I have a system with customers waiting in a queue to be serviced. Each customer will be serviced by a server out of a pool of servers $s$ which takes a fixed amount of time $r$ and I wish to calculate the size of the queue I will need to cope with all the customers.
At time $t=0$ the rate of customers arriving is 0 and the rate of customer arrivals will accelerate to $\lambda_{max}$ at $t=1$. After this point in time no further customers will be accepted into the queue.
All the resources I've been able to find have the average rate of customer arrivals being uniform over time with some sort of distribution. However the system I'm dealing with there is a deadline point and the rate of customers entering the system accelerates up to that point in time.
My question is three fold.
Is there a formulae for this problem?
What is the correct terminology to describe this sort of increasing traffic (so I can search for it and ask better questions)?
Are there reference materials (books/journals/papers etc) for queuing theory where a point in time is special?