JPL's Wireless Communication Reference Website

Chapter: Data Networks
Section: Random Access, ALOHA, Throughput


Uniformly Offered Traffic in Mobile ALOHA Networks

If user terminals are uniformly spread over an infinitely large area, and if the message arrive rate per unit area is a constant G_0, the total arrival rate G is unbounded. In an ALOHA system without capture, the throughput would become zero. However, if capture occurs, data packets from terminals sufficiently close to the receiver are likely to be successful anyhow.

Capture Probability versus Range


Figure: The maximum acceptable offered traffic load (in message attempts per time slot per unit area) for a certain required probability of success at unity distance. Various receiver thresholds.

Total Throughput


Figure: Total throughput (in message attempts per time slot per unit area) versus signal to noise ratio at unity distance.


Figure: Total throughput (in message attempts per time slot per unit area) versus path loss law ("20 log d" to "50 log d").

ALOHA Net with Uniformly Distributed Users in Free Space Loss

In the above Figure, we see that for free space loss ("20 log d" or beta = 2), the throughput of the ALOHA network reduces to zero. The explanation is that the total interference power almost surely becomes infinite in any time slot. The signals from users on a ring at distance r are attenuated proportionally to r^-2. The number of users on the ring is proportional to r. So the total interference power is proportional to 1/r. If we assume that users are present in an infinitely extended area, the total interference power is proportional to the integral from 0 to infinity over 1/r, which is known to diverge.

Hence, every packet sees an infinitely large interference power, so it has zero probability to capture the receiver.

Why it gets dark at night

For centuries, astronomers have wondered why it gets dark at night. Assuming that
  • the universe were infinitely large,
  • the universe existed already for an infinite period of time, and
  • the density of stars were uniform,
then one can conclude that the amount of light seen from the stars would be tremendous. In every direction, we would see a star shining as bright as our sun.

This can be understood as the intensity of the light from any star attenuates according to free space loss, i.e., inversely proportional to the square of distance, while the angular surface area at which we observe the star also decreases with distance squared. As a result, the amount of light per steradian remains independent of the distance of the star. Whether we look at the sun or at any remote star, the intensity of light would be equal. Now under the conditions mentioned, in any direction we see a star (almost surely, i.e., with probability one).



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