What are the key $M/M/1$ formulas for queue length, waiting time, and utilization?

Queuing theory is the mathematical study of waiting lines, used to analyze and optimize the performance of service systems.

Conditions: The context is operations research.; The system involves arrivals, waiting, and service.

The first M stands for Markovian (Poisson) arrivals, the second M stands for Markovian (Exponential) service times, and the 1 stands for a single server.

Conditions: The notation refers to a queueing model.; The model is the $M/M/1$ queue.

For an $M/M/1$ queue to be stable, the utilization factor $\rho$ must be strictly less than 1. This means the arrival rate $\lambda$ must be less than the service rate $\mu$.

Conditions: The model is $M/M/1$.; $\rho = \lambda / \mu$.