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Wireless signals appear Poisson
Under the standard statistical propagation model, researchers have mathematically shown that wireless networks can appear (in terms of received signal powers) to any single observer as an inhomogeneous Poisson point process on the real line, provided there are sufficient random propagation effects, such as multi-path fading or shadowing. In other words, the signal powers form a point process on the positive real line, which is statistically close to a Poisson point process. This has been shown for a quite general propagation model with independent propagation effects and, more recently, correlated shadowing. These results imply, through the Poisson mapping theorem, that network transmitter layouts that do not appear Poisson, such as lattices or other configurations, can be modelled as realizations of Poisson processes, as a Poisson network with matching intensity measure would produce the same inhomogeneous Poisson point process on the positive real line.
The source of randomness that makes a non-Poisson network behave more Poisson is the random propagation effects of fading, shadowing, randomly varying antenna gains, and so on, or some combination of these. These Poisson results were originally derived for independent log-normal shadowing, but then they were greatly extended so they still hold true under a propagation model with a general path loss function and (sufficiently large and independent) propagation effects, such as Rayleigh or Nakagami fading, and this behavior is more likely for the stronger signals. In the case of log-normal shadowing and a power law path loss function, this Poisson behaviour still holds true if there is correlation between the random propagation effects. More specifically, signals can still appear Poisson when the propagation effects are modelled with a correlated Gaussian field.
For mobile (or cellular) phone networks and other wireless network, a very popular stochastic geometry model consists of base stations (that is, transmitters) as a Poisson process, a simple power law as the path loss function, and iid random variables as the random propagation effects such as fading and shadowing. Borrowing an expression from physics, this Poisson model could be called the "standard model" due to its popular use. Researchers have used it to derive closed-form expressions, sometimes with surprisingly simple forms, for the probability distributions of the signal-to-interference ratio (SIR) in the downlink, which gives the probability of a user being covered in the network. Under various models, these coverage probability expressions have been derived, where the model assumptions depend on considerations such as whether a user connects to the closest base station or the base station with the strongest signal.
My colleagues and I derived an expression for the probability that a user can connect to k base stations in a single-tier Poisson network. Using point process techniques, we then extended these k-coverage results to the case of multi-tier Poisson network models, which are often used to model heterogeneous networks. Our results can also be used to calculate the coverage probabilities under certain signal management schemes, such as successive interference cancellation, so generalizing previous results based stochastic on geometry.
I originally studied physics and electronic engineering before changing over to applied mathematics and completing an honours thesis in fluid mechanics. I completed my PhD in applied mathematics at the University of Melbourne under the supervision of Peter G. Taylor whose detailed websites can be found here and here. I then spent two years, partly being funded by Orange Labs, in Paris as a post-doc in the Inria (formerly INRIA) group DYOGENE * headed by Marc Lelarge (the group was known as TREC, headed by François Baccelli, during my first few months there). Then I became a member of the Weierstrass Institute (or WIAS), Berlin, in a group headed by Wolfgang König.
* DYOGENE is actually a reference to ancient Greek philosopher Diogenes of Sinope (who famously lived in a giant jar) as his name in French is pronounced the same as DYOGENE, all of which had to be explained to me.
** ł is not a Latin "l", but a Slavic letter, and it's pronounced more like a "w". The "z" also plays a different role, so in English the name "Błaszczyszyn" sounds more like "Bwasht-chy-shyn" with a long "a" sound.