Which Breaks First

Question

A feed handler fails after an exponential time with mean 3 hours; an independent gateway fails after an exponential time with mean 6 hours. What is the probability the feed handler is the first to fail?

Accepted Answer

Work with rates: the feed handler fails at rate lambda_F = frac13 per hour, the gateway at lambda_G = frac16. For independent exponentials, P(X_F < X_G) = int_0^infty lambda_F e^-lambda_F t, e^-lambda_G t, dt = fraclambda_Flambda_F + lambda_G = frac1/31/3 + 1/6 = frac23. The race is won in proportion to the rates: the component that fails twice as fast loses the race twice as often. Approach 2, competing clocks Think of failures as the first arrivals of two independent Poisson alarm clocks. Merge them: events arrive at total rate frac12, and each arriving event is the feed handler's with probability fraclambda_Flambda_F + lambda_G = frac23, independent of when it arrives. This "which clock rang" decomposition also gives a bonus fact for free: the time of the first failure is textExp!left(frac12right), mean 2 hours, independent of which component failed. The rate-proportional race is a workhorse: which venue fills first, which of two orders gets hit, which leg of a spread executes. Knowing fraclambda_1lambda_1 + lambda_2 cold turns many two-sided timing questions into one-line answers.

Which Breaks First

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