Online load balancing and correlated randomness

This paper looks at online load balancing, in a setting where each job can only be served by a subset of the servers. The subsets are revealed only on arrival, and can be arbitrary. The cost of an allocation is the sum of cost for each server, which in turn is a convex increasing function of the number of jobs allocated to it. There are no departures. A natural class of policies are those which always put a job into one of its servers that has the least load at the time of arrival. However, it turns out that not all (randomized) ways of breaking ties is the same. We propose an algorithm - TIERED RANKING - that breaks ties in a very particular, correlated random way; it is inspired by the online matching work of Karp, Vazirani and Vazirani. We show that it is optimal (in terms of competitive ratio) in the above class, for all convex cost functions that grow slower than e^x (which includes all l_p norms). We also prove it strictly outperforms any deterministic algorithm; simulations show that it also visibly outperforms the naive randomized algorithm, that breaks ties among lowest-loaded servers uniformly at random.