Binary Component Reliability with Repair

A Two-State Model for a Component

A component observed repeatedly can be described by a binary state: working or failed. In a reliability model with repair, the state can change in either direction from one observation to the next. The transition matrix records the probabilities of those changes over a specified time step. Defining the observation interval and the meaning of “failure probability” and “repair probability” precisely is essential: these are probabilities of the state at the next observation, not necessarily counts of every event that occurred in between.

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space, and observe the component at times \(n=0,1,2,\ldots\), with a fixed interval between observations. Let \(X_n\in S=\{W,F\}\), where \(W\) means working at observation \(n\) and \(F\) means failed. Write \(\mathcal{F}_n=\sigma(X_0,\ldots,X_n)\) for the history of observed states up to time \(n\). Suppose that, given this history, the probability of being failed at the next observation is \(a\) when the component is currently working, and the probability of being working at the next observation is \(b\) when it is currently failed. Assume \(0\le a,b\le1\), and that these conditional probabilities do not depend on \(n\) or on earlier observations.

The conditional assumptions say more than merely specifying the two marginal probabilities. They say that the next-state distribution depends on the observed past only through the current state, and that the same transition probabilities apply at each step. Thus they specify a time-homogeneous Markov chain. If the component may fail and then be repaired between observations, the state at the next observation can still be \(W\); such within-interval events are already reflected in the endpoint transition probabilities. A model that instead records whether any failure occurred during the interval needs a different state definition or different probabilities.

Constructing and Checking the Transition Matrix

Given \(X_n=W\), the component is working at the next observation with probability \(1-a\) and failed with probability \(a\). Given \(X_n=F\), it is working with probability \(b\) and remains failed with probability \(1-b\). Therefore, in the state order \(W,F\), the transition matrix is

Every entry is nonnegative because \(0\le a,b\le1\). The first row sums to \((1-a)+a=1\), and the second sums to \(b+(1-b)=1\). Thus \(P\) is row-stochastic, as required for a transition matrix. In particular, the first row describes outcomes from a currently working component; the second describes outcomes from a currently failed component.

Proof. To prove the forward statement, consider each possible current state. On \(\{X_n=W\}\), the conditional probabilities of \(X_{n+1}=F\) and \(X_{n+1}=W\), given \(\mathcal{F}_n\), are \(a\) and \(1-a\), respectively; the second follows because the two states exhaust \(S\). On \(\{X_n=F\}\), the corresponding probabilities are \(b\) and \(1-b\). Hence, for either state \(j\in S\), the conditional probability \(\mathbb{P}(X_{n+1}=j\mid\mathcal{F}_n)\) depends on the history only through \(X_n\), and its value is independent of \(n\). This is the Markov property with a time-independent transition matrix whose rows, in order \(W,F\), are \((1-a,a)\) and \((b,1-b)\). For the converse, a time-homogeneous Markov chain has a fixed row of transition probabilities for each current state. The stated failure probability fixes the \(W\)-to-\(F\) entry at \(a\), so the other entry in that row must be \(1-a\). The repair probability fixes the \(F\)-to-\(W\) entry at \(b\), so the other entry must be \(1-b\). This gives exactly the displayed matrix. \(\square\)

If the initial distribution is \(\mu_0=(q_0,1-q_0)\), where \(q_0=\mathbb{P}(X_0=W)\), the row-vector convention gives \(\mu_1=\mu_0P\). More generally, one-step prediction gives a useful scalar update.

Proof. The events \(\{X_n=W\}\) and \(\{X_n=F\}\) are disjoint and together have probability one. For the component to be working at \(n+1\), either it was working at \(n\) and remains working, or it was failed at \(n\) and is repaired. These disjoint cases have probabilities \(q_n(1-a)\) and \((1-q_n)b\), respectively, by the transition probabilities. Adding them gives the stated formula. \(\square\)

Worked Reliability Calculations

Interpreting the Model and Avoiding a Common Error

The transition matrix answers a question about observed states at consecutive inspection times. It does not, without further assumptions, specify the exact time of a failure or repair within an interval. If repair and failure can both occur during one interval, the probabilities \(a\) and \(b\) must be supplied or derived for the endpoint states under the chosen observation scheme. Treating a raw “failure occurs” probability as \(a\) is only justified when that probability really is the probability that the next observed state is failed, given a working state.

A common mistake is to reverse the interpretation of the second row. For example, with repair probability \(b=\tfrac{1}{3}\), the failed-state row is sometimes incorrectly written as \((\tfrac{2}{3},\tfrac{1}{3})\), on the grounds that a failed component stays failed with probability \(1-b=\tfrac{2}{3}\). In the state order \(W,F\), however, the first entry means transition to \(W\), so that row would claim repair probability \(\tfrac{2}{3}\), not \(\tfrac{1}{3}\). The correct row is \((b,1-b)=(\tfrac{1}{3},\tfrac{2}{3})\). Its entries sum to one and its first entry matches the defined repair probability.

Another modeling check is whether the probabilities remain constant across time and depend on the past only through the current state. If, for example, failure risk increases with component age, the binary state \(\{W,F\}\) may not contain enough information for a time-homogeneous Markov model. One may need to include age or another relevant variable in the state. For the two-state model here, the reliability calculation is valid precisely under the stated endpoint and conditional assumptions.