Category: Point of view

# A dense debate

In recent years, many research efforts were dedicated towards modeling and analyzing denser and denser wireless networks, in terms of the number of devices per km2 or m2. The terminology used ranges from “ultradense” to “hyperdense”, “massively dense”, and “extremely dense”.

IEEE Xplore lists more than 500 journal papers on “ultradense” networks, 25 on “hyperdense” networks (generally published more recently than the ultradense ones), and about 90 on “extremely dense” networks. There even exist 15 on “massively dense” networks. The natural question is how they are ordered. Is “ultradense” denser than “hyperdense” or vice versa? How does “extremely dense” fit in? Are there clear definitions what the different levels of densities mean? And what term do we use when networks get even denser?

Perhaps we can learn something from the terminology used for frequency bands. There is “high frequency” (HF), “very high frequency” (VHF), “ultrahigh frequency” (UHF), followed by “super high frequency” (SHF), “extremely high frequency” (EHF), and “tremendously high frequency” (THF). The first five each span an order of magnitude in frequency (or wavelength), while the last ones spans two order of magnitude, from 300 GHz to 30 THz.
So how about we follow that approach and classify network density levels as follows:

HD: 1-10 km-2
VHD: 10-100 km-2
UHD: 100-1’000 km-2
SHD: 1’000-10’000 km-2
EHD: 10’000-100’000 km-2
THD: 0.1-10 m-2

So, who will be the first to write a paper on tremendously dense networks?

What comes after THD? Not unexpectedly, there is a mathematical answer to that question. A dense set has a well-defined meaning. So in the super tremendously extreme case, we can just say that the devices are dense on the plane, without further qualification. This is achieved, for instance, by placing a device at each location with rational x and y coordinates. This is a dense network model, and almost surely there is no denser one.

# Tractable, closed-form, exact

The attributes “tractable”, “closed-form”, and “exact” are frequently used to describe analytical results and, in the case of “tractable”, also models. At the time of writing, IEEE Xplore lists 4540 journal articles with “closed-form” and “wireless” in their meta data, 650 with “tractable” and “wireless”, and 220 with “closed-form”, “wireless” and “stochastic geometry”.

Among the three adjectives, only for “exact” there is general consensus what is means exactly. For “closed-form”, mathematicians have a clear definition: The expression can only consist of finite sums and products, division, roots, exponentials, logarithms, trigonometric and hyperbolic functions and their inverses. Many authors are less strict, using the term also for expressions involving general transcendent functions or infinite sums and products. Lastly, the use of “tractable” varies widely. There are “tractable results”, “tractable models”, “tractable analyses”, and “tractable frameworks”.

“Tractable” is defined by Merriam-Webster as “easily handled, managed, or wrought”, by the Google Dictionary as “easy to deal with”, and by the Cambridge Dictionary as “easily dealt with, controlled, or persuaded”. Wikipedia refers to the mathematical use of the term: “ease of obtaining a mathematical solution such as a closed-form expression”. These definitions are too vague to clearly distinguish a “tractable model” from a “non-tractable” one, since “easy” can mean very different things to different people.

We also find combinations of the terms; in the literature, there are “tractable closed-form expressions” and even “highly accurate simple closed-form approximations”. But shouldn’t all “closed-form” expressions qualify as “tractable”? And aren’t they also “simple”, or are there complicated “closed-form” expressions?

It would be helpful to find an agreement in our community what qualifies as “closed-form”. Here is a proposal:

• Use “closed-form” in its strict mathematical understanding, allowing only elementary functions.
• Use “weakly closed-form” for expressions involving hypergeometric, (incomplete) gamma functions, and the error and the Lambert W functions.
• Any result involving integrals, limits, infinite sums, or general transcendent functions such as generalized hypergeometric and Meijer G functions is not “closed-form” or “weakly closed-form” (but may exact of course).

Thus equipped, we could try to define what a “tractable model” is. For instance, we could declare a model “tractable” if it allows the derivation of at least one non-trivial exact closed-form result for the metric of interest. This way, the SIR distribution in the Poisson bipolar network with ALOHA, Rayleigh fading, and power-law path loss is tractable because the expression only involves an exponential and a trigonometric function. The SIR in the downlink Poisson cellular with Rayleigh fading and path loss exponent 4 is also tractable; its expression includes only square roots and an arctangent. In contrast, the SIR in the uplink Poisson cellular network is not tractable, irrespective of the user point process model.
A result could be termed “tractable” if the typical educated reader can tell how the expression behaves as a function of its parameters.

Going a step further, it may make sense to be more formal and introduce categories for the sharpness of a result, such as these:

A1: closed-form exact
A2: weakly closed-form exact
A3: general exact
B1: closed-form bound
B2: weakly closed-form bound
B3: general bound
C1: closed-form approximation
C2: weakly closed-form approximation
C3: general approximation

Alternatively, we could use A+, A, A-, B+, etc., inspired by the letter grading system used in the USA. We could even calculate a grade point average (GPA) of a set of results, based on the standard letter grade-to-numerical grade conversion.

Such classification allows a non-binary quantification of “tractability” of a model. If the model permits the derivation of an A1 result, it is fully “tractable”. If it only allows C3 results, it is not “tractable”. If we can obtain, say, an A3, a B2, and a C1 result, it is 50% “tractable” or “semi-tractable”. Such a sliding scale instead of a black-and-white categorization would reflect the vagueness of the general definition of the term but put it on a more solid quantitative basis. Subcategories for asymptotic results or “order-of” results could be added.

This way, we can pave the way towards the development of a tractable framework for tractability.

# What is “coverage”?

In the literature, the probability that the signal-to-interference ratio (SIR) at a given location and time exceeds a certain value is often referred to as the coverage probability. Is this sensible terminology, consistent with the way cellular operators define “coverage”? All publicly accessible cellular service coverage maps are static, i.e., their view of “coverage” is purely based on location and not on time. This seems natural since a rapidly changing coverage map, say at the level of seconds, would not be of much use to the user, apart from the fact that it would be very hard to collect the information at such time scales.

In contrast, the event SIR(x,t)>θ depends not only on the location x but also on the time t. A location may be “covered” at SIR level θ at one moment but “uncovered” just a little bit (one coherence time) later. Accordingly, a “coverage” map based on this criterion would have to be updated several times per second to accurately reflect this notion of coverage. Moreover, it would have to have a very high spatial resolution due to small-scale fading – one location may be “covered” at time t while another, half a meter away, may be “uncovered” at the same time t. Lastly, there is no notion of reliability. For each x and t, SIR(x,t)>θ either happens or not. It seems natural, though, to include reliability in a coverage definition; for example, by declaring that a location is covered if an SIR of θ is achieved with probability 95%, or 95 times out of 100 transmissions.

Hence there are three disadvantages of using SIR(x,t)>θ as the criterion for coverage:

• The event depends on time (at the level of the coherence time)
• The event depends on space at a very small scale (at the granularity of the coherence length of the small-scale fading)
• The event does not allow for a reliability threshold to define coverage.

How can we define “coverage” without these shortcomings? First, we interpret coverage as a purely spatial term, consistent with the coverage maps we find on the web; it should not include a temporal component, at least not in the short-term – hopefully cellular coverage keeps improving over the years, but it should not vary randomly many times per second. Put differently, coverage should only depend on the network geometry (locations of base stations relative to the position x) and shadowing, but not on the rapid signal strength fluctuations due to small-scale fading. The solution to eliminate the temporal component is fairly straightforward – we just need to average over the temporal randomness, i.e., the small-scale fading. Such averaging eliminates the other two shortcomings as well. For a base station point process Φ, we define the conditional SIR distribution at location x as $\displaystyle P(x)=\mathbb{P}(\text{SIR}(x,t)>\theta\mid\Phi).$

Here, the probability is taken over the small-scale fading, which eliminates the dependence on time (assuming temporal ergodicity of the fading process, which means that the ensemble average here could be replaced by a time average over a suitable long period). If shadowing is present, it can be incorporated in Φ. Then, introducing a reliability threshold ν, we declare $\displaystyle \{x \text{ covered}\}\quad\Leftrightarrow\quad \{P(x)>\nu\}$

The reliability threshold ν appears naturally in this definition. The probability that P(x)>ν is the meta distribution of the SIR, since it is the distribution of the conditional SIR distribution given Φ. For stationary and ergodic point processes Φ, it does not depend on x and gives the area fraction that is covered at SIR threshold θ and reliability threshold ν. The figure below shows a coverage map where the colors indicate the reliability threshold at which locations are covered, from dark blue (ν close to 0) to bright yellow (ν close to 1).
So if P(SIR>θ) is not the coverage probability, what is it? It is simply the complementary cumulative distribution (ccdf) of the SIR, often interpreted as the success probability of a transmission. Visualization of coverage at SIR threshold 1. Red dots indicate base stations. The colors indicate the reliability threshold. For example, for ν=0.8, regions colored orange and yellow are covered.