# M/G/$\infty$ queue with different servers

Consider an M/G/$\infty$ queue. Each server has a different service time $\mu_i$. Jobs arrive at the system with rate $\Lambda$. I would like to know the average service time of this queue to use the little law. The number of job being processed in the system is $\Lambda$ times the average service time. However, I'm not sure how to derive the average service time.

The average service time depends on the probabilities of what is the lowest index server available upon arrival (according to the stationary distribution). So you need to decide which servers are activated first, for example fastest server first. If $\pi_n$ is the probability that $n$ is the lowest index server available and $W$ is the stationary waiting time, then $$E[W]=\sum_{n=1}^\infty \pi_n\frac{1}{\mu_n}.$$ But calculating $\pi_n$ is hard, especially when the service times are not exponential. In the M/M/$\infty$ case (in fact for G/M/$\infty$) it can be done recursively: