2. Model Description
Setting:
Assume a large scale wireless multihop network:
- a large number of nodes, and
- a typical path between two nodes consists of large number of hops.
The network is said to be dense
when the number of nodes tends to infinity, n →∞.
In this case it is impractical to keep book of each node or
path individually.
Modelling approach: (continuum approach)
- Nodes exist everywhere as a continuous medium with certain density,
ρ(r) (measured in nodes/m²).
- Rate of packets exchanged by two locations is defined by
traffic demand density, λ(r,x)
(measured in pkts/s/m²/m²).
- Path between two locations is described by a continuous curve
p(r,x).
Objective: minmax traffic load Φ(r)
- Find such paths that minimize the maximum traffic load in the network.
- Traffic load in this setting can be defined as scalar packet flux,
Φ(r). (in analogy, e.g., with neutron transport theory).
- Scalar packet flux Φ(r)
corresponds to the arrival rate of packets into
a small d-disk about r multiplied by 2d,
λ′(r,x) = 2d ⋅ Φ(r)
when d → 0
⇒
Φ(r) represents the spatial forwarding load.
In the applet we assume uniform traffic demands in given area, e.g.,
for unit disk λ(r,x) = Λ / π².
Note that when the task is to minimize the scalar packet flux,
i.e., "traffic load per unit area",
the node density ρ(r) just needs to be high enough to
ensure full connectivity and
that neighbouring nodes exist in every direction (dense network).
However, it is worth noting that several other relevant problem formulations
have been proposed.
For example, the task may be to maximize the expected life time of a sensor network,
or to deploy a given number of nodes into a certain area in order to maximize some
quantity such as the expected rate of information.
In such cases the optimal solution depends also on the node density.
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