Just bumped at this thread, and although a very old one, I'd like to share my thoughts..
This device at the picture is probably one inspired by the chandelier of Ibn Yunus, made for educational purposes in order to explain the functionality and not an exact replica.
The initial device was used to measure night time and involved 12 separate lamps in which according to the time of the year, a specific amount of oil was measured and inserted to each one of them, x at the first, 12x at the last. The mass/volume of x was determined by the time of the year and there were specific tables created for each week of the year at a specific place.. So, filling up this device was a very tedious job and this is probably why the image above shows a chandelier engineered as to work on the communicating vessels principle in order to ease the filling.
At the time of the invention, the day was divided in 24 hours, 12 daylight, 12 at night, starting from dusk till dawn and were not equal between them. Today this type of time measurement is known as Byzantine time and even nowadays some orthodox monasteries as the ones in Mt.Athos continue to use it.
The device pictured can only be used for a fixed date, and I presume is used to measure modern time, ie x equals to the oil required to burn for exactly 1 hour.