Hard disk datasheet: MTTF of 1M to 1.5M hours, suggesting nominal failure rate of 0.58 to 0.88%. But annual disk replacement rate observed >1%, usually 24% and up to 13% in some system.
Time between replacement as proxy for time between failure:
 Many sites follow “better safe than sorry” policy in replacing disks
 Manufacturer may see a disk as healthy while consumer declare it faulty
 Mean time to replacement > mean time to failure
 not well modeled by exponential distribution, with significant levels of autocorrelation and long range dependency
Presumed lifecycle failure pattern for hard drives ^{1}
 high early failure (infant mortality) in first year
 steady failure rate in years 25
 higher failure due to wearout in years 57
Observation findings:
 wearout may start much earlier than expected
 does not agree with the common assumption that after the first year of operation, failure rates reach a steady state for a few years
 Chisquare test does not suggest disk replacement per month follows a Poisson distribution
 due to failure rates are not steady over the lifetime
 Time between failure distribution: Agree with Weibull or gamma distribution, with Chisquare test at significant level of 0.05
 Distribution of time between replacements exhibits decresing hazard rate
 Correlation is significant for lags in the rage of up to 30 weeks
 Autocorrelation function (ACF): measures the correlaton of a random variable with itself at different time lags $\ell$
 number of failrues in one day is correlated with the number of failures observed $\ell$ days later
 Autocorrelation function (ACF): measures the correlaton of a random variable with itself at different time lags $\ell$
 Squared coefficient of variation: Variance divided by mean $C^2 = \sigma^2 / \mu$
 exponential distribution: $C^2 = 1$
 observation: $C^2 = 2.4$
 Longrange dependence: how quickly the autocorrelation coefficient decays with growing lags
 strength quantified by Hurst exponent $H$, LRD if $0.5 < H < 1$
 observation: Hurst exponent between 0.6 to 0.8 at weekly granularity
Notes
Annualized failure rate (AFR) is used to characterize the reliability of hard disk drives:
where $8766 = 365.25 \times 24$ is number of hours in a year, MTTF is mean time between failure. $1\textrm{AFR}$ is the fraction of devices or components that will show no failure over a year. If AFR is small, this can be approximated by:
Example: 1 million hours as MTTF, and exponential arrival model for failure. Then failure model is:
where $\lambda^{1} = 10^6$ and $t$ is in hours. Taking annualized failure rate, it is
AFR data from backblaze (2018 Q1): https://www.backblaze.com/blog/harddrivestatsforq12018/. But the AFR is expressed in probability estimate. However, the accuracy of estimate, especially in the case of failure is relatively rare, depends on number of samples. Beta distribution should be a good way to estimate the accuracy of the probability estimate.

J. Yang and F.B. Sun. A comprehensive review of harddisk drive reliability. In Proc. of the Annual Reliability and Maintainability Symposium, 1999 ↩
Bibliographic data
@inproceedings{
title = "Disk failures in the real world: What does an MTTF of 1,000,000 hours mean to you?",
author = "Bianca Schroeder and Garth A. Gibson",
booktitle = "Proceedings of FAST'07: 5th USENIX Conference on File and Storage Technologies",
year = "2007",
}