A probability model of system downtime with implications for optimal warranty design

Traditional approaches to modeling the availability of a system often do not formally take into account uncertainty over the parameter values of the model. Such models are then frequently criticised because the observed reliability of a system does not match that predicted by the model. Instead this paper extends a recently published segregated failures model so that, rather than providing a single figure for the availability of a system, uncertainty over model parameter values are incorporated and a predictive probability distribution is given. This predictive distribution is generated in a practical way by displaying the uncertainties and dependencies of the parameters of the model through a Bayesian network. Permitting uncertainty in the reliability model then allows the user to determine whether the predicted reliability was incorrect due to inherent variability in the system under study, or instead due to the use of an inappropriate model. Furthermore, it is demonstrated how the predictive distribution can be used when reliability predictions are employed within a formal decision-theoretic framework. Use of the model is illustrated with the example of a high-availability computer system with multiple recovery procedures. A Bayesian network is produced to display the relations between parameters of the model in this case and to generate a predictive probability distribution of the system’s availability. This predictive distribution is then used to make two decisions under uncertainty concerning offered warranty policies on the system: a qualitative decision, and an optimisation over a continuous decision space. NOTATION k number of recovery levels F set of failure types F j set of failures served at level j F typej set of failures of type j p recj conditional probability of recovery at level j p nextj conditional probability of progressing to service at level j + 1 p lastj conditional probability of immediately progressing to service at level k p typej probability that failure is of type j ⌧ j time taken to serve failure at level j (mins) μ j restoration rate expected number of failures per year typej expected number of type j failures per year v actual number of failures in next year v typej actual number of type j failures in next year ⌧ typej total downtime due to a type j failure (mins) t dj expected total yearly downtime due to failures of type j (mins) T d expected total yearly downtime (mins) This work was supported by grants from Science Foundation Ireland (SFI). B. Flood ([email protected]), B. Houlding ([email protected]) and S.P. Wilson ([email protected]) are with the Centre for Telecommunications Value-Chain Research (CTVR), Department of Statistics, Trinity College Dublin, Ireland. S. Vilkomir is with the Software Quality Research Laboratory (SQRL), Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee, USA. He is a senior member of the IEEE.