7.4.6. Use of IOR data and/or tests results#

As presented, a reliability model can be built based on IOR data and/or on test data (manufacturers, user) or combined with IOR data and/or on test data (manufacturers, user).

The basic failure rate or the basic probability of failure in Table 7.4.8 is based on such IOR data. The data was compiled from the IOR background in ADS & TAS fleets.

Per miscellaneous items, the anomalies are collected regarding representative products (quality, range of operational conditions, orbit as relevant) and the failures are classified in random failures (\(RF\)) and systematic failures (\(SF\)). To avoid duplication of systematic anomalies, they are only counted once. That means that the derived failure rates include as relevant systematic failure contribution.

The IOR data recorded are:

  • Reference: miscellaneous item number

  • Satellite subsystem

  • Miscellaneous item description (e.g. Li-Ion Battery)

  • Cumulated time in-orbit and/or tests for the relevant sample of items

  • Number of failures

  • Type of failure: random, systematic

  • Point estimate: to provide a point of comparison with one-sided interval estimator

  • File: the file addresses the events (failures) listed to derive a failure rate

    • Description of the event

    • Classification of the failure, either random or systematic

    • Time to failure

    • Statistical weight: representing the percentage of the items fleet at the time of the failure (number of items operational at failure time divided by the number of the items in the fleet).

    • Example: the percentage is nearly 100% for low values of \(TTF\) and the percentage decreases as the \(TTF\) increases in time.

7.4.6.1. Failure rate estimator#

The basic failure rate estimation is a one-sided upper bound interval estimation, see Section 2 for details.

7.4.6.2. Proportion of failure estimator#

To determine a proportion of failures based on the observation of failures within a sample of n elements (e.g. IOR data for “one-shot” device), it is possible to use:

  • Point estimate

  • Bayesian estimator

  • Estimation by interval.

It is recommended to use the interval estimate, see Chapter 6 Section 1 for details.

7.4.6.3. Gamma Bayesian estimator#

The failure rate is no longer considered as a real but as a random variable \(\Delta\).

In the case of the exponential distribution, a natural prior distribution is the Gamma distribution (conjugate of the exponential distribution), see Section 2 for details.

The Bayesian inference estimator is defined as the expected value of the random variable \(\Delta\) knowing that a failure has been observed at \(t_{1}\), \(t_{2}\)\(t_{n}\)). Then, with a sample encompassing \(n\) items and \(n\) associated times to failure:

Equation

(7.4.9)#\[E(\Delta | T_{1} = t_{1}, T_{2} = t_{2}, ..., T_{n} = t_{n}) = \frac{\alpha + n}{\beta + \sum_{i} (t_{i})} = \frac{\frac{\alpha}{n}}{\frac{\beta}{n} + \frac{\sum_{i} (t_{i})}{n}}\]

The parameters of the gamma prior distribution are selected as per the knowledge of the item.

Let us assume for a certain miscellaneous item that a basic failure rate is calculated.

Let us assume that there are some items already in flight with a certain cumulated time, not sufficient enough to provide a result in the order of magnitude of the specification (without prior knowledge).

Let us consider \(\lambda_{0}\) as a first estimation of the failure rate (e.g. provided by Current document or existing model).

Let us consider a Gamma (\(\alpha\), \(\beta\)) distribution assumed as “prior” distribution of the failure rate \(\Delta\).

It is necessary to provide rules for determining the two parameters \(\alpha\) and \(\beta\) with two equations.

We determine \(\alpha\) so that the expected value of the gamma distribution is \(\lambda_{0}\).

Equation

(7.4.10)#\[E(\Delta) = \frac{\alpha}{\beta} = \lambda_{0}\]

This provides a first equation with \(\alpha\) and \(\beta\). A second equation is provided by the level of confidence\(a\)” attributed to the value \(\lambda_{0}\):

Equation

(7.4.11)#\[F_{\Delta}(\lambda_{0}) = P(\Delta \leq \lambda_{0}) = \int_{0}^{\lambda_{0}} \frac{\beta^{\alpha} e^{- \lambda \beta} \lambda^{\alpha - 1}}{\Gamma (\alpha)} = \frac{1}{\Gamma (\alpha)} \gamma (\alpha, \alpha) = a\]

Ideally, the prior should be defined based on two different estimates for the failure rate \(\lambda_{0}\), one representing an “average” value (point estimate, entering Eq. (7.4.10)) and one representing an estimate at a given level of confidence (e.g. conservative estimate, entering Eq. (7.4.11)).

Once \(\alpha\) and \(\beta\) are determined it allows to derive the Bayesian inference estimator with n being the number of failures and \(\sum t_{i}\) the cumulated time.