Two servers both done within the hour

Question

A job is sent to two independent servers at the same time. Server A's finish time is exponential with rate 2 per hour (a 30-minute average). Server B's finish time is exponential with rate 1 per hour (a 60-minute average).

What is the probability that both servers finish within 1 hour?

Accepted Answer

"Both finish within 1 hour" means the slower of the two, the maximum finish time, is at most 1 hour. Because the two servers are independent, the chance they both meet the deadline is the product of their individual chances. For one exponential finish with rate \lambda, the chance of finishing by time t is 1 - e^{-\lambda t} (the complement of still running). So: Server A by 1 hour: \;1 - e^{-2 \cdot 1} = 1 - e^{-2} \approx 1 - 0.1353 = 0.8647. Server B by 1 hour: \;1 - e^{-1 \cdot 1} = 1 - e^{-1} \approx 1 - 0.3679 = 0.6321. Multiply, since independent "and" means multiply: P(	ext{both done…

Two servers both done within the hour

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