# Throughput of Aloha Networks

Published by Arun Isaac on

Tags: math

A derivation of the mathematical expression for the throughput of an aloha network using the exponential and Poisson random distributions

Throughput is defined as the average number of successful transmissions. It is a measure of how many transmitted frames successfully reach the receiver without being damaged by collision. The mathematical estimate of the throughput of aloha networks is obtained from the exponential probability distribution. A quick recap of exponential and Poisson distributions:

**Exponential and Poisson Distributions**

Exponential distributions and Poisson distributions are intricately related.

The Poisson distribution measures the number of occurences of a random event in unit time, say, the number of buses that arrive at a bus stop in an hour. Conversely, the exponential distribution measures the amount of time before the first occurence of the random event, that is, the number of hours before the arrival of the first bus.

Let λ be the average number of occurences of the random event in unit time.

Then, the Poisson distribution defines the probability of n occurences of the random event in unit time to be

\[ P(n) = \frac{\lambda^n e^{-\lambda}}{n!}, \, n \gt 0 \]

The exponential distribution defines the probability of the first occurence of the random event in time t to be

\[ F(t) = 1 - e^{-\lambda t} \]

**Pure Aloha Networks**

Modelling the transmission of frames in an aloha network by an exponential distribution, we have the probability of transmission of a frame to be

\[ F(t) = 1 - e^{-Gt} \]

where G is the average number of frames generated by the system during one frame transmission time.

In a pure aloha network, \( 2T_{fr} \) is the time during which a frame is vulnerable to collision from another frame, where \( T_{fr} \) is the time taken for transmission of one frame.

Therefore, the probability of collision is

\[ P(\text{collision}) = 1 - e^{-2G} \]

And the probability of a frame being successfully transmitted without collision is

\[ P(\text{no collision}) = e^{-2G} \]

This gives the percentage of frames that will be successfully transmitted. Therefore, on an average, the expected number of frames that would be successfully transmitted is

\[ S = G e^{-2G} \]

This gives the throughput of a pure aloha network.

**Slotted Aloha Networks**

In a slotted aloha network, a frame is vulnerable to collision with another frame only for one frame transmission time. Therefore, the probability of collision is

\[ P(\text{collision}) = 1 - e^{-G} \]

Following from the same logic used in the case of a pure aloha network, we have the throughput of a slotted aloha network to be

\[ S = G e^{-G} \]