There are many, beginning with the "easy" case where both time between successive arrivals (only one entity needing service per arrival event) and service times (single server to start with) are both exponentially distributed.  As these specifications are eased, the mathematical models rapidly become more complex.  Soon, they become intractable to the point that discrete-event process simulation (my own specialty) takes over in practical work.  Meanwhile, a superb reference, though aged by a generation, is "Queueing Systems Volume 1:  Theory" by Leonard Kleinrock, published by Wiley.  It is in the bookcase in my office as I type this response.

Dear Edwards,

I attach an article by Little that will give you a sense of the math involved in a proof of a Queue. 

Good reference, thanks!  The result is often called "Little's Law" for a reason -- as shown here.

Lall B. RamrattanEdward Williams, yes prof, I am reading it. Honestly it helps a lot and gives a good background about queuing theory.