# Random Thoughts on Stochastic Geometry

## Taming the meta distribution

The derivation of meta distributions is mostly based on the calculation of the moments of the underlying conditional distribution. The reason is that except for highly simplistic scenarios, a direct calculation is elusive. Recall the definition of a meta distribution $\displaystyle\bar F(t,x)=\mathbb{P}(P_t>x),\qquad\text{\sf where }\;\;P_t=\mathbb{P}(X>t\mid\Phi).$ Here X is the random variable we are interested in,…

## How well do distributions match? A case for the MH distance

Papers on wireless networks frequently present analytical approximations of distributions. The reference (exact) distributions are obtained either by simulation or by the numerical evaluation of a much more complicated analytical expression. The approximation and the reference distributions are then plotted, and a “highly accurate” or “extremely precise” match is usually declared. There are several issues…

## Meta visualization

In this post we contrast the meta distribution of the SIR with the standard SIR distribution. The model is the standard downlink Poisson cellular network with Rayleigh fading and path loss exponent 4. The base station density is 1, and the users form a square lattice of density 5. Hence there are 5 users per…

## Realistic communication

Today’s blog is about realistic communication, i.e., what kind of performance can realistically be expected of a wireless network. To get started, let’s have a look at an excerpt from a recent workshop description: “Future wireless networks will have to support many innovative vertical services, each with its own specific requirements, e.g. End-to-end latency of…

## What to expect (over)

In performance analyses of wireless networks, we frequently encounter expectations of the form $\displaystyle\mathbb{E}\log(1+{\rm SIR}),\qquad\qquad\qquad\qquad\qquad\qquad (*)$ called average (ergodic) spectral efficiency (SE) or mean normalized rate or similar, in units of nats/s/Hz. For networks models with uncertainty, its evaluation requires the use stochastic geometry. Sometimes the metric is also normalized per area and called…

## Signal-to-interference, reversed

Interference is the key performance-limiting factor in wireless networks. Due to the many unknown parts in a large network (transceiver locations, activity patterns, transmit power levels, fading), it is naturally modeled as a random variable, and the (only) theoretical tool to characterize its distribution is stochastic geometry. Accordingly, many stochastic geometry-based works focus on interference…

## Path loss point processes

Naturally the locations of wireless transceivers are modeled as a point process on the plane or perhaps in the three-dimensional space. However, key quantities that determine the performance of a network do not directly nor exclusively depend on the locations but on the received powers. For instance, a typical SIR expression (at the origin) looks…

## The transdimensional approach

In vehicular networks, transceivers are inherently confined to a subset of the two-dimensional Euclidean space. This subset is the street system where cars are allowed to move. Accordingly, stochastic geometry models for vehicular networks usually consist of two components: A set of streets and a set of point processes, one for each street, representing the…

## To be connected or not to be

These days, “connectivity” is a very popular term in wireless networking. Related to 5G, typical statements include “5G will be the main driver of wireless connectivity.””5G is designed to provide more connectivity.””5G provides 1 million connected devices per square km.” There is also talk about “massive connectivity”, “poor connectivity”, “intermittent connectivity”, “high-speed connectivity”, “dense connectivity”,…

## Wishful thinking

Today we are listening in to a conversation between Achill and the Turtle. Achill: I have been conducting research on the performance of wireless links for a while now, and I learnt that analyzing a fixed deterministic channel does not lead to insightful and general results. To capture a variety of channel conditions and obtain…