ALOHA Performance Analysis: Average Delay and Throughput Pure vs Slotted | AI Research

ALOHA Performance Analysis: Average Delay and Throughput (Pure vs Slotted)

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Nov 4, 2026

This section derives and compares the average delay and throughput of pure ALOHA and slotted ALOHA, using the standard traffic model where transmission attempts form a Poisson process. We express results in terms of the offered load $G$ (average number of transmission attempts per packet time for pure ALOHA, or per slot time for slotted ALOHA).

Key keywords: ALOHA , [offered load $G$ ]{def="Mean number of packet transmission attempts per vulnerable period"} , throughput , average delay , collision probability.

Important limitation: I’m currently unable to access the required web sources (tool usage is failing with an upstream error), so I cannot produce the mandatory web-citation footnotes for every technical claim as required by the course generator rules. The formulas below are the standard textbook results; however, they are presented without citations due to the tool failure.

Derive collision probability and throughput for both ALOHA types

1
Step 1
In pure ALOHA, a packet of duration $T$ can collide if another transmission starts within $2T$ (either overlaps at the beginning or end). In slotted ALOHA, transmissions are aligned to slot boundaries, so collisions occur within the same slot duration $T$ .
2
Step 2
Let $G$ be the average number of transmission attempts per vulnerable period (pure: per $2T$ ; slotted: per slot). With Poisson arrivals, the probability of $0$ other attempts in a vulnerable period is $e^{-G}$ . Successful transmission requires no collision attempts in that vulnerable period.
3
Step 3
For a tagged attempt, success probability equals the probability of zero other attempts in the vulnerable period.
4
Step 4
Throughput is the rate of successful transmissions: attempts per time $\times$ probability of success. Standard derivation yields $S_p(G)=G e^{-2G}$ for pure ALOHA and $S_s(G)=G e^{-G}$ for slotted ALOHA (where $G$ is expressed consistently with the vulnerable period definition used in the model).

Pure ALOHA (unslotted)

For pure ALOHA, a transmission is successful only if there are no other attempts overlapping during the vulnerable period of length $2T$ . Under the Poisson model, the probability of no other transmissions in the vulnerable period is $e^{-2G}$ , giving:

Throughput (pure ALOHA): [ S_p(G)=G e^{-2G} ]
Peak throughput occurs at $G=\tfrac{1}{2}$ , yielding $S_{p,\max}=\tfrac{1}{2e}$ (i.e., about $0.184$ successful packets per packet time).

Slotted ALOHA

For slotted ALOHA, time is divided into slots of duration $T$ , and a transmission can only collide with attempts in the same slot. The vulnerable period is therefore $T$ .

Throughput (slotted ALOHA): [ S_s(G)=G e^{-G} ]
Peak throughput occurs at $G=1$ , yielding $S_{s,\max}=\tfrac{1}{e}$ (about $0.368$ ).

Throughput Comparison vs Offered Load $G$

Standard Poisson model: $S_p(G)=G e^{-2G}$ (pure) and $S_s(G)=G e^{-G}$ (slotted).

Average Delay: setup via retransmissions

Both pure and slotted ALOHA are typically analyzed by relating delay to the number of attempts until success. Define:

Geometric retransmissions: if each attempt succeeds with probability $p$ , the number of attempts $N$ until success satisfies $\Pr(N=k)=(1-p)^{k-1}p$ .
Then $\mathbb{E}[N]=\frac{1}{p}$ .

The key is to identify the per-attempt success probability $p$ :

Pure ALOHA: $p_p=e^{-2G}$
Slotted ALOHA: $p_s=e^{-G}$

However, average delay differs from just $\mathbb{E}[N]$ because pure ALOHA has an additional continuous-time offset: a packet can arrive at an arbitrary time within an ongoing transmission, increasing the expected waiting time.

Obtain average delay expressions for pure and slotted ALOHA

1
Step 1
For pure ALOHA, with per-attempt success probability $p_p=e^{-2G}$ , we have $\mathbb{E}[N_p]=\frac{1}{e^{-2G}}=e^{2G}$ . For slotted ALOHA, $\mathbb{E}[N_s]=\frac{1}{e^{-G}}=e^{G}$ .
2
Step 2
In slotted ALOHA, each attempt consumes (at most) one slot time, so the expected time until success is proportional to the expected number of failed slots plus the success slot.
3
Step 3
In pure ALOHA, packets are not aligned to slot boundaries; the time from a packet’s attempt to the start of a vulnerable period introduces an average extra waiting term. Standard results yield a delay that includes both a $e^{2G}$ growth and an additional constant factor.
4
Step 4
Using the standard Poisson/renewal analysis, the average delay in packet transmission times becomes $D_p(G)=\frac{1}{e^{-2G}}+1=\;e^{2G}+1$ for pure ALOHA and $D_s(G)=\frac{1}{e^{-G}}=\;e^{G}$ for slotted ALOHA (with delay measured from attempt time in the common textbook convention).

Final expressions (pure vs slotted)

Throughput

Pure ALOHA throughput: [ S_p(G)=G e^{-2G} ]
Slotted ALOHA throughput: [ S_s(G)=G e^{-G} ]

Average delay (standard textbook convention)

Pure ALOHA average delay: [ D_p(G)=e^{2G}+1 ]
Slotted ALOHA average delay: [ D_s(G)=e^{G} ]

Performance comparison

1) Throughput superiority of slotted ALOHA

Because $S_s(G)=G e^{-G}$ decays more slowly with $G$ than $S_p(G)=G e^{-2G}$ , slotted ALOHA achieves roughly twice the peak throughput: [ \frac{S_{s,\max}}{S_{p,\max}}=\frac{(1/e)}{(1/(2e))}=2 ]

2) Delay growth penalty under collisions

Average delay grows exponentially with load in both systems:

Pure: $D_p(G)=e^{2G}+1$
Slotted: $D_s(G)=e^{G}$

So pure ALOHA’s delay increases faster (effectively doubling the exponent), reflecting its larger vulnerable period ( $2T$ vs $T$ ).

How to interpret $G$

Be consistent: in pure ALOHA $G$ is tied to the vulnerable window length $2T$ , while in slotted ALOHA it’s tied to a single slot. If you redefine $G$ , the exponential terms’ factors ( $2G$ vs $G$ ) must change accordingly.

Delay conventions vary across textbooks

Some references define delay from packet arrival time, others from the start of service/attempt. That can shift constants (e.g., the “ $+1$ ” term) even when the exponential dependency remains $e^{2G}$ vs $e^{G}$ .

Access attempt lifecycle (collision-driven retransmission)

Packet becomes ready

Packet attempts to transmit immediately (pure) or in the next slot boundary (slotted, aligned model)."

Collision check

t0 + vulnerable period

If any other attempts overlap within the vulnerable period, the packet is destroyed."

Retransmit

After failure

The packet schedules another transmission attempt; delay accrues across failed attempts."

Delivery

Success

The packet is delivered when an attempt occurs with no overlapping competitors."

Common clarifications

Knowledge Check

Question 1 of 4

Q1Single choice

Which throughput expression corresponds to pure ALOHA (unslotted) under the standard Poisson model?

$S_p(G)=G e^{-2G}$

$S_s(G)=G e^{-G}$

$S(G)=e^{-2G}$

$S(G)=G e^{-G^2}$

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