The Math of Collisions: Pure ALOHA vs Slotted ALOHA Efficiency

In shared medium networks, multiple autonomous stations compete for access to a single communication channel without a central scheduler.

Developed at the University of Hawaii in 1970, Pure ALOHA and Slotted ALOHA are the foundational random access protocols that established the mathematical framework for modern Ethernet and Wi-Fi.

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1. Pure ALOHA: S = G · e^-2G

In Pure ALOHA, stations transmit frames whenever data is generated. If a frame collides with another transmission—even by a single bit—the entire frame is destroyed.

The Vulnerability Period (2 · T_f):

Let T_f be the frame transmission time. Suppose station A starts transmitting at time t₀:

• If station B began transmitting between t₀ - T_f and t₀, its end overlaps with A's start.
• If station C begins transmitting between t₀ and t₀ + T_f, its start overlaps with A's end.

The Vulnerability Window is 2 · T_f.

Assuming frame generation follows a Poisson distribution with attempt rate G frames per slot time:

P(k frames in 2T_f) = frac(2G)^k · e^-2Gk!

The probability of a successful transmission (k = 0 collisions) is e^-2G.
Throughput S is:

S = G · e^-2G

Differentiating (dS / dG) = e^-2G(1 - 2G) = 0 implies G = 0.5.

S_max = (1 / 2e) ≈ 18.4\%

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2. Slotted ALOHA: S = G · e^-G

Slotted ALOHA introduces a centralized clock dividing time into discrete slots of length T_f. Stations can only begin transmission at the beginning of a slot boundary.

Halving the Vulnerability Window:

Because all transmissions start simultaneously:

• A frame either collides completely or passes completely.
• The vulnerability period shrinks from 2 · T_f down to T_f.

P(0 frames in T_f) = e^-G

S = G · e^-G

Differentiating (dS / dG) = 0 implies G = 1.0.

S_max = (1 / e) ≈ 36.8\%

Synchronizing transmissions doubles the channel capacity from 18.4\% to 36.8\%.

Explore complete protocol derivations and framing simulations at Binary Dose.