# How do I calculate the distribution of number of events in the busiest period?

I've got an estimate of the number of site visitors I'll see in a 1 hour period clicking email links in a large email campaign. I need to make sure I've got the required server capacity. That means I need to estimate the peaks in usage on a smaller time scale.

1. How do I calculate the expected distribution for the number of visitors I should expect in any given smaller time period? eg for a given 1 minute period.

2. How do I calculate the expected distribution for number of visitors in the busiest time period of a given length?

I figure it's good enough for estimation purposes to assume that each visitor arrives at a random time that's evenly distributed within the one hour period.

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EDIT: (in response to rocinante) OK, so lets consider the problem redefined in terms of the assumptions of the poisson distribution. There's some minor differences, but they don't matter to me. Also, lets not worry about possible notions like sliding windows of time for the period in 2. So long as the period in 2. is short, that doesn't matter either.

Supposing I expect visitors to arrive at a rate of 10K visitors per hour over a one hour period. If I divide that hour up into 720 adjacent 5 second intervals, I expect the number of visitors in each of those 5 second periods to follow a poisson distribution.

Part 2 of my question can then be restated: I can take 720 samples from a poisson distribution, and find the largest value of those 720 sample values. If I repeatedly took sets of samples like that, calculating the maximum value in each set, what would the distribution of those maximums be like?

Having graphed some poisson distributions, I'm comfortable that I know enough about what to expect for planning purposes, but I'd still like to how to do part 2 of my problem as posed.

Also, for large lambda, I presume some other calculation of the distribution should be used? Calculating factorials becomes impractical at some point.

• a standard statistical approach in this setting is not to look at the distribution of visitors but rather to look at the distribution of queueing visitors given the number of servers. There is a whole field called queueing theory devoted to this. In the past I've used some of the simple results to get ball-park estimates for wait times etc given arrival frequency at busiest period and number of servers. Commented Mar 18, 2014 at 19:03