Reading the Collision Probability the Section Writes Down

Reading the Collision Probability the Section Writes Down

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

Reading the Collision Probability the Section Writes Down

A room sensor that uploads only ten bytes every minute still has to share the channel, and the chapter warns that once hundreds of devices wake at the same scheduled time, those same small reports can queue, collide, and retry. The under-the-hood section behind that warning gives the DCF collision model only symbolically, as a per-slot transmit probability tau and a contender count n, without instantiating either. This audit asks the question those symbols invite: once tau and n are given concrete values, how large does the collision probability actually get, and does the result confirm that contention scales with how many radios contend, not with how little each one sends?

Companion to the chapter Wi-Fi MAC Layer and Protocols — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is contending radios. The middle card applies this page's rule. The green card is collision probability. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only contending radios, so the numeric fixture does not switch without explanation.

Walk the arrows. Even a small per-slot attempt chance compounds when many radios wake together.

Derive the baseline in four named moves

1. 1

   Name the input. The chapter baseline is 100 radios.

2. 2

   Name the relationship. collision probability = (1 - (1 - 0.02)^(radios - 1)) x 100

3. 3

   Substitute with units. (1 - 0.98^99) x 100 = 86.5%

4. 4

   Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change contending radios

Try Predict the direction of collision probability = (1 - (1 - 0.02)^(radios - 1)) x 100. Test another contending radios, then compare collision probability.

Contending radios

100 radios

Chapter baseline

Collision probability

Observe Even a small per-slot attempt chance compounds when many radios wake together. Reset contending radios to 100 and compare collision probability.

Explain Even a small per-slot attempt chance compounds when many radios wake together.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only contending radios moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only contending radios. It is not a field prediction.

Try

Insert tau = 0.05 and n = 10 into p = 1 - (1 - tau)^(n-1), then evaluate retry stages through K = 3.

Observe

Collision probability reaches about 37%, expected attempts rise to 1.56 per frame, and the illustrative backoff window grows from 8 to 16 to 32 slots.

Explain

A transmission collides when any of the other n-1 stations chooses the same slot; adding contenders compounds that probability even when every payload is small.

Ada: The under-the-hood section gives the DCF collision model symbolically – p = 1 - (1 - tau)^(n-1) for a per-slot transmit probability tau and n contenders – but never puts numbers to it. Let me instantiate it with clearly illustrative inputs (these are my example values, not chapter measurements) so the shape of the result is visible: take tau = 0.05 and n = 10 stations.

• Collision probability: p = 1 - (1 - 0.05)^(10-1) = 1 - 0.95^9 = 1 - 0.630249 = 0.369751, about 37%.
• Expected attempts for a retry limit K = 3, using the section’s 1 + p + p^2 + p^3: 1 + 0.369751 + 0.136716 + 0.050556 = 1.5570, so roughly 1.56 transmissions per delivered frame.
• Backoff growth from b_k = 2^k CWmin / 2 with an illustrative CWmin = 16: stage 0 gives 8 slots, stage 1 gives 16, stage 2 gives 32 – the window doubling the section describes.

Notice what drives the 37%: it is n, the number of stations waking together, not the payload size. Ten small sensors reporting at the same instant already collide over a third of the time and average 1.56 attempts each. The design meaning is exactly the chapter’s opening warning – a few-byte payload does not make MAC behaviour safe, because contention scales with how many radios contend, not with how little each one sends.

Every number above is taken from the chapter’s own material and re-derived step by step.

Technical boundaries: This illustrative DCF calculation assumes independent fixed tau and identical contenders; it omits hidden terminals, capture effect, variable contention windows, rate adaptation, correlated wakeups, channel errors, and queue dynamics.

Ready: work the ledger before checking it.