What condition must hold for an $M/M/1$ queue to be stable?

The key formulas are: Average Queue Length $L = \frac{\lambda}{\mu - \lambda}$, Average Waiting Time $W = \frac{1}{\mu - \lambda}$, and Utilization $\rho = \frac{\lambda}{\mu}$.

Conditions: The model is $M/M/1$.; $\lambda$ is the arrival rate.; $\mu$ is the service rate.; $\lambda < \mu$ for stability.

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.