ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 3, pp. 193–203.
⃝ Pleiades Publishing, Inc., 2009.
Original Russian Text
⃝ P. Jacquet, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 3, pp. 3–14.
Shannon Capacity in Poisson
Wireless Network Model
INRIA Paris-Rocquencourt, France
Received January 11, 2009
Abstract—We consider a realistic model of a wireless network where nodes are dispatched in
an inﬁnite map with uniform distribution.Signals decay with distance according to attenuation
factor .At any time we assume that the distribution of emitters is per square unit area.
From an explicit formula of the Laplace transform of a received signal, we derive an explicit
formula for the information rate received by an access point at a random position, which is
per Hertz.We generalize to network maps of any dimension.
Wireless networks are being extensively deployed in densely populated urban or semi-urban
areas.The question is how the wireless networks can ﬁt the increasing demand in capacity that
is expected in the future.In  it is shown that the information rate received per node is ﬁnite
regardless of the network density.However, this is done under the restrictive hypothesis that a node
can only receive from a single neighbor node at a time.The problem is that a correct assessment
of the capacity of a wireless network in its most general setting is at the crossing of
Physics, for wave propagation and attenuation in medium;
Geometry, for positioning of the nodes;
Information theory, for extraction of information from a signal.
This paper addresses analytical evaluation of the wireless network capacity in a realistic model,
which is surprisingly tractable.This model involves the three aspects mentioned above.Attenuation
is a function of the distance of the form
and of random fadings; nodes are randomly dispatched
and have any given nominal power; signals superpose; information is extracted in parallel ﬂows as in
the multiple-input-multiple-output (MIMO) technology; nodes transmit independent information
ﬂows.The model was primarily introduced in  and then partially developed in [3, 4].
We assume a central receiver node (e.g., an access point), to which all nodes in the network
transmit data.We assume that information ﬂows are independent, so that every ﬂow is impacted
by other ﬂows as if they were pure Gaussian noises.This allows us to make use of the celebrated
Shannon formula  for the wireless capacity in presence of Gaussian noise.Our main ﬁnding is
that in the absence of noisy sources and in the presence of any fading and any nominal power
distribution, the average information rate in bit per Hertz received by the access points is exactly
where is the attenuation coeﬃcient and is the dimension of the network map (for instance,
= 2 for a planar map).The formula is remarkable in the sense that it is simple and collects the