Zachary Taylor

Analyzing Confidence Levels

To increase the predictive usefulness of reliability models, we need good estimates of reliability parameters, such as Mean Time between Failures (MTBF) and (Mean Time to Repair) MTTR. A key source of information that can be applied to verify or improve our reliability models is test data. These test data may be obtained from projects with similar characteristics as our model or be based directly on test data or field data from the system we are modeling. GoF tests provide us with a measure of how well our assumed probability distributions for our model match the actual test data. If a large disparity is identified, then our model may need to be amended to improve its accuracy of modeling the actual system behavior. Confidence levels, on the other hand, tell us the range of values for reliability estimates, such as MTBF, that fall within a certain confidence level, for example, minimum and maximum values that fall within 95% confidence levels. The more data we have, the more narrow the range between the maximum and minimal values. And the tighter the confidence level, the wider the range will be. The confidence level provides us with feedback on how likely our model will be in predicting future reliability based on data we have??-??including the case where no failure data exist but the system(s) has been operational for a given period of time. In Chapter 18, we will apply these techniques to a case study.

Application of DFMEA to Real-Life Example

This chapter provides a case study of a Design Failure Modes and Effects Analysis (DFMEA) for an actual telecommunications project. As a result of conducting this DFMEA, design faults were identified that otherwise may not have been discovered. Design changes were introduced to prevent these faults from being discovered after product introduction. Using the initial and revised RPNs, the design improvements can be quantified. The lessons learned from this project were identified and used to improve the effectiveness and efficiency of the subsequent DFMEAs for this project.

A Game of Dice: An Introduction to Probability

This chapter introduces some of the fundamentals of probability theory illustrated by our analysis of a game of dice and coin flip experiments. By analyzing these famil??iar examples in greater depth, using familiar illustrations, we have hopefully been able to shed some light on some fundamental concepts of probability theory. These concepts are the building blocks for more practical and varied applications of reli??ability analysis.

Updating Reliability Estimates: Case Study

This chapter describes some practical techniques that can be used to analyze field data and revise the availability model and predictions. This analysis can be continued during the life of the system as and when more field data are obtained. The availability analysis can be periodically updated based on potential new outages and/or additional field exposure. Even if we have zero outages over a long period of time, the system unavailability will not remain the same. This will actually increase the availability. On the contrary, if we have more outages, the overall availability of the system will decrease. The system availability is dynamic and must be updated as more information becomes available. It is a good practice to monitor and track system availability regularly to ensure that a system is healthy and operational at satisfactory levels.

Initial Considerations for Reliability Design

Prior to designing a high availability system, we must have a set of availability require??ments or goals for our system. We then set out to design a system that meets these requirements. By decomposing the system into appropriate components, we can create a system reliability model and mechanisms for ensuring high availability. For each component in this model, we allocate specific MTBF, MTTR, and other reli??ability information. We describe a few general methods can be used to estimate and improve this reliability information. The more knowledge we have regarding the reli??ability of these components, the maintenance plan for the system, and the number of systems we expect to deploy, the more accurate our model will be in predicting actual system reliability and availability once deployed to the field. Now that we have introduced the mechanics of obtaining initial reliability informa??tion, the next several chapters will dwell on basic mathematical concepts that will set the foundation for more advanced techniques and applications to build, predict, and optimize high availability systems.

Continuous-Time Markov Systems

In this chapter, we expand on the concept of a continuous-time Markov processes previously introduced in Chapter 8. Given the Markov properties and comparing with the Discrete Time Markov Chain (DTMC), the state transition rate matrix is derived. With this canonical form of the continuous-time Markov process, we are able to derive the dynamic properties of each defined state and explore how they evolve over time. A two-state Markov model for a single component repairable reliability model is derived. We walk through the steps to create a Markov model and considered the characteristics of this model, including asymptotic behavior. We also provide a brief introduction to Markov reward models. The continuous-time Markov process is one of the most powerful tools we have for modeling system behavior. With the appropriately defined model, we are able to extract many useful characteristics of the system under analysis. We will apply this powerful technique to a broad variety of problems in the latter half of this text. One of the drawbacks of this technique is state-space explosion. We will investigate this problem in the following two chapters and discuss ways in which we can mitigate this to make the problems more tractable.