In the previous post I discussed the signal-to-interference-plus ratio or SIR in wireless networks. If noise is included, then then signal-to-interference-plus-noise ratio or just SINR. But I will just write about SIR, as most results that hold for SIR, will also hold for SINR without any great mathematical difficulty.
The SIR is an important quantity due to reasons coming from information theory. If you’re unfamiliar with it, I suggest reading the previous post.
In this post, I will describe a very popular mathematical model of the SIR, which I like to call the standard model. (This is not a term used in the literature as I have borrowed it from physics.)
Definition of SIR
To define the SIR, we consider a wireless network of \(n\) transmitters with positions located at \(X_1,\dots,X_n\) in some region of space. At some location \(x\), we write \(P_i(x)\) to denote the power value of a signal received at \(x\) from transmitter \(X_i\). Then at location \(x\), the SIR with respect to transmitter \(X_i\) is
$$
\text{SIR}(x,X_i) := \frac{P_i(x)}{\sum\limits_{j\neq i} P_j(x)} .
$$
Researchers usually attempt to represent the received power of the signal \(P_i(x)\) with a propagation model. This mathematical model consists of a random and a deterministic component given by
$$
P_i(x)=F_i\ell(|X_i-x|) ,
$$
where \(\ell(r)\) is a non-negative function in \(r\geq 0\) and \(F_i\) is a non-negative random variable. The function \(\ell(r)\) is often called the path loss function. The random variables represent random fading or shadowing.
Standard model
Based on the three model components of fading, path loss, and transmitter locations, there are many combinations possible. That said, researchers generally (I would guess, say, 90 percent or more) use a single combination, which I call the standard model.
The three standard model assumptions are:
- Singular power law path loss \(\ell(r)=(\kappa r)^{-\beta}\).
- Exponential distribution for fading variables, which are independent and identically distributed (iid).
- Poisson point process for transmitter locations.
Why these three? Well, in short, because they work very well together. Incredibly, it’s sometimes possible to get relatively a simple mathematical expression for, say, the coverage probability \(\mathbb{P}[\text{SIR}(x,X_i)>\tau ]\), where \(\tau>0\).
I’ll now detail the reasons more specifically.
Path loss
The \(\ell(r)=(\kappa r)^{-\beta}\) is very simple, despite having a singularity at \(r=0\). This allows simple algebraic manipulation of equations.
Some, such as myself, are initially skeptical of this function as it gives an infinitely strong signal at the transmitter due to the singularity in the function \(\ell(r)=(\kappa r)^{-\beta}\). More specifically, the path loss of the signal from transmitter \(X_i\) approaches infinity as \(x\) approaches \(X_i\) .
But apparently, overall, the singularity does not have a significant impact on most mathematical results, at least qualitatively. That said, one still observe consequences of this somewhat physically unrealistic model assumption. And I strongly doubt enough care is taken by researchers to observe and note this.
Fading and shadowing variables
Interestingly, the original reason why exponential variables were used is because it allowed the SIR problem to be reformulated into a problem of a Laplace transform of a random variable, which for a random variable \(Y\) is defined as
$$
\mathcal{L}_Y(t)=\mathbb{E}(e^{- Y t}) \, .
$$
where \(t\geq 0\). (This is essentially the moment-generating function with \(-t\) instead of \(t\).)
The reason for this connection is that the tail distribution of an exponential variable \(F\) with mean \(\mu\) is simply \(\mathbb{P}(F>t)= e^{-t/\mu}\). In short, with the exponential assumption, various conditioning arguments eventually lead to Laplace transforms of random variables.
Transmitters locations
No prizes for guessing that researcher overwhelmingly use a (homogeneous) Poisson point process for the transmitter (or receiver) locations. When developing mathematical models with point processes, if you can’t get any results with the Poisson point process, then abandon all hope.
It’s the easier to work with this point process due to its independence property, which leads to another useful property. For Poisson point process, the Palm distribution is known, which is the distribution of a point process conditioned on a point (or collection of points) existing in a specific location of the underlying space on which the point process is defined. In general, the Palm distribution is not known for many point processes.
Random propagation effects can lead to Poisson
A lesser known reason why researchers would use the Poisson point process is that, from the perspective of a single observer in the network, it can be used to capture the randomness in the signal strengths. Poisson approximation results in probability imply that randomly perturbing the signal strengths can make signals appear more Poisson, by which I mean the signal strengths behave stochastically or statistically as though they were created by a Poisson network of transmitters.
The end result is that a non-Poisson network can appear more Poisson, even if the transmitters do not resemble (the realization of) a Poisson point process. The source of randomness that makes a non-Poisson network appear more Poisson is the random propagation effects of fading, shadowing, randomly varying antenna gains, and so on, or some combination of these.
Further reading
A good starting point on this topic is the Wikipedia article Stochastic geometry models of wireless networks. This paper is also good:
- 2009, Haenggi, Andrews, Baccelli, Dousse, Franceschetti, Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks.
This paper by my co-authors and I has some details on standard model and why a general network model behaving Poisson in terms of the signal strengths:
- 2018, Keeler, Ross and Xia, When do wireless network signals appear Poisson?.
Early books on the subject include the two-volume textbooks Stochastic Geometry and Wireless Networks by François Baccelli and Bartek Błaszczyszyn, where the first volume is on theory and the second volume is on applications. Martin Haenggi wrote a very readable introductory book called Stochastic Geometry for Wireless networks.
Finally, I co-wrote with Bartek Błaszczyszyn, Sayan Mukherjee, and Martin Haenggi a short monograph on SINR models called Stochastic Geometry Analysis of Cellular Networks, which is written at a slightly more advanced level. This book has a section on why signal strengths appear Poisson.