# Optimal scheduling policy for two server model

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 :

1. Throughput optimal that is it achieves the maximum capacity region.

2. 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.

• Suppose one server is so much faster that it can complete three jobs before the other finishes one. Wouldn't it make sense for small $\lambda$ to leave a single job in the queue even if the slow one is idle? – Ross Millikan Mar 19 '15 at 20:22
• yes you are right. I was thinking about that. But my question is whether we can derive an optimal policy, which will be dependent on the values of $\mu_1, \mu_2$ and $\lambda$. – rajatsen91 Mar 20 '15 at 16:49
• The answer is "yes," and this problem has been well-studied in the literature. Cf. kellogg.northwestern.edu/research/math/papers/592.pdf for the optimal policies where stalling of either server is allowed. – Math1000 Feb 23 '16 at 21:29