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

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?

• The steady-state behavior of M/M/c queues has been extensively studied. Balance eqns can be solved to give the steady-state distn based on the arrival rate $\lambda$ and the service rate $\mu\, (\lambda < c\mu)$ of each of the $c$ servers. Then one can find the avg nr $L$ of people in the system, the avglength $W = L/\lambda$ of time a customer spends in the system, the avg nr of people waiting for service and the avg length of time a customer spends waiting for service. The eqn $L = \lambda W$ is called Little's Law. See books and sites on queues, Operations Rsch, and Stochastic Processes. – BruceET Jun 16 '18 at 3:49