Throughput and delay optimality of power-of-d choices in inhomogeneous load balancing systems

It is well-known that the power-of-d choices routing algorithm maximizes throughput and heavy-traffic optimal in load balancing systems with homogeneous servers. However, if servers are heterogeneous, optimality does not hold general. We find necessary sufficient conditions for of when we prove almost same to show optimality. Additionally, generalize condition a larger class policies.