⏱️ The Memoryless Property

About this simulation

Every glowing unit in the grid is a component whose lifetime is drawn from an exponential distribution with rate λ: P(T > t) = e^−λt. As the clock runs, units wink out at random — not in a wave, but scattered, because each one fails independently with the same constant hazard rate at every instant of its life, regardless of age.

How it works

• The population's survival curve (yellow markers) traces the fraction of units still alive over time, and should track the analytic curve e^−λt (the wire curve) closely.
• The focus unit (large sphere, right pedestal) never dies on its own — you age it forward with the button. Its "expected remaining life" readout stays pinned at 1/λ no matter how old it gets.
• That's the memoryless property: P(T > s+t | T > s) = P(T > t). A component that has already survived 10 hours is exactly as likely to survive another hour as a brand-new one — it carries no memory of its age. The exponential is the only continuous distribution with this property.
• This is why exponential inter-failure and inter-arrival times underpin reliability engineering (constant hazard rate = "no wear-out") and queueing theory (Poisson arrivals, where the wait for the next customer never depends on how long you've already waited).

Controls

• Failure rate λ — hazard rate per second; mean lifetime is 1/λ.
• Clock speed — how fast simulated time advances.
• Population — grid size of the simulated fleet.
• ↺ Reset population — resamples every lifetime and restarts the clock.
• ⏩ Age focus unit — jumps the spotlighted unit's age forward by 2s without killing it, so you can watch its remaining-life estimate stay flat.

Did you know?

Because the exponential distribution is memoryless, it's the unique continuous distribution for which the "time since last event" tells you nothing about the "time until the next event" — the mathematical reason light bulbs, radioactive decay and Poisson process arrivals are all modelled the same way.