---
title: "Non-coherent systems: cold standby"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Non-coherent systems: cold standby}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

```{r setup}
library(dist.structure)
library(algebraic.dist)
```

## Why "non-coherent"

The structure-function framework assumes a monotone binary function
`phi: {0, 1}^m -> {0, 1}` that maps per-component states to a system
state. This captures a huge class of reliability topologies but not
everything practitioners build. Cold standby is a canonical example.

A **cold standby** system has m components but only one is active at a
time. When the active component fails, a dormant spare instantaneously
takes over. System lifetime is the sum of component lifetimes, not an
order statistic.

The topology is temporal succession, not a structure function on
states. So cold standby:

- is a valid distribution object (you can sample, compute survival, ask
  for mean),
- is NOT a `dist_structure`: it has no `phi`, no `min_paths`, no
  signature, no Birnbaum importance.

dist.structure provides `cold_standby_dist` for this case, with its
own class chain that deliberately does not inherit `dist_structure`.

## Constructing a cold standby

```{r}
sys <- cold_standby_dist(list(
  exponential(1),
  exponential(2),
  exponential(0.5)
))
class(sys)
dist.structure::is_dist_structure(sys)   # FALSE -- not a coherent system
```

## Mean and sampling: exact

`mean` is exact when every component has an exact mean method: it sums
the component means.

```{r}
mean(sys)                              # 1/1 + 1/2 + 1/0.5 = 3.5
```

The sampler is also exact: independent samples from each component,
summed.

```{r}
set.seed(1)
x <- algebraic.dist::sampler(sys)(5000)
mean(x)                                # empirical ~ 3.5
```

## Survival and CDF: Monte Carlo with caching

For general components, the density of a sum-of-independent-RVs has no
closed form (it's a convolution). dist.structure computes `surv` and
`cdf` via Monte Carlo sampling. The closure caches the samples on the
first call, so repeated evaluations at different `t` values are
deterministic given the same `mc`:

```{r}
set.seed(42)
S <- algebraic.dist::surv(sys)
S(c(1, 2, 3, 5), mc = 5000)           # survival at multiple t values
```

A different `mc` triggers a fresh draw:

```{r}
S(2, mc = 5000)                        # uses the same cache
S(2, mc = 10000)                       # new cache with 10000 samples
```

Because the cache lives inside the closure, using the same `S` closure
keeps results consistent. `cdf` has its own closure and its own
cache: `cdf(sys)(t) + surv(sys)(t)` computed independently is not
exactly 1. If you need that identity, use one `surv` closure and
compute the CDF from it:

```{r}
S <- algebraic.dist::surv(sys)
F_t <- function(t) 1 - S(t)
F_t(2) + S(2)                          # exactly 1
```

For reproducibility across separate closure constructions, seed before
each one:

```{r}
set.seed(1); S1 <- algebraic.dist::surv(sys); s1 <- S1(2)
set.seed(1); S2 <- algebraic.dist::surv(sys); s2 <- S2(2)
all.equal(s1, s2)
```

## Specialized closed-form case: iid exponentials

A cold standby of m iid `Exp(rate)` components has system lifetime
`Gamma(shape = m, rate = rate)`:

```{r}
m <- 4; rate <- 2
sys <- cold_standby_dist(replicate(m, exponential(rate), simplify = FALSE))
# Monte Carlo survival at t = 3
set.seed(1)
mc_est <- algebraic.dist::surv(sys)(3, mc = 10000)
# Exact via Gamma
exact <- pgamma(3, shape = m, rate = rate, lower.tail = FALSE)
cat(sprintf("MC: %.4f, Exact: %.4f\n", mc_est, exact))
```

If you hit this case often, you can avoid the Monte Carlo by
constructing an `algebraic.dist::gamma_dist` directly.

## What cold standby does NOT support

Generics requiring topology or component-level `density`/`sup` fail
cleanly:

```{r, error = TRUE}
dist.structure::phi(sys, c(1, 1, 1, 1))
```

```{r, error = TRUE}
dist.structure::min_paths(sys)
```

Methods inherited from `algebraic.dist::univariate_dist` that require
`density` or `sup` (notably `vcov`, `expectation`) are also not
supported by default. A specialized subclass with a closed-form
aggregate (e.g., iid exponential -> Gamma) can provide them by
overriding.

## When to reach for cold standby

- Sequential redundancy with dormant spares (perfect switching, zero
  standby hazard).
- Modelling sum-of-lifetimes directly as a distribution.
- Monte Carlo as a first-pass answer when no closed form is available.

When you need the full dist.structure protocol (topology, importance,
composition), cold standby is the wrong tool; use a coherent system
instead.