Where the 0.459 Safe-Count Rule Comes From

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

ADA · CALCULATION AUDIT

Where the 0.459 Safe-Count Rule Comes From

This chapter distills collision safety into a single shortcut: multiply the square root of the address space by 0.459 to get the device count that stays under a 10% collision chance. For a 16-bit Zigbee space of 65,536 addresses that works out to about 117 devices. This audit asks the two questions the shortcut invites: where does the number 0.459 actually come from, and does 117 devices really land on the 10% line?

Companion to the chapter Collision Mitigation Strategies — every number here comes from that chapter.

Ada: This chapter leans on one design rule: a 16-bit address space is safe for “approximately 117 devices” at a 10% collision threshold, from n ~ sqrt(S) x 0.459. Two numbers deserve a check — where 0.459 comes from, and whether 117 really lands on 10%.

Start from the birthday probability P = 1 - e^(-n^2 / (2S)) and solve for n at P = 0.10, so e^(-n^2 / (2S)) = 0.90:

• Take logs: -n^2 / (2S) = ln(0.90) = -0.105360516.
• Rearrange: n = sqrt(2S x 0.105360516) = sqrt(S) x sqrt(0.210721033) = sqrt(S) x 0.459044.

That is the chapter’s 0.459 — it is sqrt(-2 x ln(0.90)), not a fudge factor. For a 16-bit space, sqrt(65536) = 256, so the safe count is 256 x 0.459044 = 117.50, which floors to 117 devices. Verifying forward closes the loop:

• P(117) = 1 - e^(-(117^2) / (2 x 65536)) = 1 - e^(-13689 / 131072) = 1 - e^(-0.104439) = 1 - 0.900830 = 0.099170, i.e. 9.9%.

The same derivation at P = 0.50 gives sqrt(-2 x ln(0.50)) = 1.177410, so the 50% point sits at 256 x 1.177410 = 301.4, about 302 devices. The design meaning lives in the square root, not the space size: because n scales with sqrt(S), adding 16 address bits multiplies the space by 65,536 but the safe device count by only sqrt(65536) = 256 — which is exactly why a 16-bit short address exhausts its safety margin at a few hundred devices while a 64-bit address effectively never does.

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