Normal length of time taken to arrive at a phd admission decision?
In my experience it's common for academic departments to have a graduate admissions committee that meets periodically to make admissions decisions and decisions on financial aid.  Sometimes decisions on financial aid are made separately from decisions on admission.  

In our department we review applications as they come in and typically respond with a decision on admission within a few weeks.  However, decisions on financial aid are made only a few times per year.  For example, we'll meet in early April to decide on assistantship awards for the fall semester.    

Under this system, if you applied for admission now, we'd review your application and reach an admissions decision by the end of January, and you'd either be told "no", or "you've been admitted but we'll make decisions about financial aid in early April."  

I've seen other institutions where all of the applications are held until one meeting where both admissions and financial aid decisions are made at the same time.  Under that system, you probably wouldn't hear anything at all until the committee met and made its decisions.  

A couple of other comments:


Applications are often sent to a central office ("graduate studies" or something similar)  and then distributed to the departments to make admissions decisisons.  In my experience, there are many incomplete applications received by our graduate office.  We don't see them in the department until and unless all of the required materials have been submitted.  You can and probably should check with the office where you sent your application to make sure that the complete application has been received.  
Christmas (December 25) and New Year's Day (January 1) are important holidays in the US.  Traditionally, fall semester classes end before Christmas and spring semester classes don't start until after New Year's Day.  Many colleges and universities are effectively closed for a few weeks around these holidays.  You shouldn't expect to hear anything from any university in the US until after New Year's Day because of these holidays.