I don't know of any research articles, but I'm sure there is market research out there on the matter. As an observation, city gyms have 3 peak periods: 6am-8am, 12pm-2pm, and about 4:30pm-6:30pm. That's all give or take, of course. Suburban gyms don't usually have a lunch time peak period, but they seem to have longer morning and evening peaks, say 6am-10am and 3pm-7pm.
I honestly couldn't say what the average duration of each gym session is, or the average frequency. Seems to depend on people's attitudes. The under-ambitious people who are only there because they've got doctors orders will often do a half-arsed 30min session, say they're 30min is up, and leave ASAP. The super-ambitious people (usually highly strung, obsessive types) will often be there for multiple hours, every day of the week, some training multiple times at such a rate. These are also the same people who are most likely to quit the second something goes wrong, because they have a very high standard that they set for themselves on performance, and falling short of that standard even by the smallest margin consitutes complete and utter failure in their minds. Then there's the athletic crowd, who definitely don't make up the majority of gym members. They'll also train for a couple hours, often faily, sometimes multiple times per day, but that's because they're actually required to train like that to be thoroughly conditioned for their sport.