Expected time for all four tasks to finish

Question

Four independent tasks run at once. Each task's completion time is exponential with a mean of 10 minutes. What is the expected time until the last task finishes (all four are done)?

Accepted Answer

The last completion is the maximum of the four times. Track the waits between successive finishes. Until the first finishes, all four are still running, so that wait is exponential with rate 4\lambda and mean \mu/4. After one finishes, three remain (mean \mu/3), then two (mean \mu/2), then the last one alone (mean \mu). Memorylessness makes each stage start fresh, so the expected total is the sum: E[	ext{last finish}] = \mu\left(\frac14 + \frac13 + \frac12 + 1ight) = \mu \cdot \frac{25}{12}. With \mu = 10 minutes, E[	ext{last finish}] = 10 \cdot \frac{25}{12} = \frac{250}{12} \approx 20.8…

Expected time for all four tasks to finish

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