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I have a problem from my Stochastic Processes class I have no idea how to solve. We have been studying Poisson processes including models of queues, but this question involves a small amount of sample data and reverse engineering. We are modelling an ATM that opens at 7:30. It can track how long each operation takes a client but does not know when they joined the queue. From this data we are meant to calculate the expected time of the first client arrival. The data is

The machine opens at 7:30

At 7:34 the first transaction starts. At 7:40 it ends

At 7:40 the second transaction starts. At 7:42 it ends

At 7:45 the third transaction starts. At 7:50 it ends

From this I am meant to calculate the rate $\lambda$ I think. And then work out mean of that exponential distribution which is just 1/$\lambda$ but I have no idea how to work this rate out from the data as I cannot see how the arrival times link to the service times. As this is an assessed question, I would prefer hints rather than full solutions. Many thanks!

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  • $\begingroup$ One possibility would be to track the gaps between a transaction finishing (or the opening) and the next transaction starting, and drop all the information where this is $0$. Another might be to consider total arrivals (i.e. transactions) in a day, but this assumes nobody gives up because of a queue. $\endgroup$
    – Henry
    May 10, 2021 at 12:07

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My best shot at an answer would be to say,

In 15 mins from opening, 3 people arrived and thus the rate is 12 clients an hour and thus the expected value of the first arrival is 7:35 or 5 mins.

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