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eduKateSG · Why Mathematics? · Standby Redundancy Reliability: Cold, Warm and Hot Spares, Switching Coverage and Markov Models
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 6–10
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 11–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important in system design becomes clear when a spare component waits to take over after the active one fails. Cold, warm and hot standby arrangements differ in how much the spare operates, how quickly it ages and whether detection and switching succeed. Reliability therefore depends on component failure rates, dormancy, switch coverage, repair and mission time, not simply on the number of boxes.
This guide develops the ideal cold-standby Erlang formula, active-parallel reliability, imperfect switching and continuous-time Markov models. Worked examples show why standby can improve mission reliability while still creating single points of failure. The practical lesson is optimistic but careful: redundancy helps when failure modes are independent enough, switching is covered, dormant failures are controlled and the full system is tested.
Full chapter index
Chapters 1–5 · Understand the question
Chapters 6–10 · Reproduce the example
Chapters 11–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–25 · Study and verify
A standby unit is reserved to take over after an active unit fails. A backup pump may start only when the duty pump trips. This is the central mathematical mechanism in why a spare changes the system. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A spare that cannot detect, start or connect provides no protection. Draw active, spare and switch states. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind cold standby. An ideal cold spare is assumed not to fail while dormant and begins ageing only after successful activation. In the illustrative case, One active unit plus one perfect cold spare provides two sequential exponential lives. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Real dormant components can deteriorate or fail to start. Treat ideal cold standby as a benchmark.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat ideal two-unit formula as a model-checking problem rather than a vocabulary exercise. For identical exponential active lifetimes with rate lambda and perfect instantaneous switching, R(t)=exp(-lambda t)(1+lambda t). The total life is Erlang with shape two. Recompute the example from the raw values and label every unit. The formula assumes zero dormancy rate and perfect coverage. Derive it by conditioning on first failure time.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of a cold-standby calculation becomes useful only when its mechanism is explicit. With lambda=10^-4 per hour and t=1000 hours, lambda t=0.1 and R≈exp(-0.1)×1.1≈0.99532. Consider the worked illustration: A single unit would have R≈0.90484. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The large gain is conditional on ideal switching and independence. Show both benchmark values.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Primary source: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19720006837.pdf. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use hot standby or active parallel to practise sensitivity analysis. Hot redundant units operate simultaneously, so either can survive the mission in a simple one-out-of-two system. For independent identical units, R=1-[1-exp(-lambda t)]^2. Before accepting the result, identify the quantity that would change the conclusion most. Shared load and common causes violate the independent formula. Define whether load changes after failure.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for comparing hot and cold. At lambda t=0.1, ideal hot parallel reliability is about 0.99094, while ideal cold standby is about 0.99532. The initial example is: The cold spare avoids active ageing before demand. Then create a nearby case where the usual shortcut becomes unreliable. Switching risk can reverse the comparison. Compare with realistic coverage.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect switching coverage to a real decision while protecting the boundary between calculation and judgement. Coverage c is the conditional probability that failure is detected, isolated and the spare is successfully activated under the model. A switch may fail mechanically or the controller may misdiagnose the event. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Coverage is not merely switch hardware reliability. Define every required transition.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
For the ideal two-unit cold model with switch coverage c, R(t)=exp(-lambda t)[1+c lambda t]. With c=0.9 and lambda t=0.1, R≈0.98627. This is the central mathematical mechanism in imperfect-switch formula. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. This compact formula still assumes no dormancy failure. Check c=0 and c=1 limits. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind warm standby. A warm spare has a nonzero dormant failure rate lower than or different from its active rate. In the illustrative case, Electronics may be energised at reduced load while waiting. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. One label cannot capture startup shocks or environmental ageing. Use separate dormant and active rates.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat continuous-time markov states as a model-checking problem rather than a vocabulary exercise. A Markov model represents active/spare combinations and transition rates between them. States can include both healthy, active failed with successful switch, and system failed. Recompute the example from the raw values and label every unit. Memoryless rates may not fit wear-out or scheduled tests. Write the generator matrix and states.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of state-probability equations becomes useful only when its mechanism is explicit. Kolmogorov forward equations track how probability flows among Markov states over time. Consider the worked illustration: The derivative of each state probability equals incoming flow minus outgoing flow. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Sign or indexing mistakes can violate total probability. Check that probabilities sum to one.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use dormant failure to practise sensitivity analysis. A spare can fail unseen while waiting, especially without diagnostics or proof tests. A warm spare with dormancy rate lambda_s may be unavailable when demanded. Before accepting the result, identify the quantity that would change the conclusion most. Assuming zero dormancy can overstate reliability. Include inspections and detection.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
A second primary source is https://ntrs.nasa.gov/citations/20020077965. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for failure detection. The system must recognise the active-unit failure quickly enough to initiate takeover. The initial example is: A sensor threshold and diagnostic delay affect coverage. Then create a nearby case where the usual shortcut becomes unreliable. False trips can cause unnecessary switching and wear. Model missed and false detections.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect switching delay to a real decision while protecting the boundary between calculation and judgement. The load may be unsupported during the time between active failure and spare delivery. A process with no buffering may fail during a two-second gap. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Instantaneous switching is an idealisation. Compare delay with system tolerance.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Shared power, environment, software or maintenance can defeat both active and standby units. A flood can disable duty and backup pumps together. This is the central mathematical mechanism in common-cause failure. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Adding a nominally independent spare does not remove shared causes. Diversify and quantify common causes. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind load sharing after takeover. The spare may see a different load after activation, changing its active failure rate. In the illustrative case, A backup generator may operate at full capacity rather than partial standby. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Using one lambda for all states can be wrong. Assign rates by operating state.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat repairable standby systems as a model-checking problem rather than a vocabulary exercise. If failed units can be repaired during the mission, availability and reliability answer different questions. Repair can restore redundancy before the next failure. Recompute the example from the raw values and label every unit. Availability credit is inappropriate for missions where any interruption is failure. Define the success criterion.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of multiple spares becomes useful only when its mechanism is explicit. With m ideal cold spares and exponential active lives, total life follows an Erlang distribution with shape m+1. Consider the worked illustration: Reliability is exp(-lambda t) times a finite Poisson sum. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Switches, logistics and dormant failure grow more complex with each spare. Validate the state model.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use sensitivity and design trade-offs to practise sensitivity analysis. Vary coverage, dormant rate and common-cause contribution to see which assumption controls benefit. A small coverage reduction can outweigh a low component failure rate. Before accepting the result, identify the quantity that would change the conclusion most. Local sensitivity does not include unmodelled failure modes. Review architecture and test evidence.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for practice activity. Calculate single-unit, hot-parallel and ideal cold-standby reliability at lambda t=0.2, then set coverage to 0.8. The initial example is: Predict which formula is most sensitive to coverage. Then create a nearby case where the usual shortcut becomes unreliable. The exercise excludes repair and common cause. Check boundary cases numerically.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect parent and educator guidance to a real decision while protecting the boundary between calculation and judgement. Standby reliability connects exponentials, conditional probability, state diagrams and engineering design. Ask why an unused spare can still fail to help. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Redundancy should be taught with switching and shared-cause limits. Start with a three-state diagram.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
A spare does not double reliability automatically, cold standby is not truly failure-free and hot redundancy is not always independent. Coverage includes detection, isolation and successful takeover. This is the central mathematical mechanism in common misconceptions. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Availability is not the same as uninterrupted mission reliability. Write the event sequence. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind when a simple markov model is inadequate. Nonexponential ageing, scheduled tests, deterministic delays, software logic and repair queues may need semi-Markov, Petri-net or simulation models. In the illustrative case, A battery spare loses capacity with calendar age rather than a constant memoryless rate. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Forcing exponentials can distort long missions. Choose the model from failure physics.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat research and next reading as a model-checking problem rather than a vocabulary exercise. NASA’s reliability-block-diagram report treats active and standby combinations with dormancy and switching, while NASA work on fault-tolerant control highlights coverage in Markov analysis. Read both with a modern system architecture and test plan. Recompute the example from the raw values and label every unit. Historical examples do not supply current failure rates. Benchmark formulas against simulation.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
Continue with the Mathematics Learning Hub and the previous reliability article. These links own broader probability ideas and the immediately preceding rare-event geometry.
The mathematics of frequently asked questions becomes useful only when its mechanism is explicit. FAQs should cover cold, warm and hot standby, coverage, dormancy, switching, common cause, repair and Markov states. Consider the worked illustration: Each answer should name the mission success rule. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Avoid promising reliability from the spare count alone. Publish the state diagram and rate assumptions.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Quick answers
What should I calculate first? Start with the estimand or failure event, then a simple baseline.
Does the advanced method guarantee accuracy? No. It needs input, numerical and model checks.
Which school mathematics matters? Algebra, graphs, probability, functions and careful units all contribute.
How can a student practise safely? Use fictional or open benchmark data and preserve every intermediate value.
What makes a career claim responsible? Describe how the mathematics is used without promising admission, salary or employment.
Extended practice laboratory
Practice case 1: Why a spare changes the system
Setup: Use this chapter’s illustrative case: A backup pump may start only when the duty pump trips.
Task: Draw active, spare and switch states.
Checking target: A spare that cannot detect, start or connect provides no protection.
Practice case 2: Cold standby
Setup: Use this chapter’s illustrative case: One active unit plus one perfect cold spare provides two sequential exponential lives.
Task: Treat ideal cold standby as a benchmark.
Checking target: Real dormant components can deteriorate or fail to start.
Practice case 3: Ideal two-unit formula
Setup: Use this chapter’s illustrative case: The total life is Erlang with shape two.
Task: Derive it by conditioning on first failure time.
Checking target: The formula assumes zero dormancy rate and perfect coverage.
Practice case 4: A cold-standby calculation
Setup: Use this chapter’s illustrative case: A single unit would have R≈0.90484.
Task: Show both benchmark values.
Checking target: The large gain is conditional on ideal switching and independence.
Practice case 5: Hot standby or active parallel
Setup: Use this chapter’s illustrative case: For independent identical units, R=1-[1-exp(-lambda t)]^2.
Task: Define whether load changes after failure.
Checking target: Shared load and common causes violate the independent formula.
Practice case 6: Comparing hot and cold
Setup: Use this chapter’s illustrative case: The cold spare avoids active ageing before demand.
Task: Compare with realistic coverage.
Checking target: Switching risk can reverse the comparison.
Practice case 7: Switching coverage
Setup: Use this chapter’s illustrative case: A switch may fail mechanically or the controller may misdiagnose the event.
Task: Define every required transition.
Checking target: Coverage is not merely switch hardware reliability.
Practice case 8: Imperfect-switch formula
Setup: Use this chapter’s illustrative case: With c=0.9 and lambda t=0.1, R≈0.98627.
Task: Check c=0 and c=1 limits.
Checking target: This compact formula still assumes no dormancy failure.
Practice case 9: Warm standby
Setup: Use this chapter’s illustrative case: Electronics may be energised at reduced load while waiting.
Task: Use separate dormant and active rates.
Checking target: One label cannot capture startup shocks or environmental ageing.
Practice case 10: Continuous-time Markov states
Setup: Use this chapter’s illustrative case: States can include both healthy, active failed with successful switch, and system failed.
Task: Write the generator matrix and states.
Checking target: Memoryless rates may not fit wear-out or scheduled tests.