Final answer:
The throughput of a Slotted Aloha network with three active stations A, B, and C, having probabilities pa = 0.2, pb = 0.3, and pc = 0.4, is calculated by summing the probabilities of each station transmitting successfully. The throughput is approximately 0.332.
Step-by-step explanation:
To calculate the throughput of the network using Slotted Aloha, we need to consider the probabilities of each station (A, B, and C) successfully transmitting a frame without a collision. The throughput is the product of the probability that a given station will attempt to transmit in a slot and the probability that none of the other stations will attempt to transmit in the same slot. Given the probabilities, pa = 0.2, pb = 0.3, and pc = 0.4 for stations A, B, and C respectively, we can calculate the throughput as follows:
- First, calculate the probability that no station transmits: P(no transmission) = (1 - pa) * (1 - pb) * (1 - pc)
- Next, calculate the probability that exactly one station transmits, which leads to a successful transmission.
- So we have three cases for a successful transmission:
- Now, add up the probabilities from these three scenarios to find the total probability of a successful transfer: P(success) = pa * (1 - pb) * (1 - pc) + pb * (1 - pa) * (1 - pc) + pc * (1 - pa) * (1 - pb)
- The throughput of the Slotted Aloha network is then equal to P(success).
When you plug in the given probabilities into the formula, you get P(success) = 0.2 * (1 - 0.3) * (1 - 0.4) + 0.3 * (1 - 0.2) * (1 - 0.4) + 0.4 * (1 - 0.2) * (1 - 0.3). Doing the calculations, P(success), and hence the throughput, is approximately 0.332.