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Peter arrives at ATM and there are 5 clients waiting for use it, one of them is being attended and the other 4 are in a queue. Peter are at the end of the queue, if the time of service for each client are exponential, independent and identically distributed with mean $1\over \lambda$. Find the distribution of the waiting time that Peter expend at the ATM.

If I knew that the person at ATM just start to being attended, I think that The Waiting time only be the sum of five exponential variables i.i.d. but I don't have any information about the time of service of the first client when Peter arrives. How can I considere it in the model?

Edit: For $n$ clients instead of 5, by the memoryless property, I can consider that the distribution of the waiting time is $\Gamma(n,{1\over\lambda})$?

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Basic insight. Since service times are exponentially distributed, they are memoryless, and it is the same as though the person currently in service just got there. In other words, Peter's waiting time is the sum of five exponentially distributed random variables with mean $1/\lambda$.

ETA: You probably shouldn't change your question like that, after it's been answered (it's better to add to it, instead), but yes, with the shape/scale characterization, those are the right parameters.

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  • $\begingroup$ Which is indeed $\Gamma(n,1/\lambda)$ (taking the shape,scale parametrization). $\endgroup$
    – Ian
    Oct 14 '16 at 22:35
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    $\begingroup$ @Ian: Yeah, OP just changed the question. Duly noted! :-) $\endgroup$
    – Brian Tung
    Oct 14 '16 at 22:37

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