The rapid rate of increase in data traffic means that future wireless networks will have to support a large number of users with high data rates. A promising way to achieve this is by spectrum reuse through the deployment of cells with small range, such that the same time-frequency resources may be reused simultaneously in multiple cells. At the same time, the traditional coverage requirement for wireless users (supporting a modest rate at cell-edge users) is most economically met with cells having large range, i.e. the traditional macrocellular architecture. Thus the wireless cellular networks of the future are likely to be heterogeneous, i.e. have one or more tiers of small cells overlaid on the macrocellular tier.

Let us look at network design from the point of view of a service provider considering a deployment of a network in a certain region. Throughout this book, we only consider the downlink, i.e. the links from the BSs to the user terminals. The principal metrics we shall focus on are coverage and capacity.

(1) Coverage Intuitively, a user (or, more precisely, a user location) is covered if the communication link from the BS serving that user is sufficiently “good” that the user terminal can correctly receive both the control signaling and the data traffic at some minimum rate from the BS. Here, “good” means that (a) the received signal from the BS is “strong,” i.e. the received signal power from the BS exceeds some threshold, and also that (b) the received signal to interference plus noise ratio (SINR) at the user exceeds some minimum value.

Introduction

We have so far studied only the distribution of the SINR at an arbitrarily located user in a single-tier cellular network or a multi-tier HCN. The CCDF of the SINR directly yields a measure of system performance called coverage, as discussed in Section 5.2.4. However, of equal and possibly greater interest to the network designer is another metric variously called throughput, spectral efficiency, and capacity. We favor the use of the term spectral efficiency.

The spectral efficiency can be calculated for each tier, and is defined as the total number of bits of information that an arbitrary BS in that tier can transfer to the set of its served users per use of the wireless “channel” to the users, i.e. per second and per hertz of bandwidth. We shall see that, because the users served by each BS are distributed over some region (the “cell”), this measure of spectral efficiency is an area-averaged quantity, and we shall sometimes refer to it as area-averaged spectral efficiency in order to emphasize this. Further, it is dependent on the details of the scheduling algorithm employed by the BS to select served users for transmission at each transmit interval. However, for the simplest scheduling scheme, where each of the served users is selected in turn for the same number of transmit intervals (called round robin or RR scheduling), we shall show that the area-averaged spectral efficiency is the same as the spectral efficiency on the link to a single randomly selected user served by the BS.

3GPP and LTE

The scale and complexity of design, manufacture, and deployment of wireless cellular systems make it all but impossible for a single-company proprietary architecture to gain traction in the global telecommunications market. Instead, cellular network operators and equipment vendors from all over the world have joined to form standards bodies at the national and international level in order to facilitate the evolution of cellular communication systems. At the present time, six national standards bodies (ARIB and TTC from Japan, CCSA from China, ATIS from North America, ETSI from Europe, and TTA from Korea) have combined to form a single international standards organization for wireless cellular communications, called the Third Generation Partnership Project, or 3GPP for short. The 3GPP partnership was first created to further the UMTS standard, which defined the so-called “third generation” of cellular communications using code division multiple access (CDMA) on the air-interface, but the name 3GPP was retained for the “fourth generation” using OFDMA. This fourth generation of cellular systems came to be known as the “long-term evolution” of the 3GPP standard, and is simply abbreviated as 3GPP-LTE, or just LTE for short.

Support for HCNs in LTE

3GPP standards are published as “Releases,” with a typical interval of 12 to 18 months between releases. As the universal mobile telecommunications system (UMTS) standard became mature and the standards activity ramped up on LTE, the same release (Releases 8 and 9) contained standards specifications for both UMTS and LTE.

The ever-rising demand for wireless data means that conventional cellular architectures based on large “macro” cells will soon be unable to support the anticipated density of high-data-rate users. Thus future wireless network standards envisaged by standards bodies like the Third Generation Partnership Project (3GPP), such as LTE Release 12 and later, rely on the following three ways of increasing system capacity: (a) additional spectrum; (b) enhanced spectral efficiency; and (c) offloading from the cellular network onto, say, WiFi.

So-called “small” cells are an attractive method of increasing spectral efficiency by means of spatial reuse of resources. Small cells can also exploit the fact that additional spectrum in the coming years will be freed up at higher frequencies, where path loss is higher than in the frequencies currently employed in macrocellular networks. A dense deployment of small cells can achieve the desired objective of high system capacity. However, such deployments are unlikely to be found outside of high-traffic areas such as major population centers. Thus, basic connectivity and mobility support will continue to be handled by macrocells. In other words, the wireless cellular network of the future is likely to be a heterogeneous cellular network (HCN), with more than one class of base station (BS).

How HCNs are studied and designed today

One of the most important metrics of network performance and user experience is the signal to interference plus noise ratio, or SINR, defined as the ratio of received signal power to the total received power from all sources other than the desired transmitter (i.e. from all interferers), plus the thermal noise power at the receiver (which is always present, even in the absence of any interferer).

Introduction

We have already discussed why the wireless cellular networks of the future are going to be heterogeneous in nature, with a mix of BSs with different capabilities and characteristics. The spectral efficiency advantages obtained by dense spatial reuse of resources are driving the deployment of small-cell (i.e. small in terms of range or cell size relative to macrocells) tiers as overlays on the existing macrocellular tier.

It is interesting to observe that, as cell sizes shrink, they become comparable to the range of WiFi (IEEE 802.11) access points. This has led to a resurgence of interest in the peer-to-peer (P2P) mode of operation of WiFi, now enhanced and extended to cellular links as device-to-device (D2D) communication, as a means of further enhancing area-wide spectral efficiency. Thus, future heterogeneous wireless networks are also expected to support such D2D communication. As a result, the wireless cellular networks of the future will be not only heterogeneous, but also not even fully cellular – direct transmissions between user devices will be not only permitted, but also enabled by the BSs in order to support ever greater numbers of simultaneously active links within each cell.

Analysis of a network with D2D links

With the renewal of interest in D2D links, it can be said that the analysis of wireless networks has now come full circle, back to its original roots in P2P links (Hunter et al., 2008; Weber & Andrews, 2012). However, a wireless cellular network permitting D2D transmissions among user devices exhibits some novel features not seen before.

We begin with the simple SINR calculation problem of Chapter 1 and generalize it to a deployment with more than one tier of BSs. We then examine the features of the problem that permit us to evaluate the CCDF of the SIR in terms of Laplace transforms of fading coefficients on the links to the user location from the BSs in the tiers. Then we abstract the problem formulation slightly in order to define a general probability calculation of a vector of random variables, which we call the canonical problem. We derive general results and conditions under which the canonical probability may be expressed in terms of Laplace transforms of certain random variables. This is one half of the mathematical core of the book. The other half is the study of stochastic models that yield tractable analytic expressions for the Laplace transforms in the canonical probability calculation, and we discuss that topic in Chapter 3.

Statement of the SINR calculation problem

Let us return to the problem of computing the distribution of the SINR at an arbitrarily located user somewhere in a network with one or more tiers of BSs. We begin by defining the candidate serving BSs, the criteria for their selection, the criterion for choosing the BS that will serve the user, and some notation to represent received power from the candidate serving BSs and from all interferers.

Candidate serving BSs and the serving BS

Consider a snapshot of the wireless network at a particular moment in time.

Introduction

We are interested in the distribution of instantaneous SINR at an arbitrary user location in a multi-tier deployment of BSs, where the locations of the BSs of each tier are modeled as points of a PPP. In this chapter, we restrict ourselves to a simpler problem, namely a single-tier BS deployment. We show how to derive the distribution of the SINR at the user location (assumed, without loss of generality, to be the origin) under different serving BS selection criteria (also called association rules, because they determine the BSs to which the user associates itself). The derivations will use results from the theory of Poisson point processes in Chapter 3. We shall also require some advanced results from the theory of PPPs that were not covered in Chapter 3, and these results will be introduced as needed in this chapter.

Before we study the distribution of instantaneous SINR at a user, let us begin with an even simpler problem, namely the distribution of the instantaneous total received power at the user from all BSs in the tier.

Distribution of total interference power in a single-tier BS deployment

PPP of received powers at user from BSs in a tier

Consider a marked PPP Φ with intensity function λ(·, ·), where the marks associated with the points (x, y) of λ are i.i.d. with common PDF fH(·), say.

Introduction

In Chapters 4 and 5, we analyzed the distribution of the SINR at a user in single-tier and multi-tier deployments, respectively, when all BSs in each tier transmitted with the same, fixed, power. This is a good model for the pilot channels in a system such as LTE, because the pilot channels are used by the UEs to identify BSs, and it makes sense for the power in these pilot channels to be as high as possible all the time. We also know that the distribution of maximum SIR (without selection bias) in an interference-limited deployment is not a function of the (assumed fixed) transmit powers in the tiers if either (i) there is a single tier (see (4.74)), or (ii) the path-loss exponents are the same in all tiers and all tiers are open (see (5.60)).

If serving BS selection is made after applying selection bias (usually in order to modify the loading of the tiers in some way), we know that overall coverage is not the same as with the max-SINR selection criterion without bias (see Remark 5.6). In fact, some users may suffer from poor SINR due to interference from the tier that would have served them in the absence of selection bias, as discussed in Section 5.4. In these circumstances, operating the system at the desired level of performance calls for interference control, usually realized by means of transmit power control (or just power control, for short) with different levels of coordination across BSs, both in a tier and across tiers.

In a stochastic control problem one observes and then seeks to favorably influence the behavior of some stochastic system. Such problems involve dynamic optimization, meaning that observations and actions are spread out in time. In this chapter several simple but fundamental stochastic control problems will be solved directly from first principles, with heavy reliance on the ubiquitous Ito formula. For each of the closely related problems to be considered, the optimal policy involves the imposition of control barriers; these may be either jump barriers or reflecting barriers, depending on the cost structure assumed.

Our first problem can be informally described as follows. Consider a controller who continuously monitors the contents of a storage system, such as an inventory or a bank account. In the absence of control, the contents process Z = {Zt, t ≥ 0} fluctuates as a (μ,σ) Brownian motion. The controller can at any time increase or decrease the contents of the system by any amount desired, but is obliged to keep Zt ≥ 0, and there are three types of costs to be considered. First, to increase the contents from x to x + δ, the controller must pay a fixed charge K plus a proportional charge kδ. Similarly, it costs L + lδ to decrease the contents from x to x – δ. Finally, inventory holding costs are continuously incurred at rate kZt.