I have data for $n \approx 500$ objects, and for each object I have between ~50 and ~200 observations. Each observation consists primarily of a timestamp when an event happened (and includes some minor data about the event, but I don't think that's hugely important). I'm interested in clustering the $n$ objects into groups based on the patterns of when the events happened.
A simple parallel example: You have 500 entrances to parking lots surrounding a stadium. At each entrance you record the time whenever a car enters the parking lot (and some minor auxiliary data like how the car weighs). There are some common patterns they all share, particularly that very few cars arrive after the event starts. There are also different patterns, e.g. one may have lots of cars spread out through time, while another may have only a small number at the last minute. How can you cluster the data such that entrances with similar entrance patterns are grouped together?
Here's a visual example. x-axis is time. y-axis is arbitrary sorting of the objects, such that each object is a row. I drew some colors indicating possible clusterings, but my selections were fairly arbitrary.