Remaining Useful Life Estimation [/rɪˈmeɪnɪŋ ˈjusfəl ˈlaɪf ˌɛstəˈmeɪʃən/] n - Remaining useful life is the span expected before a component fails, or declines below an acceptable threshold. It is the figure that maintenance and reliability engineering require, so that replacements may be scheduled in foresight rather than in haste. The mathematics draws upon survival analysis, time series, and Bayesian updating.

Of Survival and the Weibull Model

The survival function $S(t)$ gives the probability that a unit still serves at time $t$:

$$S(t)=P(T>t)=\exp \left(-{\left(\frac{t}{\lambda }\right)}^{\kappa }\right)$$

The Weibull distribution, with shape $\kappa$ and scale $\lambda$, is the standard model for the suppleness of its hazard: $\kappa <1$ gives a falling failure rate (infant mortality), $\kappa =1$ a constant rate (chance failures), and $\kappa >1$ a rising rate (wear-out). Most mechanical things in a data center dwell in the wear-out regime.

The parameters are estimated by maximum likelihood from the annals of past failures; even a modest fleet of fifty units with ten years of records furnishes data enough.

The remaining useful life at the present age $t$ is:

$$\text{RUL}(t)=\mathbb{E}[T-t\mid T>t]=\frac{1}{S(t)}{\int }_t^{\infty }S(u)du$$

Of Degradation, Measured by Condition

For components that betray a measurable decline (battery capacity, bearing vibration, compressor efficiency) a degradation-path model is surer than one built on time alone.

A common form is the linear model with random effects:

$$Y(t)={\varphi }_0+{\varphi }_{1}t+ϵ(t)$$

where $Y(t)$ is the degradation signal and ${\varphi }_0$, ${\varphi }_1$ are random effects proper to that one unit, estimated from its own history. Failure comes when $Y(t)$ crosses a threshold $D_f$.

The time to failure is the first crossing of $D_f$:

$$T=\inf \{t:Y(t)\ge D_f\}$$

Given the measurements gathered to time $s$, the distribution of the remaining life $T-s$ is reckoned from the posterior distribution of the path’s parameters. This is a direct application of Bayesian Updating for Beliefs: the prior upon the parameters is updated as observations accrue, and the posterior yields the RUL distribution.

Of Joining Many Signals

True equipment gives off many signals of decline: temperature, vibration, current draw, the chemistry of its oil. To join them into a single index of health betters the estimate.

One road is a multivariate regression:

$$H(t)=w_1{\tilde{y}}_1(t)+w_2{\tilde{y}}_2(t)+\dots +w_k{\tilde{y}}_k(t)$$

where each ${\tilde{y}}_i$ is a normalized, detrended form of signal $i$ and $w_i$ a weight learned from past failures. The health index then stands in place of the several signals within the degradation model.

A more principled road uses the latent structure: a state-space model whose hidden state is the true condition and whose observations are the many signals, the very framework of State Space Models and the Kalman Filter. The Kalman filter fuses the signals into one smooth estimate of health.

Of Speaking the Uncertainty

An RUL estimate without its bounds deceives. The distribution of remaining life should be reported, at the least the 10th, 50th, and 90th percentiles. An operator who must order parts a fortnight ahead needs to know the P10 RUL is three weeks, not merely that the median is six.

Bayesian credible intervals furnish a coherent frame for this. As observations arrive the interval narrows; a unit running below its expected curve earns a lower and tighter estimate, one running above, a longer.

See Predictive Maintenance for CRAC and CRAH Units, Predictive Generator Maintenance, and UPS and Battery State Prediction for how this frame is fitted to particular machines in a data center.