Sources

Reliability Math Lab is a calculator set. This file says where each formula, constant and preset comes from. Status words: primary (a standard, paper or data sheet says it), model (a standard mathematical model the lab applies), derived (computed by the lab from a primary rule), and teaching value (chosen by the lab to show the shape of an effect, not a published figure). The page labels teaching values as such.

Formulas

Calculator	Formula	Source	Status
Codeword failure	P(fail) = sum_{k>t} C(n,k) p^k (1-p)^(n-k)	The binomial distribution: n independent bit errors with probability p each, more than t failing a bounded-distance code	model
Parity	BCH parity is about m x t bits, with 2^m - 1 at least the codeword length	The standard binary BCH construction	model
UBER	UBER = P(fail) / data bits, one bit error counted per failed codeword	The lab’s normalization. It is a floor, since a failed codeword leaves several bad bits	model
AFR	AFR = 1 - exp(-8760 / MTBF)	The exponential (constant failure rate) reliability model, 8,760 hours in a non-leap year	model
Sample size	drives >= UCL(f)/FFR and drives >= UCL(0)/(min(TBW,TBR) x 8e12 x UBER)	The JEDEC SSD requirements: FFR and UBER shown at 60 percent confidence. The two-bound form follows the standard’s sample-size rule. The JESD218 text itself was not opened for this lab, so treat the form as a lead to check against the standard	model
UCL	Solve P(X <= k) = 1 - 0.6 for a Poisson mean, then round up to two decimals: 0.92, 2.03, 3.11 for 0, 1, 2 events	Derived by the lab from the 60 percent rule	derived
Arrhenius	acceleration = exp((Ea/k)(1/T_use - 1/T_stress))	Arrhenius equation	model
Tmax	Set t_S [F e(T_SH) + (1-F) e(T_SL)] = t_U [F_U e(T_UH) + (1-F_U) e(T_UL)] and solve for T_SH, with e(T) = exp(-Ea/kT)	The lab’s solution of the Arrhenius dose balance. Client Tmax comes out at 57.6 C, and a stress at 50 C needs a 100 hour delay at 67.3 C. These are computed here, not quoted from a standard	derived
Retention	RBER(t) = RBER0 + B0 x t^m, m usually 1 to 2	A power-law retention model	model

Constants and presets

Value	Where it comes from	Status
Client class: 40 C, 8 hours a day active, 1 year at 30 C retention, FFR at most 3%, UBER at most 1e-15	A. Cox (Seagate, chair of JC-64.8), “JEDEC SSD Specifications Explained”, slides summarizing JESD218	primary (author’s summary of the standard)
Enterprise class: 55 C, 24 hours a day, 3 months at 40 C retention, FFR at most 3%, UBER at most 1e-16	Same deck	primary (author’s summary of the standard)
60 percent statistical confidence	Same deck	primary
Ea = 1.1 eV for retention checks	Same deck, “temperature-accelerated mechanism (1.1eV)”. A JEDEC convention	primary (convention)
Ea = 1.04 eV for 3D NAND	Luo et al., “HeatWatch: Improving 3D NAND Flash Memory Device Reliability by Exploiting Self-Recovery and Temperature Awareness”, HPCA 2018 (fit to 3D charge-trap NAND data)	primary
1,000 hours of direct endurance stress	Same Cox deck: extrapolation is allowed when the direct method needs more than 1,000 hours	primary
8,766 hours in a year (365.25 days) for the use side of the Tmax balance	Arithmetic	derived
Boltzmann constant 8.617333e-5 eV/K	CODATA	primary
8e12 bits per terabyte	10^12 bytes x 8 bits	derived
MTBF 2.0 M h, Optane SSD DC P4800X	Intel ARK specifications page, “Mean Time Between Failures (MTBF) 2 million hours”	primary (vendor claim)
MTBF 2.0 M h, Micron 7450	Micron 7450 product brief, “MTBF 2 million device hours”	primary (vendor claim)
MTBF 2.5 M h, Exos X20 hard drive	Seagate Exos X20 data sheet, “MTBF rating 2,500,000 hours”	primary (vendor claim)
MTBF 1.0 M h preset	The lab	teaching value
Codeword sizes of 512 B to 4 KiB, t from 1 to 120, RBER from 1e-6 to 1e-2	The lab	teaching value
Fleet size of 1,000 drives, TBW of 100 TB	The lab	teaching value
Stress low-temperature lot at 25 C, half the stress hours at the high temperature	The lab	teaching value
RBER0 = 0, B0 = 7.95e-10, m = 1.1, ECC limit 4e-5, safety factor 2	The lab, chosen so the curve crosses the design limit near 10,000 hours	teaching value

What was checked

The tests recompute each published or derived figure on the page and in the walkthrough: 31 drives at FFR 3%, UBER 1e-16 and 100 TB (the FFR bound is 30.67 and the UBER bound 11.5); 0.92, 2.03 and 3.11 for the upper limits; 57.6 C for client Tmax; 67.3 C for the delay case; 0.96 years for 96 hours at 66 C; AFR of 0.437% at 2 M h; and a retention crossing near 10,000 hours.

Open items

• The JESD218 standard itself was not opened (JEDEC requires an account). The class table, the 60 percent rule, 1.1 eV and the 1,000 hour limit are from the author’s slide deck. The two-bound sample-size form and the Tmax equation structure should be checked against JESD218B before anyone cites this lab as following the standard.
• The retention panel has no measured data behind its starting values.