Expected extra wait at an idle help desk

Question

Customers reach a help desk as a Poisson process at rate \lambda = 6 per hour. The desk has already sat idle for 12 minutes. On average, how many more minutes until the next customer arrives?

Accepted Answer

The wait between customers is exponential and memoryless, so the 12 idle minutes tell you nothing about the future. The expected remaining wait equals the expected wait of a brand-new exponential, which is 1/\lambda. With \lambda = 6 per hour, the mean wait is E[T] = \frac{1}{\lambda} = \frac{1}{6}	ext{ hour} = 10 	ext{ minutes}. So on average about 10 more minutes, exactly the same as if you had just walked up. The idle stretch does not shorten the expected wait. Takeaway. For a memoryless process the expected remaining wait is always 1/\lambda, regardless of how long you have already wait…

Expected extra wait at an idle help desk

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