Given that lim_{N→∞} (1 − 1/N)^N = 1/e, derive the maximum efficiency of slotted ALOHA.
Try an answer before revealing the guidance below.
Key Concepts
- N active nodes
- Transmit probability p
- Success probability
- Maximising over p
- Limit as N → ∞
Answer Approach
- Write the probability that one given node succeeds in a slot.
- Multiply by N for the probability that any node succeeds.
- Differentiate with respect to p to find the maximising p*.
- Substitute p* and take the limit as N grows, using the given identity.
Full Answer
Answer status: Draft answer (unofficial). Revision notes, not an official marking scheme.
With N nodes each transmitting in a slot with probability p, one given node succeeds with probability p(1 − p)^(N−1), so the efficiency (probability some node succeeds) is E(p) = Np(1 − p)^(N−1). Setting dE/dp = N(1 − p)^(N−2)[(1 − p) − (N − 1)p] = 0 gives p* = 1/N. Then E(p*) = (1 − 1/N)^(N−1) = (1 − 1/N)^N / (1 − 1/N). As N → ∞, (1 − 1/N)^N → 1/e and (1 − 1/N) → 1, so the maximum efficiency is 1/e ≈ 0.37.
E(p) = Np(1 − p)^(N−1)p* = 1/NE_max → 1/e ≈ 0.37Shortcuts: K concepts · A approach · F answer · R reviewed · B bookmark · ← / → previous / next