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When I observe the popular times of a store / place / online shop on Google, sometimes the bar graph has one peak (maximum) but sometimes it has two (global maximum and a local maximum). I was thinking that the graph should approximate the Poisson distribution but it seems this is not the case for the graphs which have two peaks. So, I considered that the average rate λ is not constant for this case. Is there a distribution such as Poisson with a variable time rate λ ? What other distribution could I use to model / approximate the graphs which have two peaks ?

† See this Google Maps link for a steak restaurant in Paris and scroll down the left to see the barchart for "Popular times".

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  • $\begingroup$ Welcome to Cross Validated! This sounds like it will be an interesting question, but you'll need to explain what "the bar graph" is, as there are several things you might be plotting. Is "time" here "time of day"? $\endgroup$ – Scortchi - Reinstate Monica Feb 14 at 20:34
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    $\begingroup$ What is possible is a Poisson process with a rate which varies over time. That could give the sort of graph you are seeing on Google. If you integrate that rate over a particular time period, then the actual number of arrivals in the whole of the time period would then follow a Poisson distribution with a mean equal to that integrated rate. In practice, life is likely to be more complicated: for example some locations may have a limited capacity with people turning away when they see the place is too crowded. $\endgroup$ – Henry Feb 14 at 20:34
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    $\begingroup$ @Henry Thank you. Unfortunately I cannot vote up your comment because the question was closed but this is what I was looking for. The full name for anyone looking for this, is nonhomogeneous poisson process where λ = λ(t). $\endgroup$ – entropyfeverone Feb 14 at 20:51
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    $\begingroup$ @Henry: Those are counts (presumably) of visitors by hour of the day over the course of a week (or perhaps tallied over several weeks). So even a single pronounced peak would be incompatible with a homogeneous Poisson process for arrivals - & it's not clear what it would mean for the graph to "approximate the Poisson distribution". The title question is clear enough, granted, but it's up to the OP to describe these graphs in the question if they want the rest clearing up. $\endgroup$ – Scortchi - Reinstate Monica Feb 14 at 21:19
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    $\begingroup$ @entropyfeverone: If those assumptions were true, then you'd expect counts in any given hour to be about the same: one pronounced maximum would be as surprising as two. But if you plot the no. hours over the week in which no people visit, the no. hours in which one person visits, & c. - then that graph will have one pronounced peak. $\endgroup$ – Scortchi - Reinstate Monica Feb 14 at 21:37
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From comment while question was closed:

What is possible is a Poisson process with a rate which varies over time. That could give the sort of graph you are seeing on Google. As you commented, this would be a nonhomogeneous Poisson process

If you integrate that rate over a particular time period, then the actual number of arrivals in the whole of the time period would then follow a Poisson distribution with a mean equal to that integrated rate.

In practice, life is likely to be more complicated: for example some locations may have a limited capacity with people turning away when they see the place is too crowded.

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