# Approximate average waiting time when a m\m\1 queue shows transient behavior

I have an m/m/1 queue and a set of arrival rate $$\{\lambda_{t_1}, \lambda_{t_2} ...., \lambda_{t_n}\}$$ and a fixed service rate $$\mu$$. I want to calculate the average waiting time during each time interval. On the book "Operations research algorithms and applications" by Wayne L. Winston, they solve a similar problem by simulating it. I was wondering if there any closed equation for approximating this value. One similar note, given an initial number of people in the queue and a fixed arrival rate and service rate and a finite time length, is there any good closed equation for approximating average writing time. And if there is no such equation what is the fastest way to approximate this?

• I think the situation you're describing is called time-dependent queuing. You'll find stuff on that topic if you do a Google search. I happen to remember that there's a section on this topic in Daskin's book Service Science, but I'm sure there are lots of other references too. – LarrySnyder610 May 3 '20 at 2:55