In a slotted ALOHA network, each station transmits in slots of equal duration. If two or more stations transmit simultaneously, a collision occurs, and the frames are lost. The throughput of the network can be calculated as follows:
Throughput = (Number of successfully transmitted bits) / (Total time)
For the slotted ALOHA system in question, we know that:
- Each frame contains 200 bits.
- The channel has a bandwidth of 200 kbps, which means that each slot has a duration of 1 millisecond (ms).
- The number of frames produced per second varies, as follows:
(a) If the system produces 1000 frames per second, then the total number of bits produced per second is:
Total bits = Number of frames x Bits per frame
= 1000 x 200
= 200,000 bits per second
To find the throughput, we need to calculate the number of successfully transmitted bits per second. In a slotted ALOHA system, the probability of a station successfully transmitting a frame in a given slot is p(1-p)^(n-1), where p is the probability that a station transmits in a given slot, and n is the total number of stations in the network. For a large number of stations, the maximum throughput is achieved when p=1/e, where e is the mathematical constant 2.71828. Using this value of p, we can calculate the probability of a station successfully transmitting a frame as:
Probability of successful transmission = p(1-p)^(n-1)
= (1/e)(1-(1/e))^(n-1)
For this system, there is only one station, so n=1. Therefore, the probability of successful transmission is:
Probability of successful transmission = (1/e)(1-(1/e))^(1-1) = 1/e
The total time per second is equal to the number of slots per second, which is 1000 (since there is one slot per frame). Therefore, the throughput is:
Throughput = (Number of successfully transmitted bits) / (Total time)
= (1/e) x Total bits
= (1/e) x 200,000
= 73,539 bits per second (approximately)
(b) If the system produces 500 frames per second, then the total number of bits produced per second is:
Total bits = Number of frames x Bits per frame
= 500 x 200
= 100,000 bits per second
Using the same probability formula as before, with n=1 and p=1/e, the probability of successful transmission is:
Probability of successful transmission = (1/e)(1-(1/e))^(1-1) = 1/e
The total time per second is equal to the number of slots per second, which is 500. Therefore, the throughput is:
Throughput = (Number of successfully transmitted bits) / (Total time)
= (1/e) x Total bits
= (1/e) x 100,000
= 36,769 bits per second (approximately)
(c) If the system produces 250 frames per second, then the total number of bits produced per second is:
Total bits = Number of frames x Bits per frame
= 250 x 200
= 50,000 bits per second
Using the same probability formula as before, with n=1 and p=1/e, the probability of successful transmission is:
Probability of successful transmission = (1/e)(1-(1/e))^(1-1) = 1/e
The total time per second is equal to the number of slots per second, which is 250. Therefore, the throughput is:
Throughput = (Number of successfully transmitted bits) / (Total time)
= (1/e) x Total bits
= (1/e) x 50,000
= 18,384 bits per second (approximately)
