An M/M/1 queue

Topic: Birth–death chains and queues · more questions on this topic
An M/M/∞ system ›

An M/M/1 queue has \lambda=2 and \mu=3 per hour. Find \pi_0, \mathbb{P}(N\ge 3), the mean number in system, and the mean time in system.

Worked answer (click to show)

The load is \rho=2/3<1, so \pi_0=1-\rho=1/3 and \mathbb{P}(N\ge 3)=\rho^3=8/27. The mean number in system is L=\rho/(1-\rho)=2. Little’s law uses the arrival rate \lambda=2, so the mean time in system is W=L/\lambda=1 hour. The same time is 1/(\mu-\lambda)=1.