Address Math and Airtime Physics

Address Math and Airtime Physics

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

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Ada ADA · CALCULATION AUDIT

Address Math and Airtime Physics

A Zigbee coordinator hands out 16-bit short addresses from a pool of 65,536 values, yet random assignment reaches a 50% chance of a duplicate at just 301 devices — because collision risk grows with device pairs, not device count. The same pairwise logic governs a city’s 2,000 LoRaWAN parking sensors, whose 190 ms packets collide far sooner than their under-3% channel use would suggest. This audit re-derives both the address math and the airtime physics to ask whether the chapter’s own numbers hold.

Companion to the chapter The Birthday Problem in IoT Networks — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is devices sharing 16-bit space. The middle card applies the page rule. The green card is collision probability. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

Devices sharing 16-bit space changes collision probability An input card leads through the rule risk = (1 - exp(-devices^2 / (2 x 65,536))) x 100 to the collision probability result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Collision risk follows the number of device pairs, not the fraction of addresses used.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 300 devices.

  2. 2

    Name the relationship. risk = (1 - exp(-devices^2 / (2 x 65,536))) x 100

  3. 3

    Substitute with units. (1 - exp(-300^2 / 131,072)) x 100 = 49.67%

  4. 4

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

Predict, then change devices sharing 16-bit space

Try Predict the direction of risk = (1 - exp(-devices^2 / (2 x 65,536))) x 100. Test another devices sharing 16-bit space, then compare collision probability.

300 devices
Chapter baseline
Collision probability

Observe Collision risk follows the number of device pairs, not the fraction of addresses used. Reset devices sharing 16-bit space to 300 and compare collision probability.

Explain Collision risk follows the number of device pairs, not the fraction of addresses used.

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 devices sharing 16-bit space moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only devices sharing 16-bit space. It is not a field prediction.

TryStart Check derivation with 23 birthdays, then load the 16-bit space containing 65,536 possible short IDs.
ObservePair comparisons reach 253 for 23 people, and the 50% short-ID collision threshold appears near 301 devices.
ExplainCollision opportunity grows with n times n-minus-one over 2, so pair count rises quadratically long before the identifier space is full.

Ready: use the stated baseline inputs, then compare each displayed result.

Ada: Pair-count math is the warning; packet airtime is the physical window that turns random traffic into collisions. The chapter’s numbers can be checked from the formulas already shown above.

  • Classic birthday check: 23 x 22 / 2 = 253 pair comparisons. Since 253 / 365 = 0.693, 1 - e^-0.693 = 50.0%, which explains why 23 people is enough for a coin-flip birthday match.
  • 16-bit short-address threshold: m = 2^16 = 65,536, sqrt(65,536) = 256, and 1.177 x 256 = 301 devices. That is the chapter’s 50% collision threshold.
  • Zigbee 300-device row: 300^2 / (2 x 65,536) = 90,000 / 131,072 = 0.687, so 1 - e^-0.687 = 49.7%.
  • Zigbee 450-device check: 450^2 / (2 x 65,536) = 202,500 / 131,072 = 1.545, so 1 - e^-1.545 = 78.7%. The risk is not “450 out of 65,536”; it is the aggregate of 450 x 449 / 2 = 101,025 device pairs.
  • LoRaWAN timing starts with the symbol: 2^10 / 125,000 = 0.008192 s = 8.19 ms. The chapter’s SF10 example then gives preamble time 8 x 8.19 = 65.5 ms, payload time about 15 x 8.19 = 122.9 ms, and total airtime about 188.4 ms, rounded to 190 ms.
  • The vulnerable overlap window is 2 x 190 ms = 380 ms. One hour has 3,600,000 / 380 = 9,474 such windows, and the offered traffic is 12,000 / 48 = 250 transmissions per channel/SF combination.
  • Slot-collision estimate: 250^2 / (2 x 9,474) = 3.30, so 1 - e^-3.30 = 96.3% chance of at least one collision per virtual channel. The expected count is 3.30 x 48 = 158 collisions per hour, and 158 / 12,000 = 1.3% of transmissions.

The conclusion is bounded: use the birthday model to expose pairwise risk, then choose the real control for the domain. Address collisions need larger identifiers, partitioning, or reassignment. Airtime collisions need scheduling, backoff, channel diversity, spreading-factor planning, or lower offered load.

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

Technical boundaries. No random assignment trial, duplicate recovery, radio contention, capture, or protocol retry is simulated; the page evaluates fixed birthday and airtime approximations.