# How is the efficiency of Pure ALOHA derived using Poisson's Distribution?

Pure ALOHA has a vulnerable time of `2T` where T is the transmission time for a single frame. Pure aloha has a maximum throughput of `18%` but how is it actually derived from Poisson's distribution?

Pure ALOHA is a random access protocol. It controls the multiple access over a shared medium. In this, the stations are allowed to transmit data when available.

In this there's a central controller station that governs the communication. Whenever a station wants to communicate data with another station, it first sends it to the central controller station, which then broadcasts it to the other stations.

In Pure ALOHA a station sends a frame and waits for an `ACK` from the receiver to proceed with the sending of the next frames. If another station transmits a frame during the same period then collision occurs. If the sender doesn't receive `ACK` signal then it re-transmits the frame/data using the Back-off Algorithm.

To avoid collision, no other station must transmit data during the same period.

##### Consider the following example: Let a frame be transmitted at time `t0` and `t` be the time required for its transmission. If, any other station sends a frame between `t0` and `t0+t` then the end of the frame will collide with that earlier sent frame. Similarly, if any other station transmits a frame between the time interval `t0+t` and `t0+2t` again, it will result in a garbage frame due to collision with the reference frame. Hence, 2t is the vulnerable interval for the frame.

##### Maximum Throughput / Efficiency Derivation:

Let `S` be the maximum throughput of the channel and `G` be the rate of arrival of new frames or the number of stations that wish to transmit the data.

The throughput of the system `S` is equal to total arrival rate `G` times the probability of successful transmission with no collision.

Therefore, `S = G x P(0)`, where `P(0)` is the probability of zero frames being transmitted during the vulnerable period, `2t`.

###### According to Poisson's Distribution formula:

Probability of `k` frames transmitted in time `t` is given by
`P(k) = G^k x e^(-G) / k!, k=0,1,2,3`

###### For pure ALOHA,

`P[k=0, during vulnerable time 2t] = (2G)^0 x e ^(-2G) / 0! = e^(-2G)`, where `k=0`

Substituting the above in the equation `S = G x P(0)`

`S = G x e^(-2G)`

From the figure attached above, it can be seen that `S` will be at peak only when `G = 1/2`

Thus,
S = 1/2 x e^(-2*1/2)
S = 1/2 x e^(-1)
S = 1/(2 x e)
S = 1/2 x 2.718, where `e=2.718`
S = 0.183958793

Hence through Poisson's Distribution we have derived that the maximum efficiency/throughput of Pure ALOHA protocol is approximately `18%`

##### Graph below shows the relationship between S and G. 