This tutorial is intended as an accessible but rigorous first reference for someone interested in learning how to model and analyze cellular network performance using stochastic geometry. In particular, we focus on computing the signal-to-interference-plus-noise ratio (SINR) distribution, which can be characterized by the coverage probability (the SINR CCDF) or the outage probability (its CDF). We model base stations in the network as a realization of a homogeneous Poisson point process of density λ, and compute the SINR for three main cases: the downlink, uplink, and finally the multi-tier downlink, which is characterized by having k tiers of base stations each with a unique density λ i and transmit power p i . These three baseline results have been extensively extended to many different scenarios, and we conclude with a brief summary of some of those extensions. 2 and fading distributions, as well as the noise. These simulations can be extremely time consuming and error-prone. Although system-level simulations will continue to be indispensable for cellular network analysis and design, the need for a complementary analytical approach for the purposes of benchmarking and comparison has long been compelling. Analysis can also illuminate key dependencies in the system, and provide guidance on what features and trends to more closely inspect. An obvious analogy is the computation of bit error probability (BEP) in noisy (possibly fading) channels for different modulation formats: although this too can be simply simulated, it has been indispensable to have formulas which characterize the expected BEP. Generally, these expressions are in the form of a Gaussian Q-function integral, a special function so common that it is often considered closed-form. As we will see, functions almost this simple can describe the outage probability of the entire cellular network despite the presence of an infinite number of random variables (BS locations, fading values), under fairly reasonable assumptions. This tutorial is intended to provide a concise introduction to the tools and techniques for computing the distribution of signal-to-interference-plus noise ratio (SINR) in three key cases for cellular networks. The key new aspect of the model is that all the base stations are located according to a Poisson point process (PPP), which intuitively means they are randomly scattered in the plane with independent locations. We focus on the calculation of the coverage probability which gives the complementary cumulative distribution function (CCDF) of SINR and is the complement of the outage probability. Either one gives the entire SINR distribution. In particular, this tutorial covers the following topics. 1) An introduction to point processes, and particularly the PPP, including key mathematical definitions and computational tools that will be needed to derive the coverage probability. 2) The cellular downlink. We first derive an expression for the coverage probability in a macrocell-only downlink cellular network where all base stations (BSs) are located according to a PPP and their signals experience independent and identically distributed (i.i.d.) Rayleigh fading in addition to standard power-law path loss. The mobile user associates with the closest BS which is equivalently the one with the strongest average power. Three special cases of increasing mathematical simplicity are given. 3) Uplink. The above approach is extended to an uplink cellular system model including transmit power control at the mobile user (i.e. handset). This problem is more difficult than the downlink due to the coupling between the handset point process and the base station 3 point process when it is assumed (realistically) that only a single handset can be active per cell (in a given time/frequency resource). Under some reasonable approximations, we are able to accurately characterize its coverage probability. 4) HetNet Downlink. We extend the downlink results to a heterogeneous cellular network (HCN, or "HetNet") with k tiers, where each tier is distinguished by a unique transmit power p i , density λ i , and SINR threshold τ i . Such a model is quite general and can describe any overlaid collection of macro, micro, pico, and femtocells, which will become increasingly relevant in the future [9] . Despite this significant additional complexity in the model, we are able to compute its coverage probability in a fairly simple form. This result can be viewed as a generalization of the main result on coverage for macrocells only. These three baseline models and their corresponding SINR derivations provide a broad framework to build upon in terms of developing tractable analytical models for cellular networks. And indeed, these results have been extended in literally hundreds of ways to date. We conclude the paper by discussing some of them. In this section, we concisely introduce a few basic concepts from stochastic geometry and some key results for Poisson point processes, with simple illustrative examples. For a more thorough exposition, the interested reader is advised to consider [10] or [11] . Notation: Throughout this tutorial, we denote the random variables by upper-case letters and their realizations or other deterministic quantities by lower-case letters. We use bold font to denote vectors and normal font to denote scalar quantities. Therefore, a random vector (e.g., a random location) will be denoted by an upper case letter in the bold font. Similarly, lower case letters in bold font will denote deterministic vectors. For instance, X and X denote onedimensional (scalar) random variable and random vector (containing more than one element), respectively. Similarly, x and x denote scalar and vector of deterministic values, respectively. The upper case letters in the san-serif font, such as A and L, will be used to denote sets. Put simply, a point process (PP) Φ = {X i , i ∈ N} is a random collection of points residing in a measure space, which for cellular networks is the Euclidean space R d . One way to interpret Φ
Therefore, SG provides a supplementary platform to produce baseline results for benchmarking, and comparative performance analysis REF .
A Primer on Cellular Network Analysis Using Stochastic Geometry
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Mathematics Computer Science