Understand Dispatch Analysis

HM / HMC: mechanical and hydromechanical failures

HM refers to the mechanical/hydromechanical contribution. ARP5107C denotes its loss-of-thrust-control failure rate by λ_HMC. It is the sum of the rates of single mechanical or hydromechanical faults that lead to loss of thrust control. The input is the assessed rate for those consequences, not the rate of every mechanical fault regardless of its effect. See the definitions following Equation 1, p. 23.

UC: undetected / uncovered electrical and electronic failures

UC refers to the undetected/uncovered electrical/electronic fault contribution. λ_UC is the rate of those faults assumed to lead to loss of thrust control under the assessed architecture. It is separate from λ_HMC and from the additional electronic exits entered for ST and LT fault conditions. The combined baseline is:

X: the multiplier used to estimate the uncovered contribution

X is a dimensionless uncovered-fault multiplier. It is a fraction, not a failure rate: X = 0.02 means two percent. In Equation 2, it multiplies the summed electrical/electronic failure rates associated with loss-of-thrust-control-critical elements in both channels. That produces a failure rate λ_UC in events per hour.

The standard permits that sum to be approximated by the total ST plus total LT occurrence rates. This is the aggregate approximation implemented by the calculator:

For example, the Section 7.1.5 inputs are X = 0.02, λ_STT = 3e-05 per hour, λ_LTT = 7e-05 per hour and λ_HMC = 4e-06 per hour. Therefore λ_UC = 0.02 × (3e-05 + 7e-05) = 2e-06 per hour, and λ_HM+UC = 4e-06 + 2e-06 = 6e-06 per hour.

Page 24 generally discusses X between 0 and 0.05, but does not assign the same uncovered percentage to every element. Its note explains that a detailed assessment normally sums uncovered rates element by element, such as the power supply, processor and sensors. The appropriate multiplier depends on architecture, coverage and channel operation. Thus 0.02 is an example value, not a universal coverage claim. X estimates UC; it does not scale the mechanical/hydromechanical rate.

In the generic Equation 20, λ_i means the occurrence rate into any one individual fault state. For a short-term fault it is λ_STi; for a long-term fault it is λ_LTi. The index i identifies an individual fault, not an average. The source also writes these as λ_ST(i) and λ_LT(i) in Equations 3, 8 and 9. Each ST sum and LT sum below runs over its own faults. λ_STT and λ_LTT are totals across the applicable faults in both channels; the final T in STT/LTT denotes the total, not a repair time.

ST and LT occurrence totals

The summation sign means add the displayed kind of term for every applicable individual fault. The subscript identifies the fault: ST1 is the first ST fault and LT1 is the first LT fault.

Short-term faults: total occurrence and weighted electronic exit

The occurrence rate tells you how often a particular ST fault is entered. Its electronic exit rate tells you how quickly an additional electronic failure can cause loss of thrust control while already in that fault state. Multiply these two rates for each ST fault, add the products, and divide by the total ST occurrence rate. Faults entered more often receive more weight. This is not an arithmetic average.

Long-term faults: total occurrence and weighted electronic exit

Repeat the same operation using LT faults only. The ST total is a sum of ST occurrence rates; the LT total is a sum of LT occurrence rates. The weighted ST and LT electronic exit rates describe operation after the first fault. They exclude HM + UC, which is accounted for separately.

Source-reading note: the printed Equation 8 numerator carries an LT label although its ST definition and explanatory prose identify the ST electronic exit. The ST formula above follows that definition and prose. This is an interpretation of the supplied text, not a publisher-issued correction.

The standard uses T_ST-REPAIR and T_LT-REPAIR for the short-term and long-term repair intervals in its time-weighted-average equations; Equation 20c shortens them to T_ST and T_LT. They refer to the same respective repair intervals. μ_ST and μ_LT are repair rates, equal to their reciprocals.

For each individual fault, the combined loss-of-thrust-control exit contains the additional electronic exit plus the common HM + UC contribution. The standard uses these combined-rate symbols in Equation 20c.

Individual-state formula in standard notation

In the generic Equation 20, λ_i means the occurrence rate into any one individual fault state. For a short-term fault it is λ_STi; for a long-term fault it is λ_LTi. The index i identifies an individual fault, not an average. The source also writes these as λ_ST(i) and λ_LT(i) in Equations 3, 8 and 9. Each ST sum and LT sum below runs over its own faults. λ_STT and λ_LTT are totals across the applicable faults in both channels; the final T in STT/LTT denotes the total, not a repair time.

Averaged ST and LT formula

Equation 18 is an averaged ST/LT state model. Equation 20 retains every individual state. Because each individual exit appears inside a denominator, a weighted average generally cannot reconstruct the individual Full Markov result. They coincide under a common repair interval when all individual electronic exits within each ST or LT category equal the corresponding representative exit. Otherwise the averaged model needs its own justification; the calculator does not infer the individual result from means alone.

Read the Full Markov derivation.

Individual-state expansion

In the generic Equation 20, λ_i means the occurrence rate into any one individual fault state. For a short-term fault it is λ_STi; for a long-term fault it is λ_LTi. The index i identifies an individual fault, not an average. The source also writes these as λ_ST(i) and λ_LT(i) in Equations 3, 8 and 9. Each ST sum and LT sum below runs over its own faults. λ_STT and λ_LTT are totals across the applicable faults in both channels; the final T in STT/LTT denotes the total, not a repair time.

ST and LT coefficient form

ST coefficients

LT coefficients

First-order expansion of the averaged ST/LT model

Equation 20c needs sums involving squared individual combined exit rates: a₁ and c₁. The ST and LT averages do not determine those sums. The first-order expansion of Equation 18 below therefore describes the averaged ST/LT model, not a reconstruction of arbitrary individual states. This distinction is about the individual rates and their weighted spread; no additional grouping symbols are needed.

Read the first-order derivation.

Individual ST and LT sums

Averaged ST and LT formula

For a common interval within ST and within LT, the occurrence-weighted averages reproduce these zero-order individual sums. This identity does not mean zero-order is always equal to Full Markov.

Read the zero-order derivation.

Individual ST and LT formula

Averaged ST and LT formula

Read the derivation from time accounting.

Single-fault model requirement: Section 7.2.4, p. 37, supports using a single-fault model when the repair rate for each ST and LT fault state is approximately ten times or more than the maximum relevant failure rate into or out of those states. Check ST and LT separately. Include further failures from a single-fault state into a two-fault operating state when establishing the maximum. This condition supports neglecting such additional operating states; it is not an unconditional claim that any single-fault model is correct.

At a chosen sensitivity maximum, a longer repair interval gives a lower repair-rate ratio. Check the entered intervals and the limiting sensitivity intervals separately. Changing flight duration does not change the implemented Markov rates or repair-rate ratios.

1. Identify the states and follow their arrows

   P_FU is the probability of being fully operational. P_STi and P_LTi are the probabilities of occupying individual short-term and long-term fault states. P_LOTC is the probability of occupying the loss-of-thrust-control state. The arrows show the failure and repair transitions between them.

   Blue arrows enter an ST or LT fault state from fully operational operation. Green arrows repair the fault and return to fully operational operation. Red arrows lead to loss of thrust control. The dashed feedback arrow closes the mathematical model; it is not an operational repair instruction.

   One ST fault and one LT fault are drawn for clarity. The ST and LT branches repeat for every entered individual fault; these two ellipses do not replace all faults with a single averaged state.

   λ_{STi-LOTC+HM+UC} =λ_STi-LOTC + λ_HM+UC
   λ_{LTi-LOTC+HM+UC} =λ_LTi-LOTC + λ_HM+UC

   For each individual fault, the combined loss-of-thrust-control exit contains the additional electronic exit plus the common HM + UC contribution. The standard uses these combined-rate symbols in Equation 20c.

2. Write probability flow as starting-state probability multiplied by transition rate

   Probability flow = probability of occupying the starting state × transition rate. For example, λ_STi P_FU enters ST fault i because its arrow starts at the fully operational state. Both outgoing ST arrows start at the ST fault state, so their flows use P_STi.

   Arrow	Probability flow	Meaning for the fault state
   FU to ST fault i	λSTi PFU	Incoming ST flow
   ST fault i to FU	μST PSTi	Outgoing ST repair flow
   ST fault i to LOTC	λSTi-LOTC+HM+UC PSTi	Outgoing ST loss-of-thrust-control flow
   FU to LT fault i	λLTi PFU	Incoming LT flow
   LT fault i to FU	μLT PLTi	Outgoing LT repair flow
   LT fault i to LOTC	λLTi-LOTC+HM+UC PLTi	Outgoing LT loss-of-thrust-control flow

   The same interpretation applies to LT. Add incoming flows and subtract outgoing flows to obtain the change of probability in each fault state:

   dP_STidt =λ_STi P_FU − μ_ST P_STi − λ_{STi-LOTC+HM+UC} P_STi
   dP_LTidt =λ_LTi P_FU − μ_LT P_LTi − λ_{LTi-LOTC+HM+UC} P_LTi

   The derivative means change in state probability per unit time. Collecting the two outgoing terms gives the repair-plus-combined-exit expression used below.

3. Identify the standard ST and LT transition symbols

   In the generic Equation 20, λ_i means the occurrence rate into any one individual fault state. For a short-term fault it is λ_STi; for a long-term fault it is λ_LTi. The index i identifies an individual fault, not an average. The source also writes these as λ_ST(i) and λ_LT(i) in Equations 3, 8 and 9. Each ST sum and LT sum below runs over its own faults. λ_STT and λ_LTT are totals across the applicable faults in both channels; the final T in STT/LTT denotes the total, not a repair time.

   λ_{STi-LOTC+HM+UC} =λ_STi-LOTC + λ_HM+UC
   λ_{LTi-LOTC+HM+UC} =λ_LTi-LOTC + λ_HM+UC

   For each individual fault, the combined loss-of-thrust-control exit contains the additional electronic exit plus the common HM + UC contribution. The standard uses these combined-rate symbols in Equation 20c.

   The standard uses T_ST-REPAIR and T_LT-REPAIR for the short-term and long-term repair intervals in its time-weighted-average equations; Equation 20c shortens them to T_ST and T_LT. They refer to the same respective repair intervals. μ_ST and μ_LT are repair rates, equal to their reciprocals.

   μ_ST =1T_ST-REPAIR
   μ_LT =1T_LT-REPAIR

   Use the fully operational state, individual ST states, individual LT states and the loss-of-thrust-control state. A closed-loop feedback arrow returns from loss of thrust control to fully operational operation; this is a mathematical closure, not an operational repair instruction. Sections 7.2.2.2-7.2.2.4, pp. 33-35, derive the closed-loop result.

4. Write probability-flow equations before taking the steady solution

   dP_STidt =λ_STi P_FU − (μ_ST + λ_{STi-LOTC+HM+UC}) P_STi
   dP_LTidt =λ_LTi P_FU − (μ_LT + λ_{LTi-LOTC+HM+UC}) P_LTi
   dP_FUdt =− (λ_HM+UC + ∑_i λ_STi + ∑_i λ_LTi) P_FU + ∑_i μ_ST P_STi + ∑_i μ_LT P_LTi + μ_FB P_LOTC
   dP_LOTCdt =λ_HM+UC P_FU + ∑_i λ_{STi-LOTC+HM+UC} P_STi + ∑_i λ_{LTi-LOTC+HM+UC} P_LTi − μ_FB P_LOTC

   Each derivative is incoming probability flow minus outgoing probability flow. The feedback rate μ_FB closes the mathematical model. Add all four kinds of equations: every transition cancels, giving conservation of total probability. For the steady closed-loop solution, set the derivatives to zero; the individual ST and LT equations then give the balances in the next step.

5. Take the steady-state limit, then rearrange each balance

   Equation 20 is the steady closed-loop result. At steady state the probabilities of occupying the states no longer change, so dPdt = 0. This does not mean failures or repairs have stopped: their probability flows continue and balance. Applying that condition to the ST equation gives zero equals inflow minus outflow. Move the outflow to the other side and divide by its total exit rate:

   (μ_ST + λ_{STi-LOTC+HM+UC}) P_STi =λ_STi P_FU

   The ST state-probability expression below follows from this rearrangement; the LT expression follows by exactly the same steps.

   0 =λ_STi P_FU − (μ_ST + λ_{STi-LOTC+HM+UC}) P_STi
   P_STi =λ_STi P_FUμ_ST + λ_{STi-LOTC+HM+UC}
   0 =λ_LTi P_FU − (μ_LT + λ_{LTi-LOTC+HM+UC}) P_LTi
   P_LTi =λ_LTi P_FUμ_LT + λ_{LTi-LOTC+HM+UC}

   Occurrence supplies probability into each fault state. Repair and the combined loss-of-thrust-control exit remove it. At steady state those flows balance.

6. Derive the loss-of-thrust-control rate from the hazard-rate definition

   For an absorbing failure state with no return transition, the probability of not having failed is 1 − P_LOTC. The probability entering the failure state per unit time is dP_LOTCdt. The conditional rate of failure is therefore:

   λ_LOTC =dP_LOTCdt1 − P_LOTCAbsorbing-state hazard definition

   Equation 13, p. 31, uses the more general definition for a repairable or feedback-closed LOTC state: incoming probability flow divided by the probability of not occupying that state. Its note explains that the incoming flow is not the same as the net derivative when probability also leaves LOTC.

   λ_LOTC =probability flow entering the LOTC state1 − P_LOTCARP5107C Equation 13
   dP_LOTCdt =incoming LOTC probability flow − μ_FB P_LOTC
   λ_LOTC =dP_LOTCdt + μ_FB P_LOTC1 − P_LOTCSame definition expressed using the net derivative and feedback outflow

   Thus the numerator in the next equation is obtained by adding all incoming arrows to LOTC: the direct HM + UC arrow from FU, the combined exits from individual ST faults, and the combined exits from individual LT faults. At steady state dP_LOTCdt = 0 because incoming and feedback flows balance; that does not make the incoming failure rate zero.

   λ_LOTC =λ_HM+UC P_FU + ∑_i λ_{STi-LOTC+HM+UC} P_STi + ∑_i λ_{LTi-LOTC+HM+UC} P_LTi1 − P_LOTCEquation 13 applied to individual ST/LT states
   1 − P_LOTC =P_FU + ∑_i P_STi + ∑_i P_LTi

   The numerator is probability flow into loss of thrust control. The denominator is the probability of not being in that state. For the repairable closed-loop failure state, do not replace the incoming flow by its net probability slope.

7. Substitute both sets of state balances

   Replace each individual fault-state probability by its expression above. Every numerator and denominator term contains the common factor P_FU. Divide both by that factor, producing:

   λ_LOTC =λ_HM+UC + ∑_i λ_STi λ_{STi-LOTC+HM+UC}μ_ST + λ_{STi-LOTC+HM+UC} + ∑_i λ_LTi λ_{LTi-LOTC+HM+UC}μ_LT + λ_{LTi-LOTC+HM+UC}1 + ∑_i λ_STiμ_ST + λ_{STi-LOTC+HM+UC} + ∑_i λ_LTiμ_LT + λ_{LTi-LOTC+HM+UC}Equation 20 expanded into ST and LT sums
   λ_LOTC =λ_HM+UC + ∑_i λ_i λ_i-LOTC+HM+UCμ_i + λ_i-LOTC+HM+UC1 + ∑_i λ_iμ_i + λ_i-LOTC+HM+UCEquation 20, p. 35

   The generic sum in Equation 20 is exactly the combined ST and LT sums. No feedback rate remains in this expression.

8. Derive the averaged ST/LT model separately

   Blue arrows enter an ST or LT fault state from fully operational operation. Green arrows repair the fault and return to fully operational operation. Red arrows lead to loss of thrust control. The dashed feedback arrow closes the mathematical model; it is not an operational repair instruction.

   Use the same inflow-minus-outflow rule, now with total occurrence into the representative ST state and total occurrence into the representative LT state:

   dP_ST / dt =λ_STT P_FU − (μ_ST + λ_ST-LOTC-AVE + λ_HM+UC) P_ST
   dP_LT / dt =λ_LTT P_FU − (μ_LT + λ_LT-LOTC-AVE + λ_HM+UC) P_LT

   Set these derivatives to zero for the steady solution. Move each outflow term to the other side, then divide by its repair-plus-combined-exit rate:

   P_ST =λ_STT P_FUμ_ST + λ_ST-LOTC-AVE + λ_HM+UC
   P_LT =λ_LTT P_FUμ_LT + λ_LT-LOTC-AVE + λ_HM+UC

   Use the same probability-flow definition with one representative ST state and one representative LT state. Substitute these two balances and divide numerator and denominator by P_FU:

   λ_LOTC =λ_HM+UC + λ_STT (λ_ST-LOTC-AVE + λ_HM+UC)μ_ST + λ_ST-LOTC-AVE + λ_HM+UC + λ_LTT (λ_LT-LOTC-AVE + λ_HM+UC)μ_LT + λ_LT-LOTC-AVE + λ_HM+UC1 + λ_STTμ_ST + λ_ST-LOTC-AVE + λ_HM+UC + λ_LTTμ_LT + λ_LT-LOTC-AVE + λ_HM+UCEquation 18, p. 34: ST and LT averaged-state model

   Equation 18 is an averaged ST/LT state model. Equation 20 retains every individual state. Because each individual exit appears inside a denominator, a weighted average generally cannot reconstruct the individual Full Markov result. They coincide under a common repair interval when all individual electronic exits within each ST or LT category equal the corresponding representative exit. Otherwise the averaged model needs its own justification; the calculator does not infer the individual result from means alone.

Single-fault model requirement: Section 7.2.4, p. 37, supports using a single-fault model when the repair rate for each ST and LT fault state is approximately ten times or more than the maximum relevant failure rate into or out of those states. Check ST and LT separately. Include further failures from a single-fault state into a two-fault operating state when establishing the maximum. This condition supports neglecting such additional operating states; it is not an unconditional claim that any single-fault model is correct.

1. Start from Equation 20, retaining the individual fault terms

   Equation 20 comes from the continuous-time Markov inflow-minus-outflow equations and the steady closed-loop balance. Read that derivation from state probabilities and transition flows before applying the approximation below.

   λ_LOTC =λ_HM+UC + ∑_i λ_i λ_i-LOTC+HM+UCμ_i + λ_i-LOTC+HM+UC1 + ∑_i λ_iμ_i + λ_i-LOTC+HM+UCEquation 20, p. 35

   In the generic Equation 20, λ_i means the occurrence rate into any one individual fault state. For a short-term fault it is λ_STi; for a long-term fault it is λ_LTi. The index i identifies an individual fault, not an average. The source also writes these as λ_ST(i) and λ_LT(i) in Equations 3, 8 and 9. Each ST sum and LT sum below runs over its own faults. λ_STT and λ_LTT are totals across the applicable faults in both channels; the final T in STT/LTT denotes the total, not a repair time.

   λ_iμ_i + λ_i-LOTC+HM+UC =λ_i T_i1 + λ_i-LOTC+HM+UC T_i

   Multiply numerator and denominator of each state term by its repair interval. The printed Equation 20a denominator omits this repair-interval factor in its numerator; the consistent form above follows directly from Equation 20. This is an algebraic interpretation, not a publisher-issued correction.

2. Expand the reciprocal, showing the remainder

   11 + λ_i-LOTC+HM+UC T_i =1 − λ_i-LOTC+HM+UC T_i + (λ_i-LOTC+HM+UC T_i)²1 + λ_i-LOTC+HM+UC T_i

   First-order retains the constant and linear terms and drops the displayed remainder. Apply this replacement in both the numerator state sum and the normalization denominator. Small combined-exit-rate times repair interval makes the omitted contribution small.

3. Substitute the expansion in both sums

   λ_LOTC ≈λ_HM+UC + ∑_i [λ_i λ_i-LOTC+HM+UC T_i − λ_i λ_i-LOTC+HM+UC² T_i²]1 + ∑_i [λ_i T_i − λ_i λ_i-LOTC+HM+UC T_i²]Equation 20b, p. 36

   The numerator correction contains the square of the individual combined exit and the square of its interval. The denominator correction contains one combined exit factor and the square of its interval.

4. Separate ST and LT and calculate the standard constants

   ST coefficients

   a₀ =∑_i λ_STi λ_{STi-LOTC+HM+UC}
   a₁ =∑_i λ_STi λ_{STi-LOTC+HM+UC}²
   b₀ =∑_i λ_STi
   b₁ =a₀

   LT coefficients

   c₀ =∑_i λ_LTi λ_{LTi-LOTC+HM+UC}
   c₁ =∑_i λ_LTi λ_{LTi-LOTC+HM+UC}²
   d₀ =∑_i λ_LTi
   d₁ =c₀

   All ST states use the ST interval and all LT states use the LT interval. Move those interval factors outside the respective sums, leaving the constants above:

   λ_LOTC ≈λ_HM+UC + a₀ T_ST − a₁ T_ST² + c₀ T_LT − c₁ T_LT²1 + b₀ T_ST − b₁ T_ST² + d₀ T_LT − d₁ T_LT²Equation 20c, p. 36

   This step only rearranges the first-order expression. The constants contain failure-rate data only: calculate them once when that data is fixed, then evaluate new ST/LT intervals without repeating the individual sums. This is the computational benefit discussed in Section 7.3.

5. Derive the averaged ST/LT version without pretending it is the same individual model

   λ_LOTC =λ_HM+UC + λ_STT (λ_ST-LOTC-AVE + λ_HM+UC)μ_ST + λ_ST-LOTC-AVE + λ_HM+UC + λ_LTT (λ_LT-LOTC-AVE + λ_HM+UC)μ_LT + λ_LT-LOTC-AVE + λ_HM+UC1 + λ_STTμ_ST + λ_ST-LOTC-AVE + λ_HM+UC + λ_LTTμ_LT + λ_LT-LOTC-AVE + λ_HM+UCEquation 18, p. 34: ST and LT averaged-state model

   For ST, replace the reciprocal of [1 + (λ_ST-LOTC-AVE + λ_HM+UC) T_ST] by 1 − (λ_ST-LOTC-AVE + λ_HM+UC) T_ST. For LT, make the corresponding replacement using λ_LT-LOTC-AVE and T_LT. Substitute into both numerator and denominator:

   λ_LOTC ≈λ_HM+UC + λ_STT (λ_ST-LOTC-AVE + λ_HM+UC) T_ST − λ_STT (λ_ST-LOTC-AVE + λ_HM+UC)² T_ST² + λ_LTT (λ_LT-LOTC-AVE + λ_HM+UC) T_LT − λ_LTT (λ_LT-LOTC-AVE + λ_HM+UC)² T_LT²1 + λ_STT T_ST − λ_STT (λ_ST-LOTC-AVE + λ_HM+UC) T_ST² + λ_LTT T_LT − λ_LTT (λ_LT-LOTC-AVE + λ_HM+UC) T_LT²First-order expansion of Equation 18; derived, not a new numbered source equation

   Equation 20c needs sums involving squared individual combined exit rates: a₁ and c₁. The ST and LT averages do not determine those sums. The first-order expansion of Equation 18 below therefore describes the averaged ST/LT model, not a reconstruction of arbitrary individual states. This distinction is about the individual rates and their weighted spread; no additional grouping symbols are needed.

Single-fault model requirement: Section 7.2.4, p. 37, supports using a single-fault model when the repair rate for each ST and LT fault state is approximately ten times or more than the maximum relevant failure rate into or out of those states. Check ST and LT separately. Include further failures from a single-fault state into a two-fault operating state when establishing the maximum. This condition supports neglecting such additional operating states; it is not an unconditional claim that any single-fault model is correct.

Single-fault model requirement: Section 7.2.4, p. 37, supports using a single-fault model when the repair rate for each ST and LT fault state is approximately ten times or more than the maximum relevant failure rate into or out of those states. Check ST and LT separately. Include further failures from a single-fault state into a two-fault operating state when establishing the maximum. This condition supports neglecting such additional operating states; it is not an unconditional claim that any single-fault model is correct.

1. Start with fully operational exposure, without choosing an observation duration

   Consider the total operating time. A fraction Fract(FU) of that time is spent fully operational. The expected number of first ST or LT faults is based on that fully operational exposure. No particular observation duration or choice of time unit is needed for this algebra.

   Fully operational exposure =Fract(FU) × total operating time

2. Derive the short-term fraction from occurrence and repair interval

   Expected number of ST fault entries =λ_STT × Fract(FU) × total operating time
   Expected time operating with ST faults =λ_STT × Fract(FU) × total operating time × T_ST-REPAIR

   Divide the time operating with ST faults by the total operating time. The same total time appears in numerator and denominator and cancels:

   Fract(ST) =λ_STT × Fract(FU) × total operating time × T_ST-REPAIRtotal operating time
   Fract(ST) =Fract(FU) λ_STT T_ST-REPAIREquation 4a, p. 24

   Thus the ST time fraction is the fully operational time fraction multiplied by the ST occurrence rate and its repair interval. Rate multiplied by interval is dimensionless when their time units are consistent.

3. Derive the long-term fraction in the same way

   Expected number of LT fault entries =λ_LTT × Fract(FU) × total operating time
   Expected time operating with LT faults =λ_LTT × Fract(FU) × total operating time × T_LT-REPAIR
   Fract(LT) =λ_LTT × Fract(FU) × total operating time × T_LT-REPAIRtotal operating time
   Fract(LT) =Fract(FU) λ_LTT T_LT-REPAIREquation 4b, p. 25

4. Apply conservation of the whole operating record

   Fract(FU) + Fract(ST) + Fract(LT) =1Equation 5

   Substitute the fractions derived from the occurrence counting, rather than assuming that fully operational operation fills the whole record:

   Fract(FU) + Fract(FU) λ_STT T_ST-REPAIR + Fract(FU) λ_LTT T_LT-REPAIR =1
   Fract(FU) [1 + λ_STT T_ST-REPAIR + λ_LTT T_LT-REPAIR] =1

   Now divide by the bracket to solve for Fract(FU). Substitute that result back into Equations 4a and 4b to obtain Fract(ST) and Fract(LT):

   Fract(FU) =11 + λ_STT T_ST-REPAIR + λ_LTT T_LT-REPAIREquation 6a
   Fract(ST) =λ_STT T_ST-REPAIR1 + λ_STT T_ST-REPAIR + λ_LTT T_LT-REPAIREquation 6b
   Fract(LT) =λ_LTT T_LT-REPAIR1 + λ_STT T_ST-REPAIR + λ_LTT T_LT-REPAIREquation 6c

   Adding these three fractions returns one because their numerators add to the common denominator. The fully operational exposure adjustment is the reason these are called the balanced fractions.

5. Derive the individual ST and LT time contributions before averaging

   The same counting works for one particular fault. Replace the ST total occurrence by its individual occurrence, and do the corresponding operation for an LT fault:

   Fraction of the record in ST fault i =Fract(FU) λ_STi T_ST-REPAIR
   Fraction of the record in LT fault i =Fract(FU) λ_LTi T_LT-REPAIR

   Multiply each individual fraction by its own electronic loss-of-thrust-control exit rate and sum the ST contributions and LT contributions separately. Adding the fully operational electronic contribution and the HM + UC baseline gives:

   λ_LOTC-AVE = λ_HM+UC +λ_FU-LOTC + T_ST-REPAIR ∑_i λ_STi λ_STi-LOTC + T_LT-REPAIR ∑_i λ_LTi λ_LTi-LOTC1 + T_ST-REPAIR ∑_i λ_STi + T_LT-REPAIR ∑_i λ_LTiEquation 11 with individual ST and LT sums substituted

6. Derive the ST and LT electronic averages from those weighted sums

   Short-term faults: total occurrence and weighted electronic exit

   λ_ST-LOTC-AVE =∑_i λ_STi λ_STi-LOTC∑_i λ_STiEquation 8 definition and prose, p. 26

   The occurrence rate tells you how often a particular ST fault is entered. Its electronic exit rate tells you how quickly an additional electronic failure can cause loss of thrust control while already in that fault state. Multiply these two rates for each ST fault, add the products, and divide by the total ST occurrence rate. Faults entered more often receive more weight. This is not an arithmetic average.

   Long-term faults: total occurrence and weighted electronic exit

   λ_LT-LOTC-AVE =∑_i λ_LTi λ_LTi-LOTC∑_i λ_LTiEquation 9, p. 26

   Repeat the same operation using LT faults only. The ST total is a sum of ST occurrence rates; the LT total is a sum of LT occurrence rates. The weighted ST and LT electronic exit rates describe operation after the first fault. They exclude HM + UC, which is accounted for separately.

   Source-reading note: the printed Equation 8 numerator carries an LT label although its ST definition and explanatory prose identify the ST electronic exit. The ST formula above follows that definition and prose. This is an interpretation of the supplied text, not a publisher-issued correction.

   Each ST electronic numerator sum equals λ_STT λ_ST-LOTC-AVE. Each LT electronic numerator sum equals λ_LTT λ_LT-LOTC-AVE. Substitute those two identities into the individual expression:

   λ_LOTC-AVE = λ_HM+UC +λ_FU-LOTC + (λ_STT T_ST-REPAIR) λ_ST-LOTC-AVE + (λ_LTT T_LT-REPAIR) λ_LT-LOTC-AVE1 + λ_STT T_ST-REPAIR + λ_LTT T_LT-REPAIREquation 11, p. 26

   This is Equation 11. It shows why occurrence weighting is needed and why the same ST interval and the same LT interval are assumed within their respective categories.

7. Explain and include the fully operational electronic term

   Section 7.1.1 considers two independent electronic failures during one flight, one affecting each redundant channel. For small channel-rate-times-flight-duration products, Equation 7 gives a conservative upper bound, using half the both-channel total as the channel-rate estimate:

   λ_FU-LOTC ≤[λ_STT + λ_LTT2]² T_FLEquation 7, p. 25: conservative upper bound

   The bound assumes channel faults conservatively count as channel loss. It is not an equality valid for every architecture. The calculator uses the right-hand side as its conservative fully operational electronic estimate. This term is multiplied by Fract(FU), so it enters the same balanced fraction as the other electronic contributions.

8. Explain where HM + UC enters, and assemble the result

   HM / HMC: mechanical and hydromechanical failures

   HM refers to the mechanical/hydromechanical contribution. ARP5107C denotes its loss-of-thrust-control failure rate by λ_HMC. It is the sum of the rates of single mechanical or hydromechanical faults that lead to loss of thrust control. The input is the assessed rate for those consequences, not the rate of every mechanical fault regardless of its effect. See the definitions following Equation 1, p. 23.

   UC: undetected / uncovered electrical and electronic failures

   UC refers to the undetected/uncovered electrical/electronic fault contribution. λ_UC is the rate of those faults assumed to lead to loss of thrust control under the assessed architecture. It is separate from λ_HMC and from the additional electronic exits entered for ST and LT fault conditions. The combined baseline is:

   λ_HM+UC =λ_HMC + λ_UCDefinitions following Equation 1, p. 23

   X: the multiplier used to estimate the uncovered contribution

   X is a dimensionless uncovered-fault multiplier. It is a fraction, not a failure rate: X = 0.02 means two percent. In Equation 2, it multiplies the summed electrical/electronic failure rates associated with loss-of-thrust-control-critical elements in both channels. That produces a failure rate λ_UC in events per hour.

   λ_UC =X × [sum of both-channel electrical/electronic failure rates associated with loss-of-thrust-control-critical elements]Equation 2, p. 24

   The standard permits that sum to be approximated by the total ST plus total LT occurrence rates. This is the aggregate approximation implemented by the calculator:

   λ_UC ≈X (λ_STT + λ_LTT)
   λ_HM+UC ≈λ_HMC + X (λ_STT + λ_LTT)

   For example, the Section 7.1.5 inputs are X = 0.02, λ_STT = 3e-05 per hour, λ_LTT = 7e-05 per hour and λ_HMC = 4e-06 per hour. Therefore λ_UC = 0.02 × (3e-05 + 7e-05) = 2e-06 per hour, and λ_HM+UC = 4e-06 + 2e-06 = 6e-06 per hour.

   Page 24 generally discusses X between 0 and 0.05, but does not assign the same uncovered percentage to every element. Its note explains that a detailed assessment normally sums uncovered rates element by element, such as the power supply, processor and sensors. The appropriate multiplier depends on architecture, coverage and channel operation. Thus 0.02 is an example value, not a universal coverage claim. X estimates UC; it does not scale the mechanical/hydromechanical rate.

   The mechanical/hydromechanical and uncovered contributions are common to the operating conditions. Weighting a common baseline by fractions that sum to one leaves that baseline once. Hence it is outside the electronic fraction in Equation 11:

   λ_LOTC-AVE = λ_HM+UC +λ_FU-LOTC + (λ_STT T_ST-REPAIR) λ_ST-LOTC-AVE + (λ_LTT T_LT-REPAIR) λ_LT-LOTC-AVE1 + λ_STT T_ST-REPAIR + λ_LTT T_LT-REPAIREquation 11, p. 26

   Do not add the same baseline again inside the electronic-only ST/LT exits. Do not add the separate fully operational bound to the Markov equations: their model expressions are different.

Single-fault model requirement: Section 7.2.4, p. 37, supports using a single-fault model when the repair rate for each ST and LT fault state is approximately ten times or more than the maximum relevant failure rate into or out of those states. Check ST and LT separately. Include further failures from a single-fault state into a two-fault operating state when establishing the maximum. This condition supports neglecting such additional operating states; it is not an unconditional claim that any single-fault model is correct.