drstrrh7rt1or7 and the cell layout. The dlstarlce IS gi\~e)t rrt tern~s o f

m7iltrples qf the cell rnd~ars.

1 0 0 5 1 1 5 2 2 5 I Distance (cell radii) , I

1 L-.- -- - _ 2

figir1.e 6.23 Var-rance of the mearz ~llO??be~' of ha17dover.s per. call vel.siis offset hehveei~ the offered traffic drstr~blrlion told the cell Inyozlt. The d~.stctirce is grverl 111 terms of

mlrltiples of the cell iau'ilts.

Comments

The impact on the expected call failure probability resulting from increasing the intra layer cell overlap from 0.3 to 0.6 total sees the minimum occur at a point closer to the origin (6, = 6, = 0) From Figure 6 20 it can be seen that the minimum occurs at

approximately 1 5 cell radii when the total overlap equals 0.6 compared to 1.75 cell radii when the total overlap equals 0 3 The dominant effect is still the overall reduction in the expected call failure probability due to the increased cell overlap

The impact of ceasing the inter layer handover at layer edges (see Section 6.5 2 for more details) on the call failure probability also sees the minimum in the expected call failure probability move toward the origin However, this is accon~panied by an overall reduction in the expected call failure probability until it begins to rise very rapidly as the distance passes the point of the minimum.

It can be noted that with the larger cell overlap that both the expectation and the variance of the call failure probability rise more sharply than for the smaller overlap as the distance increases This may result from the traffic peak approaching the edge of the network and because o f the greater overlap more calls are being forced into the cells in the network edge where the capacity is least This would see 2 number of cells with low carried trafic and overflow and a smaller number of cells heavily utilised.

There is virtually no impact on both the expectation and variance of the mean number of handovers per call (Figures 6 22 and 6 23) when the intra layer cell overlap is increase and when the inter layer handover at the layer boundaries is stopped The dominant effect seen here is the reduction in the expected mean number of handovers per call when the inter layer handover is stopped causing more calls to stay in the higher layers with a consequent reduction in the handover activity.

6.7 Conclusions

In summary, the most significant conclusions that can be drawn from the work investigated in this Chapter are

1 Overflow of new calls serves to smooth out the performance over the network compared with no new call overflow This means that all cells are more evenly loaded. The overall result is that the use of new call overflow reduces both the expected mean call failure probability and the expected mean number of handovers per call compared with not using new call overflow The best overall reduction in

the expectation and variance o f the call failure probability is achieved by policy 5, but this is closely followed by policy 4 This result is repeated for the handover activity. However, for handover activity, policy 4 clearly gives a lower expected mean number of handovers per call A mojor benefit of the smoothing out of performance is that this should compensate for non-optimum network designs and make the design outcomes less dependent on designers achieving optimum balance in the numbers of channels per cell

2 Policy 4 gives results that compare favourably with those for policy 5 This means that policy 4 becomes a reasonable choice if cochannel interference is a problem in the lowest layer and can be remedied by ceasing new call overflow in that layer

3 . Increasing the intra layer cell overlap serves to hr ther reduce the mean call failure probability This is achieved without any major change in the expected mean number of handovers per call This result is significant because it means that network designers can draw on large cell overlaps (where they exist) to reduce call failure probability or increase capacity without suffering any major draw backs because o f increased handover activity

4. A policy that restricts intra layer handover in cells at a layer edge so that calls can only handover from lower to higher layer cells results in a significant reduction in the handover activity. This is achieved with a negligible increase in the expected call failure probability This is an important result because it means that the overall handover activity can be reduced by controlling handover at the layer edge without effecting network capacity

A small offset between the peak of the trafic model and the network architecture (along the trajectory studied) sees almost no change in both the expected call failure probability and the handover activity. For offsets in the range 1 5 to 2 cell radii there is a minimum in expected call failure probability and a decreasing in the expected mean number of handovers per call This appears to be due to the traffic peak approaching an area where the overflow trafic can be shared among a group of cells in the higher layers This indicates that the use of new call overflow makes the call failure probability more insensitive to small changes'in the offset compared with using no new call overflow

This result is important because it shows that for this architecture that variations in the cell layout relative to the trafic demand peak will tend to be less important when overflow is used in this way. These variations are likely t o occur because of the practical difficulties in obtaining the exact base station locations such as geographical

122

and regulatory obstacles This should therefore improve the network designers task by increasing the range of usable cell sites It also shows that the there is at least one layout for the network relative t o the traffic demand that gives some degree of optimal expected call failure probability.

Other issues that arise from the investigations in this Chapter (and discussed briefly in Chapter 5) that remain unresolved are outlined be!ow These are presented for completeness and are not meant to comprise an exhaustive list They arise because, as the work presented in this thesis shows, the study of overflow presents a large number of variables and it is therefore necessary to constrain some of these variables in order to investigate the impact on performance of those not constrained To examine all combinations of all variables presents a task the exceeds the scope of this thesis The author has therefore tried to focus on those that give some insight into the design of layered networks and leave the others for future work

1. Restricted Overflow

The network investigated in this Chapter assumed that all calls had at least one cell that would serve as an alternate route The performance of these new call overflow policies with restricted overflow remains an open question This is an extension of the two layer work presented in Chapter 4

2. Network Architecture.

In this Chapter the network is based on a particular architecture The performance of the new call and handover overflow policies for networks wrth differed architectures remains an open question For example, place the cell boundaries at the centre of the traffic distribution or use different relative cell densities and evaluate the performance under the same offered traffics

3 Allocation of Resources to Each Cell

An extension of the architecture question is the question of how to allocate resources between the cells in the each layer and between the different layers This work would extend that of Frullone et a1 in [FRG94] to include handover and overflow

4. Offered Traffic Distribution

The performance of these handover policies when a different offered traffic distribution is used and with different network architectures remains an open question Further to this we can investigate the degree of variation in the E[X] and

Var[X] with changes in the spatial location of the traffic peak similar for other new call arrival rates What happens to the mean number of handovers per call?

5 Mobility Model

This would investigate the assumptions in the mobility model and how they impact on the performance of the new call and handover overflow policies This includes the assumption that all calls are offered to the lowest layer first Restricted overflow would result in some beiiig offered directly to each cell in each layer This leads back to the question of allocating calls to appropriate layers based on mobility See for example Jolley and Warfield in [JW9 11, Hu and Rappaport in [HR93] and Lagrange and Godlewski in [LG96]

Chapter 7

Conclusion

7.1 Summary of this Research

In this thesis we set out to investigate the performance of layered cellular networks with overflow The major contributions are the development methodologies for the approximate analysis of multilayered cellular mobile networks that take account of the multiple overflow routes and restricted overflow that arises in cellular networks and the proposal and analysis of range of overflow policies that can be used t o improve customer perceived performance

W e conclude that the use of overflow in existing schemes such as directed retry, reuse partitioning and the overlaid cells scheme can also be described by the simple concepts of intra and inter layer overflow that we introduce for modelling overflow in hierarchical networks All of these overflow models represent subsets of a general model o f overflow in networks with multiple overflow routes where the ove~flow is constrained in some way. This permits us to formulate the problem of performance in hierarchical networks with overflow as a general problem of restricted overflow with multiple overflow routes This result leads to the need for an analysis technique that can take account of these properties of multiple overflow routes and restricted overflow

The Splitting Formula (SF) Method developed in Chapter 3 is such a technique The analysis of the performance of this approximate approach shows that for the given splitting formulae it gives a very close estimation of the blocking probability for calls offered directly to the macrocell It is also concluded that a Poisson model adequately estimates the blocking probability. However, the SF Method using the splitting formulae of Akimaru and Takahashi [AT831 or Wallstrom [Pra67] is far superior in estimating the blocking probability for calls offered to the microcell that overflow t o a macrocell

The results for the mean and variance of the overflow also have application in other techniques For instance they could be incorporated into the analysis technique developed by Lagrange and Godlewski and reported in [LG95] and [LG96]

From the work presented in Chapter 2 it is concluded that the general problem of different user classes has a broad application to any layered network where overflow is restricted This was investigated in Chapter 4 and from this investigation it is concluded that it is essential t o balance the capacity and demand for each user type This leads to the maximum cell capacity depending strongly on both the numbers of users that can access microcells and those that can then overflow t o the macrocell. Network operators need to be aware that both these quantities may be outside an operator's control (for example customer choice of handset type) leading to the situat~on where the capacity and performance change as a finction of the mobility of the different user classes.

This general problem can also represent that of adding spectrum to a conventional macrocell network in the form of an overlaying network Here some users can access both the new overlaying network and the existing network An example of this occurs with the GSM and DCS standards that use 900 MHz and 1800 MHz respectively These systeins are expected to coexist using dual frequency handsets Refer to Mouly and Pautet in [MP92]

The Modified SF Method developed in Chapter 5 is an extension of the SF Method to include handover and overflow of handover calls It is concluded that this method provides a unique and valuable means for evaluating the approximate performance of

, large networks The addition of a performance metric giving user perceived performance allows the impact of new call and handover blocking to be combined into one measure

Using the Modified SF Method the following new call overflow policies working with a policy of unlimited overflows for handovers have been investigated

I No overflow 2 One overflow attempt, with no intra layer overflow for layer one 3 . One overflow attempt, with intra layer overflow in layer one 4 Unlimited overflow attempts, with no intra layer overflow for layer one 5. Unlimited overflow attempts, with intra layer overflow in layer one

From the work presented in this thesis it is concluded that policies 4 and 5 provlde the largest reduction in the probability of call failure with a complementary reduction in the expected mean number of handovers per call This is achieved through the movement of

calls into the higher layers o f the network that results in more calls being carried in the higher layer cells

Further it is concluded that disallowing handovers of calls froin cells in higher layers down to cells in lower layers as those calls enter the edge of the lower layer coverage serves t o reduce the mean number of handovers per calls without any increase in the call failure probability This provides the network operator the option of controlling handover activity (ie the number of handovers that the system must cope with) without degrading customer perceived performance

The performance of these policies for variation in the distribution of the new call arrival rate with respect to the cell locations was investigated The results lead the author t o draw the conclusion that at least one optimum location exists for placing the cells with respect to the offered traffic. This optimum appears to occur in a region of the network were there is overlap due t o cell boundaries that allow the offered traffic to be shared among the overlapping cells.

7 -2 Recommendations for Further Research

The investigations reported in this thesis cover some of the important facets of the teletraf5c modelling and performance of layered cellular networks with overflow, but still many open questions remain These open questions provide the basis for the extension of the work presented in this thesis and stimulate new areas for research into cellular

, mobile networks using overflow Specifically, the areas for fkrt-eher research can be divided in to two broad categories. The first covers hr ther research into the Modified SF Method for hierarchical and non-hierarchical networks The second covers hrther research into new problems in teletraffic modelling and network perforlnance of hierarchical networks arising from alternative mobility models, network architectures and new traffic sources These are described in more detail below

The Modified SF Method developed in Chapter 5 focuses on the hierarchically layered network architecture The application of this method to network architectures that contain a mixture of cell sizes formed in a coinpletely free structure where the only condition for overflow is the existence of cell overlap presents an interesting problem Coupled with this is the issue of convergence of this fixed point formulation In this thesis no attempt has been made to rigorously establish the necessary and sufficient conditions for convergence. This remains an open question for the Modified SF Method as formulated for a hierarchically layered network and could be extended to a more

generalised formulation applicable to a network with fewer constraints on the overflow of new calls and handovers

Another extension to the method would consider the possibility of developing the analysis method to take account of channel reservation for handover caHs This will allow the comparative performance of networks using overflow and reservation and a colnbination of these strategies to be studied It would also allow the a study of the equalisation of performance by channel reservation for the Different User Classes problem described in Chapter 4

Some of the unresolved performance issues in hierarchically layered cellular networks were raised in Chapters 4, 5 and 6 These cover the following areas

- Restricted overflow This is an extension of the two layered network with restricted overflow investigated in Chapter 4 to multiple layers with handover

Network architecture Extend the work in Chapter 6 to look at alternative architectures and cell densities

- Allocation of resources to each cell This extends the work presented in Chapter 6 into a study of the optimum resource allocation problem This investigation would extend the work of Frullone et a1 in [FRG94] to include handover and overflow

Offered traffic distribution Extend the work in Chapter 6 to deal with different offered traffic distributions

Mobility model Investigate the sensitivity of the performance to the assumptions in the mobility model.

o Multi carrier environment In Chapter 2 we introduce the problem of the rnulti carrier networks where different layers (cells) in the network are operated by different businesses Chapter 4 presents and analysis of a simple 2 layer network that considers the general problem of restricted overflow that also describes the multi carrier environment Extend this analysis beyond a two layer to also include handover

Other developments in cellular mobile communications that are applicable to hierarchically layered networks are briefly discussed below.

Data traffic. The development of data applications and mobile computing will see a move from a voice based network to a data based network With this shift comes the

need to consider trafic models that are applicable to this trafic type and its characteristics.

Application of the SF and Modified SF Methods new access techniques that use alternate routing (for example ATM based cellular networks and any associated call admission control policies).

Examples of research work in the areas of data, mobile computing and multimedia for cellular systems and the application of ATM to cellular systems can be found in [ACT961

Please note

The text in this file has been automatically extracted and may contain minor errors. For the original version please consult the paper copy

held in the Swinburne Library.

A List of the Author's Publications Resulting from this Research [Fit961 Fitzpatrick P , "Performance Analysis Of A Layered Wireless Network Serving Different User Classes", Proceeding of 46th IEEE Vehicular Techr~ology Conference, VTC-96, Atlanta, May 1996, pp 43 1-43 5.

[FL96] Fitzpatrick P. and Lee C S , "On the Performance of Splitting Formulae for the Analysis of Layered Cellular Networks with Overflow", Proceedings of the Australian Telecornmunicatio~~s Networks and Applications Conference (ATNAC) 96, Melbourne, December 1996

[FL97] Fitzpatrick P. and Lee C S., "Performance of Layered Cellular Networks with Overflow and Non-Uniform Teletraffic Demand", Proceedings of the International Conference on Telecommunications, ICT97, Melbourne, April 1997

[FLa92] Fitzpatrick P and Lambert J , "Frequency Reuse in Coexisting Microcells and Macrocells", Proceedings of Communications '92, Sydney, 20-22 October 1992, pp 2 19- 224

[FLe92] Fitzpatrick P , and Lee C S , "Teletraffic Capacity of Microcells in a Matured CMTS Network", Proceedings of ICCSIISITA '92, Singapore, 16-20 November 192, pp 686-689

[FLW] Fitzpatrick P., Lee C S. and Warfield B.. "Teletraffic Modelling and Analysis of Hierarchical Multi-Layer Wireless Networks with Ovefflow", accepted for publication in IEEE Journal on Selected Areas in Communication, Special Edition on Personal Communications - Services, Architecture and Performance Issues

[FS95] Fitzpatrick P , and Sakurai T , "An Approximate Method for the Analysis of Blocking in Layered Cellular Networks", Proceedings of the Australian Teleco~nmunications Networks and Applications Conference (ATNAC '95) 1995, Sydney, December 1995, pp 49 1-495

Please note

The text in this file has been automatically extracted and may contain minor errors. For the original version please consult the paper copy

held in the Swinburne Library.

References [ACT961 Proceedings of the ACTS Mobile Telecommunications Summit, Granada, Spain, November 1996

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