Stochastic geometry models of wireless networks
[10][11][12] Key performance and quality of service quantities are often based on concepts from information theory such as the signal-to-interference-plus-noise ratio, which forms the mathematical basis for defining network connectivity and coverage.
The use of stochastic geometry can then allow for the derivation of closed-form or semi-closed-form expressions for these quantities without resorting to simulation methods or (possibly intractable or inaccurate) deterministic models.
The broadcast channel, in information theory terminology,[13] is the one-to-many situation with a single transmitter aiming at sending different data to different receivers and it arises in, for example, the downlink of a cellular network.
In wired communication, the field of information theory (in particular, the Shannon–Hartley theorem) motivates the need for studying the signal-to-noise ratio (SNR).
In general, the use of methods from the theories of probability and stochastic processes in communication systems has a long and interwoven history stretching back over a century to the pioneering teletraffic work of Agner Erlang.
[6] Starting in the late 1970s, Leonard Kleinrock and others used wireless models based on Poisson processes to study packet forward networks.
Around the early 1990s, shot noise based on a Poisson process and a power-law repulse function was studied and observed to have a stable distribution.
[28] Independently, researchers[19][29] successfully developed Fourier and Laplace transform techniques for the interference experienced by a user in a wireless network in which the locations of the (interfering) nodes or transmitters are positioned according to a Poisson process.
[1][2][30] Moreover, the assumption of the received (i.e. useful) signal power being exponentially distributed (for example, due to Rayleigh fading) and the Poisson shot noise (for which the Laplace is known) allows for explicit closed-form expression for the coverage probability based on the SINR.
[19][31] This observation helps to explain why the Rayleigh fading assumption is frequently made when constructing stochastic geometry models.
The fading models in use include Rayleigh (implying exponential random variables for the power), log-normal, Rice, and Nakagami distributions.
[10] The Poisson process in general is commonly used as a mathematical model across numerous disciplines due to its highly tractable and well-studied nature.
Despite its simplifying nature, the independence property of the Poisson process has been criticized for not realistically representing the configuration of deployed networks.
Strong correlations also arise in the case of cognitive radio networks where secondary transmitters are only allowed to transmit if they are far from primary receivers.
Although models based on these and other point processes come closer to resembling reality in some situations, for example in the configuration of cellular base stations,[34][40] they often suffer from a loss of tractability while the Poisson process greatly simplifies the mathematics and techniques, explaining its continued use for developing stochastic geometry models of wireless networks.
Building off previous work done on an Aloha model,[44] the coverage probability for the typical user was derived for a Poisson network.
In other words, given a path loss function, using a Poisson cellular network model with constant shadowing is equivalent (in terms of SIR, SINR, etc.)
to assuming sufficiently large and independent fading or shadowing in the mathematical model with the base stations positioned according to either a deterministic or random configuration with a constant density.
This is in particular used to cope with the difficulty of covering with macro-base stations only open outdoor environment, office buildings, homes, and underground areas.
[22] Then the Laplace transform for this superimposed Poisson model is calculated, leading to the coverage probability in (the downlink channel) of a cellular network with multiple tiers when a user is connected to the instantaneously strongest base station[54] and when a user is connected to the strongest base station on average (not including small scale fading).
ALOHA is not only one of the simplest and most classic MAC protocol but also was shown to achieve Nash equilibria when interpreted as a power control schemes.
[71] Several early stochastic models of wireless networks were based on Poisson point processes with the aim of studying the performance of slotted Aloha.
[7][72][73] Under Rayleigh fading and the power-law path-loss function, outage (or equivalently, coverage) probability expressions were derived by treating the interference term as a shot noise and using Laplace transforms models,[19][74] which was later extended to a general path-loss function,[31][44][75] and then further extended to a pure or non-slotted Aloha case.
Stochastic geometry models based on this type of representation were developed to analyze the coverage areas of transmitters positioned according to a Poisson process.
In recent years, models have been developed to study more elaborate channels arising from the discipline of network information theory.
[77] More specifically, a model was developed for one of the simplest settings: a collection of transmitter-receiver pairs represented as a Poisson point process.
These codes, consisting of randomly and independently generated codewords, give transmitters-receivers permission when to exchange information, thus acting as a MAC protocol.
[78] It was also shown[77] that when using the point-to-point codes and simultaneous decoding, the statistical gain obtained over a Poisson configuration is arbitrarily large compared to the scenario where interference is treated as noise.
For further reading of stochastic geometry wireless network models, see the textbook by Haenggi,[4] the two-volume text by Baccelli and Błaszczyszyn[1][2] (available online), and the survey article.
For an introduction to stochastic geometry and spatial statistics in a more general setting, see the lectures notes by Baddeley[21] (available online with Springer subscription).
