Our work extends a previous analysis of the supermarket model, a model that abstracts a simple, efficient load balancing scheme in the setting where jobs arrive at a large system of parallel processors. In this model, customers arrive at a system of n servers as a Poisson stream of rate λ n , λ < 1 , with service requirements exponentially distributed with mean 1. Each customer chooses d servers independently and uniformly at random from the n servers, and is served according to the First In First Out (FIFO) protocol at the choice with the fewest customers. For the supermarket model, it has been shown that using d=2 choices yields an exponential improvement in the expected time a customer spends in the system over d=1 choice (simple random selection) in equilibrium. Here we examine several variations, including constant service times and threshold models, where a customer makes up to d successive choices until finding one below a set threshold.

Our approach involves studying limiting, deterministic models representing the behavior of these systems as the number of servers n goes to infinity. Results of our work include useful general theorems for showing that these deterministic systems are stable or converge exponentially to fixed points. We also demonstrate that allowing customers two choices instead of just one leads to exponential improvements in the expected time a customer spends in the system in several of the related models we study, reinforcing the concept that just two choices yields significant power in load balancing.