In a slotted ALOHA with G-stations attempting to transmit in one slot, the…

In slotted ALOHA, a station may start a transmission only at the beginning of a fixed-length time slot, so every attempt in a given slot is synchronized with every other attempt. If stations together offer G transmission attempts per slot on average, the number of attempts landing in any one slot follows a Poisson distribution with mean G: P(exactly k attempts in a slot) = (Gᵏ × e⁻ᴳ) / k!. Throughput is the probability that a slot carries exactly one attempt (k = 1), since zero attempts wastes the slot and two or more attempts collide and destroy each other.

Applying the Poisson formula for a successful (collision-free) slot:

1. Write the general Poisson probability of exactly k attempts in a slot: P(k) = (Gᵏ × e⁻ᴳ) / k!.

2. A slot is successful only when exactly one attempt lands in it, so set k = 1: P(1) = (G¹ × e⁻ᴳ) / 1!.

3. Simplify the factorial and the exponent: P(1) = G × e⁻ᴳ.

4. Throughput S is this success probability, so S = G × e⁻ᴳ.

As an independent check, differentiate S with respect to G and set the result to zero: this locates the maximum at G = 1, giving S = 1 × e⁻¹ ≈ 0.368. That matches the well-known textbook result that slotted ALOHA peaks at about 36.8% throughput — exactly double pure ALOHA's peak of about 18.4% — which confirms the formula.

So the throughput of slotted ALOHA is S = G × e⁻ᴳ.