# Why analysis of Aloha protocol uses Poisson distribution?

Pretty much in all of the analysis of the Aloha protocol that I read, it is assumed that the distribution of packet arrivals is Poisson. What is the rationale behind it? Isn't it actually binomial distribution that is approximated by Poisson because $n$ (number of users) is large and $p$ (transmission probability) is close to zero?

IMO, the biggest reason is that Poisson distribution is more tractable mathematically. Some experimental studies claim that it does follow Poisson. However, there are also other experimental studies show that it does not.

Mathematically, Poisson distribution can be viewed as the approximation of binomial distribution under certain conditions. However, it is not the rationale behind the Aloha protocol.

As a mathematical model, you should consider your own scenario carefully and decide whether to use it or not.

• Why "However, it is not the rationale behind the Aloha protocol"? Each of $n$ users transmit with probability $p$. So, the probability of $k$ transmission (=$k$ success) follows binomial distribution, wrong? But, I agree that Poisson is more mathematically tractable. – Helium Dec 18 '14 at 1:58
• @Mohsen Each of $n$ users transmit with probability $p$: this is also a model. You are seeking for the rationale behind the Aloha protocol. Now you have to seek for the rational behind the probability $p$. – hengxin Dec 18 '14 at 2:04
• Well, the rational is simple: we designed the protocol to be so. I mean, that's by definition of the Aloha protocol. – Helium Dec 18 '14 at 2:09

Usually, the Poisson distribution is chosen because the process is assumed to be memoryless.

Suppose that a packet just arrived. The arrival time for the next packet has a certain distribution of probability. Now, suppose that after observing the system for some time interval we saw no new packet. In some cases, it is reasonable to assume that this observation did not change the probability distribution at all: that is, observing no packet for a while does not make new packet more likely or less likely to arrive soon. This is called the memoryless property.

Under these assumptions, one can prove that the arrival time for the next packet must be distributed according to the Exponential distribution. Further, if we fix a time interval, the number of packets which will arrive within the interval is distributed accoring to the Poisson distribution.