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Advanced Series System Composition

Mixed Component Types

Real systems often have failure modes with different aging characteristics. Consider a server with:

library(serieshaz)

disk <- dfr_weibull(shape = 2, scale = 500)
mem  <- dfr_exponential(0.001)
psu  <- dfr_gompertz(a = 0.0001, b = 0.02)

server <- dfr_dist_series(list(disk, mem, psu))

Hazard Decomposition

Each component contributes differently to system risk over time:

h_disk <- component_hazard(server, 1)
h_mem  <- component_hazard(server, 2)
h_psu  <- component_hazard(server, 3)
h_sys  <- hazard(server)

t_grid <- seq(10, 300, by = 10)
cat("Time | Disk     | Memory   | PSU      | System\n")
#> Time | Disk     | Memory   | PSU      | System
cat("-----|----------|----------|----------|--------\n")
#> -----|----------|----------|----------|--------
for (t0 in c(50, 100, 150, 200, 250)) {
    cat(sprintf("%4d | %.6f | %.6f | %.6f | %.6f\n",
                t0, h_disk(t0), h_mem(t0), h_psu(t0), h_sys(t0)))
}
#>   50 | 0.000400 | 0.001000 | 0.000272 | 0.001672
#>  100 | 0.000800 | 0.001000 | 0.000739 | 0.002539
#>  150 | 0.001200 | 0.001000 | 0.002009 | 0.004209
#>  200 | 0.001600 | 0.001000 | 0.005460 | 0.008060
#>  250 | 0.002000 | 0.001000 | 0.014841 | 0.017841

At \(t = 50\), the constant memory hazard (\(0.001\)) is the largest contributor — the Weibull (\(0.0004\)) and Gompertz (\(0.0003\)) are still small. Memory remains dominant through \(t = 100\), but the Gompertz’s exponential acceleration (\(\propto e^{bt}\)) overtakes it by around \(t = 125\text{--}150\). By \(t = 250\), the Gompertz contributes \(0.015\) vs memory’s constant \(0.001\), dominating system risk. This crossover pattern is characteristic of mixed-type series systems: each failure mode has its “era” of dominance.

Nested Series Systems

Since a dfr_dist_series is itself a dfr_dist, it can be used as a component in another series system. This enables hierarchical modeling.

Example: Subsystem Composition

# CPU subsystem: two exponential failure modes
cpu <- dfr_dist_series(list(
    dfr_exponential(0.001),  # thermal failure
    dfr_exponential(0.0005)  # electrical failure
))

# Storage subsystem: Weibull wear + exponential shock
storage <- dfr_dist_series(list(
    dfr_weibull(shape = 2, scale = 1000),
    dfr_exponential(0.0002)
))

# Full system: CPU + storage + power supply
full <- dfr_dist_series(list(
    cpu,
    storage,
    dfr_gompertz(a = 0.0001, b = 0.01)
))

cat("Full system components:", ncomponents(full), "\n")
#> Full system components: 3
cat("Total parameters:", length(params(full)), "\n")
#> Total parameters: 7

Verifying Nested Hazard

The nested system’s hazard should equal the sum of all leaf-component hazards:

h_full <- hazard(full)

# Manually sum all leaf hazards
h_cpu1 <- component_hazard(cpu, 1)
h_cpu2 <- component_hazard(cpu, 2)
h_stor1 <- component_hazard(storage, 1)
h_stor2 <- component_hazard(storage, 2)
h_psu <- component_hazard(full, 3)

t0 <- 100
nested_sum <- h_cpu1(t0) + h_cpu2(t0) + h_stor1(t0) + h_stor2(t0) + h_psu(t0)
system_h <- h_full(t0)

cat(sprintf("Sum of leaf hazards: %.8f\n", nested_sum))
#> Sum of leaf hazards: 0.00217183
cat(sprintf("System hazard:       %.8f\n", system_h))
#> System hazard:       0.00217183
cat(sprintf("Difference:          %.2e\n", abs(nested_sum - system_h)))
#> Difference:          4.34e-19

Custom Components

The dfr_dist() constructor lets you define components with arbitrary hazard functions. These can be used in series systems alongside the built-in distributions.

Bathtub Curve Component

A bathtub-shaped hazard (high early, low middle, high late) can model products with infant mortality and wear-out:

# Bathtub: h(t) = a/t^b + c*t^d  (infant mortality + wear-out)
bathtub <- dfr_dist(
    rate = function(t, par, ...) {
        a <- par[1]; b <- par[2]; c <- par[3]; d <- par[4]
        a * t^(-b) + c * t^d
    },
    par = c(0.5, 0.5, 0.0001, 2)
)

# Series system with bathtub + exponential random shock
sys <- dfr_dist_series(list(bathtub, dfr_exponential(0.001)))

h_sys <- hazard(sys)
for (t0 in c(1, 10, 50, 100, 200)) {
    cat(sprintf("t=%3d: h_sys = %.6f\n", t0, h_sys(t0)))
}
#> t=  1: h_sys = 0.501100
#> t= 10: h_sys = 0.169114
#> t= 50: h_sys = 0.321711
#> t=100: h_sys = 1.051000
#> t=200: h_sys = 4.036355

The bathtub shape is visible: high hazard at \(t = 1\) (\(\approx 0.50\), dominated by the infant mortality term \(a/t^b\)), dropping to a minimum at \(t = 10\) (\(\approx 0.17\)), then rising steeply as the wear-out term \(c \cdot t^d\) kicks in (\(\approx 1.05\) at \(t = 100\), \(\approx 4.04\) at \(t = 200\)). The constant exponential shock (\(0.001\)) adds a negligible baseline floor.

Effect on Cumulative Hazard

When a custom component doesn’t provide cum_haz_rate, the series system falls back to numerical integration:

cat("Analytical cum_haz available:", !is.null(sys$cum_haz_rate), "\n")
#> Analytical cum_haz available: FALSE

# The system still computes survival correctly via numerical integration
S <- surv(sys)
for (t0 in c(10, 50, 100)) {
    cat(sprintf("t=%3d: S(t) = %.6f\n", t0, S(t0)))
}
#> t= 10: S(t) = 0.040534
#> t= 50: S(t) = 0.000013
#> t=100: S(t) = 0.000000

With this bathtub parameterization, the high infant mortality (\(a = 0.5\), \(b = 0.5\)) drives the cumulative hazard so high that survival drops below 5% by \(t = 10\). In practice, you would calibrate the bathtub parameters to match observed failure patterns. The numerical integration fallback handles these steep hazard functions correctly, just more slowly than the analytical path.

Failure Attribution

sample_components() generates component-level lifetimes, enabling identification of which component caused each system failure.

server <- dfr_dist_series(list(
    dfr_weibull(shape = 2, scale = 200),     # mechanical wear
    dfr_exponential(0.005),            # random shock
    dfr_gompertz(a = 0.0001, b = 0.02)       # degradation
))

set.seed(42)
n <- 10000
mat <- sample_components(server, n = n)

# System lifetime = minimum across components
t_sys <- apply(mat, 1, min)

# Identify the failing component for each observation
failing <- apply(mat, 1, which.min)

Attribution Proportions

props <- table(failing) / n
names(props) <- c("Weibull (wear)", "Exp (shock)", "Gompertz (degrad.)")
cat("Failure attribution:\n")
#> Failure attribution:
print(round(props, 3))
#>     Weibull (wear)        Exp (shock) Gompertz (degrad.) 
#>              0.376              0.512              0.112

Attribution by Time Window

Failure causes change over the system’s lifetime:

# Split into early, middle, and late failures
breaks <- quantile(t_sys, probs = c(0, 1/3, 2/3, 1))
period <- cut(t_sys, breaks = breaks, labels = c("Early", "Middle", "Late"),
              include.lowest = TRUE)

cat("\nAttribution by time period:\n")
#> 
#> Attribution by time period:
for (p in levels(period)) {
    idx <- which(period == p)
    props_p <- table(factor(failing[idx], levels = 1:3)) / length(idx)
    cat(sprintf("  %s (n=%d): Wear=%.1f%% Shock=%.1f%% Degrad=%.1f%%\n",
                p, length(idx),
                100 * props_p[1], 100 * props_p[2], 100 * props_p[3]))
}
#>   Early (n=3334): Wear=20.8% Shock=75.9% Degrad=3.2%
#>   Middle (n=3333): Wear=44.7% Shock=48.9% Degrad=6.3%
#>   Late (n=3333): Wear=47.3% Shock=28.7% Degrad=24.0%

Parameter Sensitivity

The parameter layout enables systematic sensitivity analysis: vary one component’s parameters while holding others fixed.

server <- dfr_dist_series(list(
    dfr_weibull(shape = 2, scale = 100),
    dfr_exponential(0.01)
))

S_fn <- surv(server)
base_par <- params(server)  # c(2, 100, 0.01)
t0 <- 50

cat("Sensitivity of S(50) to Weibull scale parameter:\n")
#> Sensitivity of S(50) to Weibull scale parameter:
for (scale_val in c(50, 75, 100, 125, 150)) {
    par_mod <- base_par
    par_mod[2] <- scale_val  # layout tells us par[2] is Weibull scale
    cat(sprintf("  scale = %3d: S(50) = %.4f\n", scale_val, S_fn(t0, par = par_mod)))
}
#>   scale =  50: S(50) = 0.2231
#>   scale =  75: S(50) = 0.3889
#>   scale = 100: S(50) = 0.4724
#>   scale = 125: S(50) = 0.5169
#>   scale = 150: S(50) = 0.5427

cat("\nSensitivity of S(50) to exponential rate:\n")
#> 
#> Sensitivity of S(50) to exponential rate:
for (rate_val in c(0.005, 0.01, 0.02, 0.05, 0.1)) {
    par_mod <- base_par
    par_mod[3] <- rate_val  # layout tells us par[3] is exp rate
    cat(sprintf("  rate = %.3f: S(50) = %.4f\n", rate_val, S_fn(t0, par = par_mod)))
}
#>   rate = 0.005: S(50) = 0.6065
#>   rate = 0.010: S(50) = 0.4724
#>   rate = 0.020: S(50) = 0.2865
#>   rate = 0.050: S(50) = 0.0639
#>   rate = 0.100: S(50) = 0.0052

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