We investigate variations of a novel, recently proposed load balancing scheme based on small amounts of choice. The static (hashing) setting is modeled as a balls-and-bins process. The balls are sequentially placed into bins, with each ball selecting $d$ bins randomly and going to the bin with the fewest balls. A similar dynamic setting is modeled as a scenario where tasks arrive as a Poisson process at a bank of FIFO servers and queue at one for service. Tasks probe a small random sample of servers in the bank and queue at the server with the fewest tasks.
Recently it has been shown that breaking ties in a fixed, asymmetric fashion, actually improves performance, whereas in all previous analyses, ties were broken randomly. We demonstrate the nature of this improvement using fluid limit models, suggest further improvements, and verify and quantify the improvement through simulations.