ALOHA: throughput and delay

Notation: T = frame time, G = average number of frame transmissions per frame time (offered load, including retransmissions), S = throughput (successful frames per frame time). Frame arrivals are assumed to follow a Poisson process.

1. Pure ALOHA

Stations transmit whenever they have data. A frame sent at time t₀ is successful only if no other frame starts in the vulnerable period [t₀ − T, t₀ + T], which is 2T long.

• Probability of no other transmission in 2T: P₀ = e^(−2G)
• Throughput: S = G · e^(−2G)
• Maximum: dS/dG = 0 gives G = 0.5, so S_max = 1/(2e) ≈ 0.184 (18.4%)

Delay: the probability of a frame succeeding on one try is e^(−2G), so the average number of transmissions per frame is e^(2G), i.e. e^(2G) − 1 retransmissions. If each failed attempt costs one frame time plus propagation time Tₚ plus an average back-off time B:

D_pure ≈ T + Tₚ + (e^(2G) − 1)(T + Tₚ + B)

Delay grows exponentially with load G.

2. Slotted ALOHA

Time is divided into slots of length T. A station may start transmission only at the beginning of a slot. Two frames either collide completely or do not collide at all. The vulnerable period shrinks to T.

• Probability of no other transmission in one slot: P₀ = e^(−G)
• Throughput: S = G · e^(−G)
• Maximum: G = 1, so S_max = 1/e ≈ 0.368 (36.8%)

Delay: average transmissions per frame = e^G, so retransmissions = e^G − 1. A frame also waits on average T/2 for the next slot boundary:

D_slotted ≈ T/2 + T + Tₚ + (e^G − 1)(T + Tₚ + B)

Comparison

Basis	Pure ALOHA	Slotted ALOHA
Time	Continuous	Discrete slots
Synchronisation	Not required	Required (common clock)
Vulnerable period	2T	T
Throughput	S = G·e^(−2G)	S = G·e^(−G)
Maximum throughput	0.184 at G = 0.5	0.368 at G = 1
Collision probability	Higher (partial overlaps possible)	Lower (only full overlaps)
Delay at high load	Grows as e^(2G)	Grows as e^G; lower
Delay at very low load	Lower (send immediately)	Slightly higher (wait for slot start)
Complexity	Simple	Needs slot synchronisation

Conclusion: slotted ALOHA doubles the maximum throughput of pure ALOHA and has much better delay under load, at the cost of clock synchronisation.