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eduKateSG · Why Mathematics? · System Signatures in Reliability: Order Statistics, Coherent Structures, k-out-of-n Mixtures and Comparisons
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 becomes beautifully visible in a system signature. A coherent system can fail when the first, second or later component failure occurs, depending on its structure. Under independent and identically distributed continuous component lifetimes, the signature records the probability that each ordered component failure is the one that ends the system.
This guide connects Boolean structure functions, order statistics, k-out-of-n systems and mixture representations of system reliability. It shows how the signature separates design structure from the common component lifetime distribution, which makes comparisons elegant. It also explains the limits: non-identical components, dependence, repairs, load sharing and imperfect switching require extensions rather than casual use of the classical formula.
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 series system fails at the first component failure, while a parallel system survives until the last. Two architectures with the same parts can have very different failure-order behaviour. This is the central mathematical mechanism in why failure order matters. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Counting components alone ignores structure. Define the system success rule. 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 coherent systems. A coherent structure is nondecreasing in each component state and every component is relevant. In the illustrative case, Changing a working component to failed cannot improve system success. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Systems with antagonistic logic or irrelevant parts fall outside the simple definition. Check monotonicity and relevance.
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 structure function as a model-checking problem rather than a vocabulary exercise. A Boolean function phi(x1,…,xn) maps component working states to system success or failure. For a two-component series system, phi=x1 x2. Recompute the example from the raw values and label every unit. State variables at one time do not by themselves describe repair history. Write the truth table.
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 order statistics becomes useful only when its mechanism is explicit. Sort continuous component lifetimes as X_1:n<=…<=X_n:n. Consider the worked illustration: X_1:n is the first failure time and X_n:n the last. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Ties have probability zero only under a continuous model. State the lifetime 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.
Primary source: https://doi.org/10.1016/j.jmva.2011.05.007. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use signature definition to practise sensitivity analysis. The kth signature entry is s_k=P(T=X_k:n) under the classical i.i.d. continuous setting. It is the probability that the kth ordered component failure causes system failure. Before accepting the result, identify the quantity that would change the conclusion most. It is not the failure probability at clock time k. Keep order index and time separate.
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 signature sums to one. Exactly one ordered component failure ends a coherent nonrepairable system, so s_1+…+s_n=1. The initial example is: A candidate vector (0.2,0.5,0.3) passes the basic sum check. Then create a nearby case where the usual shortcut becomes unreliable. Nonnegative entries summing to one are not automatically a realizable system signature. Check structural feasibility.
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 series and parallel signatures to a real decision while protecting the boundary between calculation and judgement. An n-component series system has signature (1,0,…,0); a parallel system has (0,…,0,1). The first failure ends series and the last ends parallel. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. This assumes all components are relevant and nonrepairable. Use these as benchmark cases.
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 system that works when at least k components work fails at the (n-k+1)th component failure. A 2-out-of-3 system has signature (0,1,0). This is the central mathematical mechanism in k-out-of-n signature. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Voting coverage and common causes are not represented automatically. Match the physical architecture. 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 mixture representation. System survival can be written as the weighted sum of order-statistic survivals: P(T>t)=sum s_k P(X_k:n>t). In the illustrative case, The signature supplies structural weights and the component distribution supplies time behaviour. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The classical separation needs i.i.d. continuous lifetimes. Verify assumptions before mixing.
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 order-statistic survival as a model-checking problem rather than a vocabulary exercise. For component survival p at time t, P(X_k:n>t) is the probability that at least n-k+1 components survive t. It is a binomial tail under independent identical components. Recompute the example from the raw values and label every unit. Dependence changes the count distribution. Derive the required survivor count.
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 2-out-of-3 example becomes useful only when its mechanism is explicit. For component survival p, system reliability is 3p^2(1-p)+p^3=3p^2-2p^3. Consider the worked illustration: At p=0.9, reliability is 0.972. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The calculation assumes independence and perfect voting. Compare with direct enumeration.
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 bridge structures to practise sensitivity analysis. Non-series-parallel coherent systems can have several nonzero signature entries. Enumerating all component failure orders reveals which rank ends the system. Before accepting the result, identify the quantity that would change the conclusion most. Factorial growth makes brute-force enumeration expensive. Use symmetry and algorithms carefully.
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://doi.org/10.1002/(SICI)1520-6750(199908)46:5%3C507::AID-NAV4%3E3.0.CO;2-D. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for permutation interpretation. With i.i.d. continuous lifetimes, every component failure order is equally likely. The initial example is: Count the permutations in which system failure occurs at rank k and divide by n factorial. Then create a nearby case where the usual shortcut becomes unreliable. Non-identical lifetimes make orders unequal. Keep the counting argument explicit.
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 tail signatures to a real decision while protecting the boundary between calculation and judgement. Cumulative or tail signature quantities summarise the probability the system survives beyond a given number of component failures. They connect naturally to reliability polynomials and comparisons. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Different conventions exist for indexing. Define the convention before using formulas.
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.
Signature orders can support stochastic comparisons between coherent structures under shared component assumptions. A design weighted toward later failure ranks tends to outlast one weighted earlier under suitable conditions. This is the central mathematical mechanism in comparing designs. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. One signature comparison does not settle cost, repair or consequence. State the theorem’s conditions. 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 separating structure and lifetime. The same signature can be combined with different common component distributions. In the illustrative case, Swap an exponential component model for a Weibull model while retaining structural weights. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The signature does not validate either lifetime family. Check component data independently.
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 non-identical components as a model-checking problem rather than a vocabulary exercise. When components have different lifetime distributions, the classical signature no longer separates structure so simply. A stronger component placed in a critical position changes failure-order probabilities. Recompute the example from the raw values and label every unit. Pretending components are exchangeable can misrank designs. Use probability signatures or direct models.
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 dependent lifetimes becomes useful only when its mechanism is explicit. Shared environments and common causes make component states dependent. Consider the worked illustration: Exchangeability may preserve some representations, but independence formulas for order-statistic survival fail. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Equal marginals do not imply independence. Model the joint distribution.
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 repairs and dynamic systems to practise sensitivity analysis. Classical signatures describe nonrepairable coherent systems with fixed structure. A repairable standby system changes state and can switch configuration. Before accepting the result, identify the quantity that would change the conclusion most. A static signature may omit repair and coverage. Use Markov or regenerative models.
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. List all six failure orders for a three-component structure and mark which failure rank ends the system. The initial example is: Count each rank and form the signature. Then create a nearby case where the usual shortcut becomes unreliable. A classroom truth table should use a small coherent system. Check that entries sum to one.
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. System signatures connect permutations, probability vectors, Boolean logic, order statistics and design. Ask why identical parts can produce different system life when connected differently. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The abstraction should be built from a visible truth table. Begin with series, parallel and 2oo3.
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 signature is not a time-to-failure distribution, not a vector of component reliabilities and not valid unchanged for every dependence model. It captures structure under specific component assumptions. 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. Later failure rank does not automatically mean better real-world safety. Separate mathematical order from consequences. 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 signature analysis needs extension. Heterogeneous components, ties, load sharing, repair, switching and dynamic logic need generalized signatures or other models. In the illustrative case, Warm standby components age differently from active ones. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Forcing exchangeability can erase the key mechanism. Choose the representation from system 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. Marichal, Mathonet and Waldhauser analyse signature-based reliability expressions and exchangeability, while Kochar, Mukerjee and Samaniego develop system comparisons. Read the definitions and assumptions before the comparison theorems. Recompute the example from the raw values and label every unit. Elegant formulas do not remove data uncertainty. Reproduce a three-component case.
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 coherence, order statistics, s_k, series, parallel, k-out-of-n, i.i.d. assumptions and extensions. Consider the worked illustration: Each answer should distinguish structure from lifetime. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Avoid applying classical signatures to repaired systems without justification. Publish the truth table and counting steps.
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 failure order matters
Setup: Use this chapter’s illustrative case: Two architectures with the same parts can have very different failure-order behaviour.
Task: Define the system success rule.
Checking target: Counting components alone ignores structure.
Practice case 2: Coherent systems
Setup: Use this chapter’s illustrative case: Changing a working component to failed cannot improve system success.
Task: Check monotonicity and relevance.
Checking target: Systems with antagonistic logic or irrelevant parts fall outside the simple definition.
Practice case 3: Structure function
Setup: Use this chapter’s illustrative case: For a two-component series system, phi=x1 x2.
Task: Write the truth table.
Checking target: State variables at one time do not by themselves describe repair history.
Practice case 4: Order statistics
Setup: Use this chapter’s illustrative case: X_1:n is the first failure time and X_n:n the last.
Task: State the lifetime assumptions.
Checking target: Ties have probability zero only under a continuous model.
Practice case 5: Signature definition
Setup: Use this chapter’s illustrative case: It is the probability that the kth ordered component failure causes system failure.
Task: Keep order index and time separate.
Checking target: It is not the failure probability at clock time k.
Practice case 6: Signature sums to one
Setup: Use this chapter’s illustrative case: A candidate vector (0.2,0.5,0.3) passes the basic sum check.
Task: Check structural feasibility.
Checking target: Nonnegative entries summing to one are not automatically a realizable system signature.
Practice case 7: Series and parallel signatures
Setup: Use this chapter’s illustrative case: The first failure ends series and the last ends parallel.
Task: Use these as benchmark cases.
Checking target: This assumes all components are relevant and nonrepairable.
Practice case 8: k-out-of-n signature
Setup: Use this chapter’s illustrative case: A 2-out-of-3 system has signature (0,1,0).
Task: Match the physical architecture.
Checking target: Voting coverage and common causes are not represented automatically.
Practice case 9: Mixture representation
Setup: Use this chapter’s illustrative case: The signature supplies structural weights and the component distribution supplies time behaviour.
Task: Verify assumptions before mixing.
Checking target: The classical separation needs i.i.d. continuous lifetimes.
Practice case 10: Order-statistic survival
Setup: Use this chapter’s illustrative case: It is a binomial tail under independent identical components.
Task: Derive the required survivor count.
Checking target: Dependence changes the count distribution.