Glossary

Short definitions of the terms used in the course, each with the lesson that teaches it. Symbols follow the notation table.

Acquisition cost. The one-off price of buying a unit, as opposed to the running costs it incurs while owned. Lesson 7

Aleatory uncertainty. The variability a model describes: which units fail, and when. More units do not reduce it. Compare epistemic uncertainty. Lesson 2

Active redundancy. Redundant units that all run at the same time, so each ages and can fail while the others work. A parallel block in a diagram. Lesson 2

Age replacement. Replace a unit when it fails or when it reaches a set age \(T\), whichever comes first; a failure restarts the clock. Lesson 8

Antithetic pairs. Simulations run in pairs, the second of each drawing \(1 - u\) wherever the first drew the random number \(u\): a late failure where the first had an early one. Each is still a correct simulation, but the two tend to err in opposite directions, so their mean is more precise than two independent simulations'. Lesson 6

Availability. The probability that a repairable system is working at a given time (point availability \(A(t)\)), or the long-run fraction of time it is working (long-run or steady-state availability \(A\)). For one unit, \(A = \text{MTTF} / (\text{MTTF} + \text{MTTR})\). Lesson 6

Barlow–Proschan importance. The share of a system's failures over a window that each component caused: the probability that the component's failure is the one that fails the system, given that the system fails. The integrated form of theta. The Greeks

B-life. The age by which a given fraction of units has failed: \(B_{10}\) is the age at which \(F(t) = 0.1\). Lesson 1

Basic event. A leaf of a fault tree: a component failure, with a probability or a lifetime model. Lesson 3

Bathtub curve. A hazard rate that falls early in life (early failures), stays flat (random failures) and then rises (wear-out). Lesson 1

Beta factor. In the beta-factor model of common-cause failure, the fraction \(\beta\) of a unit's failure probability that fails all the redundant units together. Lesson 5

Birnbaum importance. The probability that a component is critical, \(I_B(i) = R(1_i) - R(0_i)\); equivalently \(\partial R / \partial p_i\), the gain in system reliability per unit gain in the component's. Lesson 4

Block replacement. Replace units at fixed calendar times \(T, 2T, 3T, \ldots\) whatever their age, and at failures in between. Lesson 8

Censored observation. A unit still working when observation stopped: its life is known to be longer than its age, but not by how much. Lifetime fits must use such data; fitting is done in surpyval. Lesson 1

Characteristic life. The Weibull scale \(\alpha\): the age by which 63.2% of units have failed, whatever the shape. Lesson 1

Cold standby. A spare that does not run, and so does not age, until it is switched in to replace a failed unit. Warm standby ages at a reduced rate while waiting. Lesson 5

Common-cause failure. One cause (a shared design flaw, environment or maintenance error) failing several redundant units at once, so that they do not fail independently. Lesson 5

Common random numbers. Comparing two designs by simulating both with the same random numbers, component by component, so that the difference between their results comes from the designs rather than from chance. Lesson 6

Consumer's risk. In a demonstration test, the chance of passing a design that is as bad as the level the test must reject (\(\beta\)). Compare producer's risk. Demonstration testing

Cost rate. The long-run cost per unit time of running a system or a maintenance policy. Lessons 7 and 8

Critical component. A component whose state decides the system's: with the other components as they are, the system works if the component works and fails if it fails. Lesson 4

Criticality importance. The failure-oriented form, \(I_B(i)(1 - p_i)/(1 - R)\), is the probability that component \(i\) has failed and is critical, given that the system has failed: the share of system failures it is decisive in. The success-oriented form, \(I_B(i)\,p_i/R\), is 1 for every component in series with the rest. Lesson 4

Cumulative hazard. \(H(t) = \int_0^t h(u)\,du\); reliability is \(R(t) = e^{-H(t)}\). Lesson 1

Cut set. A set of components whose failure fails the system. A minimal cut set has no smaller cut set inside it; a minimal cut set of one component is a single point of failure. Lesson 3

Delta. How far a system's reliability or availability moves per unit change in a component's: its Birnbaum importance, the first of the Greeks; a lever's delta moves a parameter (a life, a repair, an interval) instead. The Greeks

Density. \(f(t) = dF/dt\): the fraction of the original population that fails per unit time around age \(t\). Lesson 1

Differential importance (DIM). The share of a small change in a system's reliability or availability that comes from each component when all change together; the shares add up to 1. The Greeks

Discount rate. The rate \(r\) at which money later is worth less now: a cost \(c\) at time \(t\) is worth \(c\,e^{-rt}\) today, its present value. Costs

Epistemic uncertainty. Not knowing a model exactly, because its parameters are estimated from limited data; more data reduces it. Lesson 2

Exponential distribution. A lifetime with a constant hazard \(\lambda\): \(R(t) = e^{-\lambda t}\), MTTF \(= 1/\lambda\). It has no memory. Lesson 1

Failure frequency. The long-run number of failures per unit time of a repairable unit, \(1/(\text{MTTF} + \text{MTTR})\), or of a system, \(\sum_i I_B(i)\,\omega_i\). Lesson 6

Fault tree. A diagram of how a system fails: a top event, worked down through OR, AND and VOTE gates to basic events. The dual of a reliability block diagram. Lesson 3

Fill rate. The fraction of demands for a spare met at once from the shelf. Compare stock-out probability, the fraction of time the shelf is empty. Spares

Fussell–Vesely importance. The share of the system's unreliability that comes through minimal cut sets containing a component. Lesson 4

Gamma. The joint (second-order) importance of two components: positive when improving both gains more than the two gains apart (complements), negative when less (substitutes). The Greeks

Hazard rate. \(h(t) = f(t)/R(t)\): the rate at which units that have survived to age \(t\) fail, per unit time. Also called the failure rate. Lesson 1

Hidden failure. A failure that nobody notices when it happens, such as a seized relief valve or a standby pump that will not start: the part is down until a proof test finds it. Also called an unrevealed or latent failure. Lesson 8

Improvement potential. \(R(1_i) - R\): the gain in system reliability if component \(i\) were made perfect. Lesson 4

Input and output nodes. The start and end of a reliability block diagram; the system works when working components connect them. Lesson 2

Junction. A node that never fails, where paths of a diagram meet: a k-out-of-n vote point, say (PerfectReliability). It is folded out of the structure, and is no component to fail, repair or stock. Repairable systems

\(k\)-out-of-\(n\). A block that works when at least \(k\) of its \(n\) units work. 1-out-of-\(n\) is parallel and \(n\)-out-of-\(n\) is series. Lesson 2

Lead time. The time a spare ordered takes to arrive. With one-for-one replenishment, the spares on order are those used in the last lead time. Spares

Load sharing. Redundant units that share a load, so that when one fails the survivors carry more of it and are more likely to fail. Lesson 5

Mean down time (MDT). The mean length of a system outage: \((1 - A)\) divided by the rate of outages. Lesson 6

Mean time between failures (MTBF). For a repairable system, the mean time from one system failure to the next: one over the system failure frequency. Not the same as a component's MTTF. Lesson 6

Mean time to failure (MTTF). The mean life, \(\int_0^\infty R(t)\,dt\): the area under the reliability curve. Often half the units or more fail before it. Lesson 1

Mean time to repair (MTTR). The mean time a failed unit is down while it is repaired or replaced. Lesson 6

Mean up time (MUT). The mean length of a period in which the system works without interruption: \(A\) divided by the rate of outages. Lesson 6

Memoryless. A lifetime whose remaining life does not depend on its age: \(\Pr(T > s + t \mid T > s) = \Pr(T > t)\). Only the exponential has this property. Lesson 1

Minimal repair. A repair that restores a unit to working order but not to new: the unit is "as bad as old". Lesson 8

Monte-Carlo simulation. Estimating a quantity by simulating many random histories and averaging them. Its answers carry sampling error, which shrinks like \(1/\sqrt{N}\). Lessons 6 and 7

Parallel system. A system that works while at least one of its units works: \(R = 1 - \prod_i (1 - R_i)\); the unreliabilities multiply. Lesson 2

Path set. A set of components whose working makes the system work. A minimal path set has no smaller path set inside it. Lesson 3

PFDavg. The average probability of failure on demand of a protective function: its long-run unavailability, when its failures are hidden until a proof test. About \(\lambda\tau/2\) for one channel tested every \(\tau\). Lesson 8

Pivotal decomposition. Conditioning on one component: \(R = p_i\,R(1_i) + (1 - p_i)\,R(0_i)\). Applied repeatedly, it computes any system's reliability exactly; it is the basis of RePyability's engine (also called the Shannon expansion). Lesson 3

Planned outage. Downtime for preventive maintenance or a proof test. It counts as downtime in availability, but not as a failure. Lesson 8

Preventive maintenance. Maintenance done before a failure, on a schedule (such as age or block replacement), to prevent failures that would cost more. It only pays for parts that wear out. Lesson 8

Producer's risk. In a demonstration test, the chance of failing a design that meets the level it was built to (\(\alpha\)). Compare consumer's risk. Demonstration testing

Proof test. A periodic inspection that finds a part's hidden failures; the part is repaired if it is found failed. Lesson 8

Rare-event approximation. Approximating a system's unreliability by the sum, over the minimal cut sets, of the probability that all their components have failed. Accurate when failures are rare. Lesson 3

Reduction. Replacing a group of blocks purely in series, purely in parallel, or \(k\)-out-of-\(n\), by one equivalent block, and repeating. A series-parallel system reduces to a single block; RePyability reduces every diagram as far as it goes before applying the pivotal decomposition to the rest. Lessons 2 and 3

Redundancy allocation. Choosing how many redundant copies of each component to fit, to maximise reliability within a budget or to meet a target at least cost; for a repairable system, to own it at the lowest total cost. Lesson 9, Lesson 7

Reliability. \(R(t) = \Pr(T > t)\): the probability that a unit (or a system) is still working at age \(t\). Also called the survival function. Lesson 1

Reliability allocation. Apportioning a system's reliability target among its components: how reliable must each part be? Lesson 9

Reliability block diagram (RBD). A diagram of a system's success logic: blocks between an input and an output, such that the system works when some path of working blocks connects them. It shows logic, not wiring. Lesson 2

Renewal-reward theorem. For a process that renews in independent, identical cycles, the long-run cost (or reward) per unit time is the expected cost of a cycle divided by the expected length of a cycle. Lessons 6, 7 and 8

Repeated event. A basic event that feeds several gates of a fault tree, such as a shared power supply; it must be counted once, not as independent copies. Lesson 3

Risk achievement worth (RAW). \(Q(0_i)/Q\): how many times more likely the system is to fail while component \(i\) is failed or out of service. Lesson 4

Risk reduction worth (RRW). \(Q/Q(1_i)\): the factor by which a perfect component \(i\) would divide the system's unreliability. Lesson 4

Series system. A system that works only while every unit works: \(R = \prod_i R_i\). Lesson 2

Single point of failure. A component whose failure alone fails the system: a minimal cut set of one. Lessons 3 and 4

Standard error. The standard deviation of an estimate: how far a simulated mean is likely to be from the true value. For the mean of \(N\) independent simulations it is their standard deviation over \(\sqrt{N}\), so it halves when \(N\) is quadrupled. Lesson 6

Structural importance. Birnbaum importance with every component working with probability ½: the fraction of the other components' states in which a component is critical. It depends on the diagram alone. Lesson 4

Structure function. \(\varphi(x)\): 1 if the system works when its components are in states \(x\), 0 otherwise. Lesson 3

Theta. The rate at which a system's availability (or reliability) is changing now, split by the component moving it. The Greeks

Top event. The undesired event at the top of a fault tree, usually the system failing. Lesson 3

Total cost of ownership. What owning a system costs over a horizon \(H\): buying it plus \(H\) times its cost rate (a life-cycle cost, undiscounted unless a discount_rate is given). Lesson 7

Unavailability. One less the availability: the probability that a system is down at a time (point unavailability), or the fraction of a mission it is down. For a safety function, its probability of failure on demand. Repairable systems

Unreliability. \(F(t) = 1 - R(t)\): the probability that a unit has failed by age \(t\). Lesson 1

Vega. How much each component's parameter uncertainty widens the uncertainty of the system's answer. The Greeks

Weibull distribution. A lifetime with \(R(t) = \exp[-(t/\alpha)^\beta]\). Its shape \(\beta\) gives a falling (\(\beta < 1\)), constant (\(\beta = 1\)) or rising (\(\beta > 1\)) hazard. Lesson 1