Suppose there are two servers with exponential arrival rates $\mu_{1}$ and $\mu_2$ such that $\mu_1 > \mu_2$. These two servers have a shared infinite buffer, where there is independent Poisson arrival with rate $\lambda$. What is an optimal scheduling policy in these two sense :
Throughput optimal that is it achieves the maximum capacity region.
It reduces the mean time spent by a customers in the system.
I have a feeling that the greedy policy should be optimal, that is choose the faster server if it is free, otherwise choose the other one. Never keep the servers idle. I have no idea as to how one would prove optimality.
In the case of throughput optimality perhaps the greedy policy can stabilize an arrival rate of $\mu_1 + \mu_2$. Then it would be throughput optimal.