Representing complex dependencies and Data with parameter networks

In the Low Pressure Coolant Injection system, there are four Pumps, two Motor Operated Valves (MOVs) and six check valves (CVs). In the example, we are concerned with whether or not these components fail during standby. For each component type, the failure rates are obviously related, this induces a dependencies among the parameters. Furthermore, the failure chances of each system are related to some global parameter, such as the length of time the system has been on standby without a test. Finally, Martz and Waller [1990] provide test data for some of the components. Graphical-Belief provides tools to support all of these aspects of parameter dependencies.

A simple Model For Unavailability

Although up to this point we have been talking about failure probability, for this (and similar) example, what we would really like to know is unavailability: the probability that the system will not be available when it is needed. A system might be unavailable for several reasons other than failure, for example it might be off-line for testing (or have been accidentally left off-line after a test, one of the contributing causes to the Three Mile Island incident). For a repairable system, the engineers must carefully balance testing unavailability with reliability (failure probability).

In order to simplify the modelling, we will only consider failures during standby. But still we find that the failure probability for each component depends on a local parameter (the failure rate for the component) and a global parameter (the amount of time the system spends on standby between tests). In Graphical-Belief parameter are always associated with objects in the model. We associate the local parameter (the failure rate) with the rule for the individual component failures. We associate the global parameter (system standby time) with the model itself.

To build the model of failure during standby, we make three simplifying assumptions: (1) a failure in any one hour is independent of the failure during a different hour, (2) during a small time interval the failure probability is approximately the failure rate times the time interval, and (3) the failure rate is constant over the entire time the system is on standby. A system which follows these rules is known as a Poisson Process. Except for (3) these are very reasonable assumptions. For a given standby-time and failure-rate the probability of failure during standby is:

1-exp(-failure-rate*standby-time)

We can relax assumption (3), but then the formula for calculating the failure probability becomes more complex.

In Graphical-Belief we represent such a model with a formula. The failure probability for the component is calculated according to the formula which depends on the values of two other parameters: the :failure-rate parameter (which belongs to the rule) and the :standby-time parameter which belongs to the model. Graphical-Belief copies the formulae for rule parameters automatically when it copies the rule. Because Graphical-Belief allows parameters to be indirect references, when we copy a Poisson process component the :failure-rate always refers to that particular rule's :failure-rate parameter (whose value may be obtained from its prototype by inheritance).

An Example: Sensitivity to standby-time.

As an example of the parameter mechanism, we study the dependence of the LPCI failure rate on the standby-time, which is a model parameter. To do this we open up an editor for the model parameters. There is currently only one global parameter, the standby-time. Figure 1 shows the parameter editor for the (:model :standby-time) parameter.


Figure 1. Parameter editor for the Model's :standby-time parameter. Red border around propagate button indicates changes which need to be propagated.

Suppose that we want to double the interval between tests of the system. Thus, the system will standby for 2 months between tests rather than 1. We set the Model's :standby-time parameter to 2.0 and press the propagate button. There are 12 rules (associated with the 12 basic components in the system) which depend on the Model's :standby-time parameter. Pressing the propagate button causes Graphical-Belief to update each of their valuations and propagate the results. Figure 2 shows the effect. Note that because of the fourfold redundancy of the pumps and the twofold redundancy of the MOVs the new system failure probability is about 4 times the old failure probability.


Figure 2. Probe showing change in failure probability from doubling Model's :standby-time parameter.

Parameter Uncertainty. Continue with this example and explore the implication of uncertainty about the parameters.

Return to the main example page.

Back to overview of Graphical-Belief.

View a list of Graphical-Belief in publications and downloadable technical reports.

The Graphical-Belief user interface is implemented in Garnet.

Get more information about obtaining Graphical-Belief (and why it is not generally available).

get the home page for Russell Almond , author of Graphical-Belief.

Click here to get to the home page for Insightful (the company that StatSci has eventually evolved into).

Russell Almond, <lastname> (at) acm.org
Last modified: Fri Aug 16 16:07:45 1996