Exponential Race

Among independent exponential clocks, each rings first with probability proportional to its rate.

What is it?

For independent $T_{i} \sim Exp (λ_{i})$ , the minimum is $Exp (\sum λ_{i})$ and $T_{j}$ is the minimum with probability $λ_{j} / \sum λ_{i}$ .

Why is it useful?

Competing Poisson events, queue servers and failure times reduce to rate ratios.

When should you think of it?

Use it when several independent exponential waits compete and you care about which ends first or when.

Think of this method when…

Several independent exponential times or Poisson streams
"Which happens first?" or "time until the first"
Rates given per unit time
Memoryless restart after the first event

Core idea

Rates add for the minimum; the winner is chosen proportionally to rate, independently of when it wins.

i min T_{i} \sim Exp (i \sum λ_{i}), P (T_{j} = min) = \frac{λ _{j}}{\sum _{i} λ _{i}}

Step-by-step playbook

1Confirm independence and exponential times.
2Add the rates for the first-event time.
3Use rate ratios for which event wins.
4After the first event, restart by memorylessness.

Worked example

Three independent machines fail at exponential rates 1, 2 and 3 per year. What is the probability the rate-1 machine fails first, and the expected time to the first failure?

1
Total rate $1 + 2 + 3 = 6$ .
2
Winner probability and mean.
$P = \frac{1}{6}, E [min] = \frac{1}{6} year$
Answer: $1/6$ and $1/6$ year

Common mistakes

Adding means instead of rates.
Using it for non-exponential waits.

Would you use this method?

Two buses arrive as Poisson processes with rates 2 and 3 per hour. Which comes first?

Practice this method

26 QuantSolve questions are tagged with this method.

Practice this method

Related lessons & methods

Exponential · Distributions Poisson Processes · Stochastic Processes Thinning & Superposition · Stochastic Processes

Memorylessness Poisson Thinning Conditioning on the First Event