Question

Consider Aloha protocol and assume that in addition to collisions a message may also experience bit errors due to noise. Message lengths are L bits and with probability p each bit may be received in error by the destination. Thus a message will be successfully received if it doesnot experience any collisions and it is received error free. Derive the throughput equation of this protocol.

Answer 1

ALOHA :

In ALOHA network one station will work as the central controller and the other station will be connected to the central station. Here, no station is contrained , any station that has data/ frame can transmit at any time.

Throughput: Throughput is defined as the average number of successful transmissions. It is a measure of how many transmitted frames successfully reach the reciever without being damaged by the collision. Throughput of ALOHA protocol can be determine by the Poisson Distribution.

Let R be the bit rate of the transmission channel and L be the length of the frame/message. Here, consider the size of the message be constant , hence it will take constant time t = L/R for transmission of each packet.

As in case of ALOHA protocol message can be sent at any time so, the probability of collision will be very high. In order to prevent a message from colliding , no other frame should be sent within its transmission time. Now, let a message transmitted at time t0 and t be the time required for its transmission. If , any other station sends a message between t0  and t0+t then the collision will occurr. Similarly , if any other station transmits a frame between the time interval   t0+t and  t0+2t , it will result in garbage frame/ message due to collision. Hence, 2t is the vulnerable time interval for the message. So, for the probability of successful transmission (Throughput), no additional message should be transmitted in the vulnerable interval 2t.

Let S be the arrival rate of new message per message time. As we find probability of no collision , S represent the troughput of the system. Let G be the total arrival rate of messages including transmission frames. The probablity of k message ,of length L , transmission in 2t seconds is given by Poission distribution as follows:

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

S = G * P   (zero frame transmission in the vulnerable interval i.e. 2t seconds)

Since , P[K frames in vulnerable interval 2t] = (2G) e-2G  / K! , K=0,1,2,3..........

Thus, P[K=0 in 2t] = -2G

Hence, S = G * P

S = G * e-2G ---------(1)

The eqaution (1) is the required Throughput equation for the ALOHA protocol.

As G is increasing , S is also increasing for small values of G. At G =1/2 , S attains its peak value i.e. S= 1/2e i.e. S= 0.18 (approx) , After that , it starts decreasing for increasing values of G. Here the average number of successful transmission attempts/ frames can be given as G/S= e2G .