Reliability, Survival and Degradation Probability

Introduction and context

Reliability analysis applies probability theory to the question: how long will a component, process, or system perform its required function under stated conditions? In Total Quality Management, reliability is not merely a design attribute. It connects engineering design, supplier quality, maintenance, warranty cost, customer satisfaction, safety, and continual improvement.

A reliability model must specify four elements: the object being studied, its intended function, the operating environment, and the definition of failure. A pump may be considered failed when it stops rotating, when flow falls below a specification, or when energy consumption becomes unacceptable. These definitions lead to different statistical models and management decisions.

This chapter develops probability models for non-repairable and repairable systems, hazard rates, competing risks, censoring, degradation, redundancy, and preventive maintenance. The emphasis is on interpretation: a mathematically correct estimate can still be operationally misleading if assumptions, censoring mechanisms, or changing operating conditions are ignored.

Learning objectives

By the end of this chapter, the reader should be able to:

1. Define reliability, survival, failure, hazard, and mean time to failure.
2. Derive and interpret relationships among the survival function, density, cumulative failure probability, and hazard rate.
3. Model series, parallel, k-out-of-n, and standby systems.
4. Analyse censored observations and competing failure modes.
5. Distinguish non-repairable reliability from availability of repairable systems.
6. Use degradation measures and first-passage concepts to predict failure.
7. Compare corrective and preventive-maintenance policies economically and statistically.
8. Critically evaluate model assumptions and governance risks in reliability decisions.

1. Fundamental reliability quantities

Let T be a non-negative random variable representing the time to failure of a non-repairable item. Time may be measured in hours, cycles, kilometres, or another appropriate exposure unit.

The reliability or survival function is

It is the probability that the item performs successfully beyond time t. The cumulative distribution function is

where F(t) is the probability of failure by time t. If T has a continuous density f(t), then

The density is not itself the probability of failure at an exact time; for a continuous variable that probability is zero. Instead, f(t) describes the rate at which probability accumulates around t.

The conditional failure intensity is the hazard rate:

This has units of inverse time. It is the instantaneous failure rate among units that have survived to t. It is not generally equal to the proportion of the original population failing per unit time.

Integrating the hazard gives the cumulative hazard H(t):

Consequently,

and, where differentiability holds,

These relationships assume a well-defined time-to-event distribution and a hazard that is integrable over the interval of interest.

The mean time to failure is

provided the integral is finite. For a repairable item, mean time to failure and mean time between failures are not interchangeable; the latter depends on the repair and renewal policy.

Conditional reliability and residual life

For an item that has already survived to age s, the probability of surviving an additional duration t is

The expected residual life at age s is

These expressions are important in warranty analysis and replacement decisions. An ageing component may have lower residual life than a new component, whereas a component subject to early-life defects may become more reliable after surviving an initial screening period.

2. Parametric lifetime models

Exponential model

The exponential model assumes a constant hazard h(t) = λ, where λ > 0. Thus,

The memoryless property is

This model is useful for random failures during a stable useful-life period, but it is inappropriate when ageing, wear-out, or infant mortality is material.

Weibull model

The Weibull distribution is widely used because it can represent decreasing, constant, or increasing hazard:

Here β > 0 is the shape parameter and η > 0 is the scale parameter, with the same unit as t. Its hazard is

If β < 1, hazard decreases, commonly representing manufacturing defects or infant mortality. If β = 1, the model reduces to the exponential distribution. If β > 1, hazard increases, representing wear-out. The mean is

where Γ denotes the gamma function. The parameters must be estimated from data rather than assigned solely from the visual appearance of a failure-rate curve.

Lognormal model

If ln(T) is normally distributed with mean μ and standard deviation σ, then T is lognormal. This model is often suitable for multiplicative ageing mechanisms and highly right-skewed lifetimes. Its mean is

The lognormal hazard is generally non-monotonic, so it can describe situations in which hazard first rises and later declines. Model selection should be based on engineering knowledge, graphical diagnostics, likelihood criteria, and predictive validation—not on fit statistics alone.

3. Reliability of systems and redundancy

System reliability depends on the logical relationship among component successes.

Series systems

A series system operates only if every one of n components operates. If component lifetimes are independent,

For identical components with reliability R(t), this becomes R(t)^n. The corresponding cumulative hazard is additive:

Independence is a strong assumption. Shared environment, common suppliers, common-cause shocks, and dependent loads can make the product formula optimistic.

Parallel systems

A parallel system operates if at least one of n components operates. Under independence,

For two components, this is R₁ + R₂ − R₁R₂. Redundancy improves reliability only if the backup is genuinely available, switching succeeds, loads are controlled, and common-cause failures are addressed.

k-out-of-n systems

A k-out-of-n system functions when at least k of n components function. For identical independent components with success probability R(t),

where C(n,j) = n!/[j!(n−j)!]. This model is common in voting systems, sensor arrays, and modular production capacity.

Standby redundancy

In cold standby, a backup is not exposed to operating stress until activated. If switching is perfect and component lifetimes are independent, the system lifetime for two units is T₁ + T₂, and reliability is the convolution survival probability. If both lifetimes are exponential with rates λ₁ and λ₂, the system lifetime is hypoexponential rather than exponential. For equal rate λ,

Imperfect switching, dormant failures, and common-cause events must be included in a realistic model.

Worked example: parallel reliability

Two independent pumps each have six-month reliability 0.90. A duty arrangement requires at least one pump to operate. The system reliability is

Thus, the idealised system has 0.99 six-month reliability. This is not a guarantee of 99% field performance because the calculation excludes common power loss, shared piping, maintenance unavailability, and failure of the automatic changeover mechanism.

4. Competing risks and failure modes

An item may fail from several mutually exclusive causes: bearing wear, electrical insulation breakdown, overload, or corrosion. Let T₁, …, T_m denote potential failure times for causes 1 through m. The observed failure time is

If cause-specific hazards are h₁(t), …, h_m(t), the all-cause hazard is

The probability of failure by time t from cause j, called the cumulative incidence function, is

where R(u) is survival from all causes. A common analytical error is to treat other causes as ordinary non-informative censoring when estimating the probability of a particular cause. That can overstate the cause-specific failure probability.

Competing-risks analysis supports decisions such as whether to improve a bearing design or an overload protection system. A reduction in one hazard may reveal another as the dominant cause; therefore, cause-specific improvements should be assessed at system level.

5. Censoring and reliability data

Reliability experiments frequently end before every unit fails. In right censoring, an item is known to have survived until a censoring time C, but its exact failure time is unknown. The observed data are

where δ equals 1 for an observed failure and 0 for a censored observation. Right censoring may arise from test termination, withdrawal, or the end of a warranty observation period.

Censoring is non-informative when, conditional on measured covariates, the censoring mechanism is independent of the failure process. If the least reliable machines are removed from service early, that assumption is violated and conventional estimates may be biased.

The Kaplan–Meier estimator is a non-parametric estimate of survival. If t₁ < … < t_k are distinct failure times, d_j failures occur at t_j, and n_j units are at risk immediately before t_j, then

Censored observations reduce the risk set after their censoring times but do not directly cause a downward step. Confidence intervals should be reported, especially in the tail where few units remain at risk.

Parametric models can incorporate censored observations through likelihood methods. For an observation with time y and indicator δ, the contribution is

and the full likelihood is the product over observations, assuming conditional independence. Covariates such as temperature, load, supplier, or operating profile can be incorporated through regression models, including proportional-hazards models:

The proportional-hazards assumption means that hazard ratios between covariate profiles remain constant over time. It must be checked rather than assumed.

6. Repairable systems, availability, and maintainability

For a repairable asset, failures are followed by restoration. Reliability, which concerns successful operation until the first failure, is not the same as availability, which concerns the proportion of calendar time the asset is operational.

If failures occur at rate λ and repairs at rate μ under a two-state continuous-time Markov model, steady-state availability is

where MTBF = 1/λ is mean time between failures under the model and MTTR = 1/μ is mean time to repair. The formula assumes constant rates, statistically stable operation, independent failure and repair processes, and a single operating and a single repair state.

Unavailability is U = 1 − A. For production systems, downtime cost may include lost output, labour, expedited logistics, quality losses, and safety consequences. A high availability estimate can therefore coexist with poor business performance if failures occur during critical production windows.

Renewal-reward analysis is more general than the two-state model. If each cycle consists of an operating period X and repair period Y, and rewards equal operating time, long-run availability under suitable renewal conditions is

This framework accommodates non-exponential lifetimes and repair durations.

7. Degradation and prognostic reliability

Failure is often preceded by a measurable degradation signal: crack length, vibration amplitude, insulation resistance, corrosion depth, or battery capacity. Let D(t) denote degradation and let L be a critical threshold. Failure time can be defined as

This is a first-passage or threshold-crossing definition. It is useful when failures are rare but condition-monitoring data are abundant.

A simple deterministic model is D(t) = D₀ + vt, where D₀ is initial degradation and v is a constant degradation rate. If v > 0 and D₀ < L,

A stochastic model may write

where W(t) is standard Brownian motion and σ controls random variation. This model is analytically convenient but may be unsuitable if degradation is monotone, bounded, or strongly autocorrelated. Gamma processes are often used for monotone cumulative damage because their increments are non-negative, while state-space models are useful when measurements are noisy and the latent degradation state must be inferred.

Prognostic decisions require more than a point estimate of remaining useful life. They should communicate a predictive distribution, uncertainty from measurement and model parameters, and the consequences of false alarms and missed failures. Artificial intelligence can assist with anomaly detection and remaining-useful-life prediction, but training-data shift, opaque decisions, sensor bias, and inadequate failure labels create governance risks. An AI model should be validated against operational outcomes and integrated with engineering thresholds rather than treated as an autonomous authority.

8. Preventive maintenance and replacement decisions

A preventive-maintenance policy replaces or overhauls an item at age τ if it has not failed; otherwise corrective replacement occurs at failure. Let c_p be the planned replacement cost, c_f the corrective replacement cost, and c_d a downtime or consequence cost associated with failure. If F(τ) = P(T ≤ τ), the expected cost per cycle is

The expected cycle length, ignoring separate planned-maintenance duration, is

Thus the long-run cost rate is

This ratio is applicable under renewal assumptions and must be modified to include replacement times, logistics, imperfect maintenance, and production opportunity cost where relevant. A preventive policy is not automatically economical: if failure consequences are low and the hazard is constant, frequent replacement may waste useful life.

A sound decision compares policies using total expected cost, risk, safety, quality impact, and resource constraints. Reliability-centred maintenance asks what function can fail, how it fails, what the consequences are, and which technically feasible task manages the consequence. Age replacement is only one policy; condition-based maintenance may be superior when degradation is observable.

9. Illustrative case: packaging-line servo motors

Background

A food-packaging plant experiences unplanned stoppages in servo motors. Each motor drives a sealing station, and a failed motor stops the associated line. The plant has vibration sensors, maintenance records, and 30 months of operating data. Some motors remain operational when the study ends, so the data are right-censored.

Problem

Management must decide whether to replace motors at a fixed age, install redundant drives, or use condition-based maintenance. Early analysis based only on average lifetime is unsatisfactory because motors operate under different loads and failure modes include bearing wear and encoder faults.

Analysis

The engineering team first defines failure as inability to meet speed and positional tolerances for two consecutive production cycles. It separates bearing and encoder causes and records load, ambient temperature, supplier, and maintenance history. Kaplan–Meier estimates are used for descriptive survival analysis. A Weibull model is then fitted with load as a covariate and censoring included in the likelihood.

Suppose the fitted shape estimate is β = 2.1. Since β > 1, the estimated all-cause hazard increases with age, supporting investigation of wear-out. However, the team checks whether the apparent trend is caused by high-load motors being older. A competing-risks analysis shows that encoder faults dominate early failures, while bearing failures dominate after extended service.

For a critical line, two independent motors in a parallel arrangement are considered. The team does not accept the ideal parallel formula without testing the shared risks. Both motors share a controller and power supply, so common-cause failure is significant. Instead, the design includes independent protection and a tested changeover circuit. Vibration trends are used for bearing condition monitoring, while encoder diagnostics trigger immediate inspection.

Approach and solution

The plant adopts a hybrid policy: condition-based replacement for motors with high predicted bearing degradation, scheduled encoder inspection, and age replacement only for motors lacking reliable sensor data. A small critical spare pool is maintained. The decision rule includes uncertainty bands and a production-loss threshold, rather than relying on a single predicted failure date.

Outcome and learning points

The main improvement is not simply a higher calculated reliability. It is better alignment between failure definitions, data collection, statistical modelling, and maintenance action. The case illustrates five principles: censoring must be retained in analysis; failure modes should not be collapsed prematurely; redundancy requires common-cause assessment; predictive models need uncertainty; and maintenance policy should optimise system consequences rather than component averages.

10. Limitations, governance, and best practice

Reliability models are conditional representations, not physical laws. Important limitations include:

• independence assumptions may fail because of common environment or shared design;
• field conditions may differ from accelerated-test conditions;
• failure definitions may change over time;
• censoring may be informative;
• small samples create unstable tail estimates;
• repair may not restore an item to an as-good-as-new condition;
• competing risks can hide the probability of a specific failure mode;
• degradation measurements may be noisy, irregular, or affected by sensor drift.

Best practice is to combine statistical evidence with physics-of-failure knowledge. Analysts should document data lineage, units, exposure measures, censoring rules, missing-data treatment, model diagnostics, and uncertainty. Reliability targets should be traceable to customer and safety requirements. Safety-critical decisions require independent review, conservative assumptions, configuration control, and compliance with applicable industry standards. Data should not be manipulated by excluding inconvenient failures or redefining failure after results are seen.

Contemporary reliability engineering increasingly combines digital twins, fleet-wide telemetry, Bayesian updating, physics-informed machine learning, and automated maintenance scheduling. These methods are valuable when they improve decisions, not merely when they increase model complexity. A simpler validated model with transparent assumptions is often preferable to a black-box model that cannot be audited.

Chapter Summary

Reliability is the probability that an item performs its required function beyond a specified time under stated conditions. The survival function, failure distribution, density, hazard, and cumulative hazard are mathematically linked, but each answers a different operational question. Exponential, Weibull, and lognormal models provide useful representations of different failure patterns.

System reliability depends on architecture. Series systems multiply component reliabilities under independence, parallel systems benefit from redundancy, and k-out-of-n systems require combinatorial analysis. Common-cause failures and imperfect switching can substantially reduce the benefit of redundancy.

Competing-risks methods distinguish failure mechanisms, while censoring methods allow incomplete life histories to be used without treating censored observations as failures. Repairable-system analysis focuses on availability and maintainability rather than first-failure reliability. Degradation and threshold-crossing models support condition-based maintenance and remaining-useful-life prediction.

Preventive-maintenance decisions should minimise long-run cost and risk, not simply maximise replacement frequency. Reliable practice requires explicit assumptions, validated data, uncertainty reporting, engineering interpretation, and ethical governance of automated predictions.

Key Terms

Reliability: Probability that an item performs its required function beyond a specified time.

Survival function: Probability that failure time exceeds a specified time.

Failure distribution: Probability that failure occurs by a specified time.

Hazard rate: Instantaneous conditional failure intensity among surviving items.

Cumulative hazard: Integral of the hazard over time.

MTTF: Mean time to failure for a non-repairable item.

MTBF: Mean time between failures under a repairable-system model.

MTTR: Mean time to repair.

Availability: Long-run proportion of time an asset is operational.

Series system: System that functions only when all components function.

Parallel system: System that functions when at least one component functions.

k-out-of-n system: System requiring at least k functioning components among n.

Competing risks: Multiple mutually exclusive causes that can produce the observed failure.

Censoring: Incomplete knowledge of an event time, commonly because observation ends first.

Kaplan–Meier estimator: Non-parametric estimator of a survival function with censoring.

Degradation process: Evolution of a measurable condition indicator toward failure.

First-passage time: Time at which degradation first reaches a critical threshold.

Preventive maintenance: Planned intervention before failure or at a specified age.

Common-cause failure: Event that disables multiple redundant components simultaneously.

Review Questions

Short Answer Questions

1. Distinguish reliability from availability.
2. What does a hazard rate condition on?
3. Interpret Weibull shape parameters β < 1, β = 1, and β > 1.
4. Why is censoring not counted as a failure in Kaplan–Meier estimation?
5. Define a competing risk and give one industrial example.

Long Answer Questions

1. Derive the relationship among survival, density, hazard, and cumulative hazard functions.
2. Compare series, parallel, k-out-of-n, and standby redundancy, including their assumptions.
3. Explain how repairable-system availability differs from non-repairable reliability.
4. Discuss the use of degradation data for condition-based maintenance and its limitations.
5. Explain how a preventive-replacement policy can be evaluated using renewal-reward logic.

Analytical / Application Questions

1. A system contains three independent components in series with reliabilities 0.98, 0.97, and 0.95 over one year. Calculate system reliability and explain why common-cause failures could invalidate the result.
2. A plant has many right-censored motor observations and two competing failure modes. Propose an analysis plan, including descriptive estimation, cause-specific analysis, and model validation.
3. A sensor-based maintenance model predicts remaining useful life but produces occasional false alarms. Design a decision framework that incorporates prediction uncertainty, maintenance cost, downtime consequence, and safety risk.

Multiple Choice Questions with Answers

Question 1

If the survival function is R(t), which expression gives the cumulative failure probability by time t?

A. R(t)

B. 1 − R(t)

C. 1/R(t)

D. −R(t)

Correct Answer: B. 1 − R(t)

Question 2

A Weibull shape parameter β greater than 1 generally indicates:

A. Decreasing hazard

B. Constant hazard

C. Increasing hazard or wear-out

D. No possibility of failure

Correct Answer: C. Increasing hazard or wear-out

Question 3

Two independent components each have reliability 0.90, and the system works if either component works. What is system reliability?

A. 0.81

B. 0.90

C. 0.91

D. 0.99

Correct Answer: D. 0.99

Question 4

In a series system, the system functions when:

A. At least one component functions

B. All required components function

C. Exactly half the components function

D. The repair rate exceeds the failure rate

Correct Answer: B. All required components function

Question 5

For competing risks with cause-specific hazards h₁(t), …, h_m(t), the all-cause hazard is:

A. Their sum

B. Their product

C. Their average only

D. The largest hazard only

Correct Answer: A. Their sum

Question 6

In right-censored reliability data, a unit censored at time c is known to have:

A. Failed exactly at c

B. Failed before c

C. Survived at least until c

D. Never been placed in service

Correct Answer: C. Survived at least until c

Question 7

Under a two-state exponential repair model, steady-state availability is:

A. λ/(λ + μ)

B. μ/(λ + μ)

C. λμ

D. 1/(λμ)

Correct Answer: B. μ/(λ + μ)

Question 8

If degradation is D(t) = D₀ + vt and failure occurs at threshold L, with v > 0 and D₀ < L, the failure time is:

A. (L + D₀)/v

B. v(L − D₀)

C. (L − D₀)/v

D. L/(D₀v)

Correct Answer: C. (L − D₀)/v

Question 9

The Kaplan–Meier estimator changes at an observed failure time by a factor based on:

A. The number failed divided by the number at risk

B. The number censored divided by total sample size only

C. The mean lifetime divided by variance

D. The hazard squared

Correct Answer: A. The number failed divided by the number at risk

Question 10

A principal danger in claiming high redundancy benefit is ignoring:

A. Component labels

B. Common-cause failures

C. The existence of time units

D. The definition of a median

Correct Answer: B. Common-cause failures