A primary life-support unit (a CO₂ scrubber, an O₂-generation cartridge) fails after a random time drawn from an exponential distribution with rate λ — the memoryless failure model standard in reliability engineering. A single cold standby sits idle until the primary fails, at which point an automatic switchover attempts to bring it online; the switchover itself succeeds only with probability p — real hardware (a stuck relay, a valve that won't seat) can fail to engage. If it succeeds, the standby then runs on its own exponential clock (rate λ) until it, too, fails. The closed-form probability the system is still delivering life support at mission time t is:
R(t) = e^(-λt) + p·λt·e^(-λt)
The first term is "the primary simply never failed." The second is "the primary failed once, the switchover succeeded, and the standby has not yet failed either" — a switchover failure ends the mission immediately, with no retry.
- Each live mission trace samples a real primary lifetime T₁ ~ Exp(λ); if T₁ falls inside the mission it draws a real Bernoulli(p) switchover outcome and, on success, a real standby lifetime T₂ ~ Exp(λ) starting from the switchover instant.
- The Monte Carlo batch repeats that same process 20,000 times, records each trial's total system lifetime L = T₁ (switch failed) or T₁+T₂ (switch succeeded), and reports the empirical mission-success probability R̂(T) = fraction of trials with L ≥ T — no shortcuts, no lookup table, just counting outcomes.
- Because failure and switchover draws are independent and memoryless, R̂(T) converges to the closed-form R(T) above as the trial count grows; the curve panel plots both so you can watch them agree.
- Drag p down and re-run the batch: the empirical R(T) measurably drops, exactly tracking the analytic formula — the switchover, not the primary's raw failure rate, is what limits the redundant system once p gets low.