# [e2e] Collaboration on Future Internet Architectures

Fri May 4 15:20:00 PDT 2007

```People who claim that increasing the density of raido nodes will increase
the per-node bandwidth (or at least leave it unchanged) are simply not
good with arithmetic.

Let's look at a plane with some distribution of radio nodes on it, with
per-node characteristic bandwidth B, achieved at signal/noise ration SN0.
Let's now scale down that plane by reducing all distances by factor F.

Node-to-node signal changes as square of distance, i.e. we can get the
same power at receiver by transmitting at 1/F^2 of the original power.

The total noise from interference from other transmitters is proportional
to the density of transmitters times their power - density increases as
F^2, power of each transmitter is adjused by 1/F^2, as above.

Thus by scaling distances down and reducing the power of transmitters
accordingly, we end up with the same S/N, and the same bandwidth per node.
Which means that we get same effective bandwidth _for the same average
number of hops_ (Beff = B/Nhops).

But each hop is now shorter; so a packet has to make F times more hops.
This makes effective bandwidth from two fixed points (same between the
original and scaled network) to decrease: BEFFscaled = BEFForig/F.

Note that this result does not depend on directionality of antennae,
method of sharing the medium capacity, quality of receivers, etc.  Just
increasing the number of nodes in a multihop system HAS to decrease the
effective bandwidth as square root of the number of nodes.

The way to achieve the claimed scaling is to have an overlay short-cut
non-interfering network (i.e. fiber-optic, etc) with a number of
interconnection points proportional to the number of radio nodes.

(Another option is to exploit smaller scale in order to make use of
higher-frequency (i.e. free-space optical) interconnects unfeasible at
longer distances).

On Thu, 3 May 2007, Christian Huitema wrote:

> > 1/ some people have claimed that one can build many-to-many multihop
> > systems that offer more capacity as the number of nodes join.
>
> Some have claimed it, but it is far from being proved. The number of
> hops tend to increase with the number of nodes -- typically scaling as
> the square root of that number if the nodes are arranged in a plane. The
> available bandwidth per node tends thus to decrease with the number of
> nodes.
>
> -- Christian Huitema
>
>
>
>

```