In the previous problem, we showed that the throughput is S = Ge^−2G for a pure Aloha network and S = Ge^−G for a slotted Aloha network. In this problem, we want to find the value of G in each network that makes the throughput maximum and find the value of the maximum throughput. This can be done if we find the derivative of S with respect to G and set the derivative to zero.

a. Find the value of G that makes the throughput maximum, and find the value of the maximum throughput for a pure Aloha network.

b. Find the value of G that makes the throughput maximum, and find the value of the maximum throughput for a slotted Aloha network.  

Previous Problem

To formulate the performance of a multiple-access network, we need a mathematical model. When the number of stations in a network is very large, the Poisson distribution, p[x] = (e^−λ × λ^x)/(x!), is used. In this formula, p[x] is the probability of generating x number of frames in a period of time and λ is the average number of generated frames during the same period of time.