How would I determine how large an M/M/c queue grows after a certain amount of time?

Question

Most of the reading on queueing theory I've done focuses on when the arrival rate is less than the service rate so that the queue doesn't explode. However, if I have a queueing system (either M/M/1 or M/M/c) where the arrival rate is greater than the service rate, how can I find properties such as the following:

• How large will the queueing system grow after a certain amount of time?
• What is the probability that the expected wait time upon arrival will be greater than a certain amount of time at time t?
• How many servers should the system have so that the queue never grows beyond a certain size in a fixed amount of time?

Answer 1

I'm many years removed from messing with queuing systems, so take this with a grain of salt. I think you can (theoretically) answer the first two questions, and the third with some trial and error (iterating over server counts), using the backward Kolmogorov equations. That requires solving a system of simultaneous PDEs. Alternatively, if you are asking about a specific system rather than asking a general theoretical question, you could probably get reasonably accurate results (while consuming fewer brain cells) doing it via discrete event simulation.

Answer 2

You can set up a Continuous Time Markov Chain to do the first two; the last one is infinite, since however many servers you have, there will always be some positive probability that there will be N customers coming (for any N). (In an M/M/c queue, the probability that 1000000 customers show up in 0.5 seconds is positive so long as the arrival rate is)