A slotted ALOHA network transmits 200 bits frames using a shared channel with…
2016
A slotted ALOHA network transmits 200 bits frames using a shared channel with 200 kbps bandwidth. If the system (all stations put together) produces 1000 frames per second, then the throughput of the system is _______.
Answer: C. 0.368 — ConceptIn slotted ALOHA, time is divided into frame-length slots. If the aggregate offered load is G = λT, the normalized throughput is S = G·e−G successful…
- A.
0.268
- B.
0.468
- C.
0.368
- D.
0.568
Attempted by 656 students.
Show answer & explanation
Correct answer: C
Concept
In slotted ALOHA, time is divided into frame-length slots. If the aggregate offered load is G = λT, the normalized throughput is S = G·e−G successful frames per slot.
Here T = L/R is the frame-transmission time, where L is the frame length and R is the channel bit rate. The expression peaks at G = 1, where S = 1/e.
Application
Compute one slot duration: T = L/R = 200 bits ÷ 200,000 bits/s = 0.001 s.
Convert the slot duration to slots per second: 1/T = 1/0.001 = 1000 slots/s.
Compute the offered load: G = 1000 frames/s ÷ 1000 slots/s = 1 frame/slot.
Apply the throughput formula: S = G·e−G = 1·e−1 = 1/e ≈ 0.367879.
Cross-check
For f(G) = G·e−G, the derivative is f′(G) = e−G(1 − G), so the maximum occurs at G = 1. The computed load is exactly this maximizing value, giving the known slotted-ALOHA maximum 1/e.
Result
The normalized throughput is 0.368 (approximately 368 successful frames per second).