Markov Modeling for Reliability



The term “Markov model”, named after the mathematician Andrei Markov, originally referred exclusively to mathematical models in which the future state of a system depends only on its current state, not on it’s past history. This “memoryless” characteristic, called the “Markovian property”, implies that all transitions from one state to another occurat constant rates. Much of the practical importance of Markov models for reliability analysis is due to the fact that a large class of real-world devices (such as electronic components) exhibit essentially constant failure rates, and can therefore be effectively represented and analyzed using Markov models. (The term “Markov model” is sometimes used in a more general sense, allowing for variable failure rates, as discussed briefly in Section 3.8, but the most common applications of Markov modeling in reliability still involve only constant failure rates, so those will be the main focus of this discussion.)


One of the notable strengths of Markov models for reliability analysis is that they can account for repairs as well as failures. This makes the technique particularly useful for assessing the long-term average reliability of one or more devices with established maintenance and repair strategies.  With the advent of high-integrity “fault-tolerant” systems, the ability to account for repairs of partially failed (but still operational) systems has become increasingly important. Markov modeling is well-suited to the task of determining inspection and repair intervals needed to achieve a desired level of safety.


However, in practice, system repairs do not typically occur at a constant rate. Instead, repairs usually occur either within some fixed length of time after a failure, or else at scheduled periodic inspection/repair intervals. Since these operations are not “Markovian” in the strict sense, there has been some confusion within the reliability community as to whether periodic and on-condition repairs could be incorporated into Markov models, and if so, how best to represent such repairs. This has been one of the main obstacles to more widespread use of Markov modeling in industry. A main focus of this article is to describe how this can be accomplished easily and accurately.


Part 2: Markov Model Fundamentals