# deriving an equation for wireless network capacity

I'm trying to derive an equation for wireless network capacity given topological constraints of the nodes and certain obstacles... but before I get that far, I want to understand the most simple case. That is, randomly distributed nodes communicating to random partners. The classic paper for this seems to be Gupta & Kumar. I understand the concepts in the paper, but unfortunately I get lost fairly quickly in their equations.

I understand that a message will be received at $$\j\$$ if the power from the transmitting node $$\i\$$ is greater than the power from all other nodes $$\k\$$ (+ noise $$\N\$$) by some margin $$\\beta\$$. That is equation 2 in the paper (and equation 5).

$$\ \frac{\frac{P_i}{|X_i-X_j|^\alpha}}{N + \sum_{k \epsilon T}\frac{P_k}{|X_k-X_j|^\alpha}} \ge \beta \$$

But I can't see how to get from there to the transport capacity. I guess it will probably depend on assumptions for

• the transmitting node's choice of intended receiver node
• where the other nodes are (their distribution in space)
• what the other nodes are doing at the time (how many nodes are interfering)

Can someone point me in the direction of a derivation which goes a bit more step-by-step than Gupta? Or if you can provide your own derivation that would be exceptional!