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.

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.