Load Balancing and Chernoff Bounds

This week, we consider a very simple load-balancing problem. Suppose you have n machines and m jobs. You want to assign the jobs to machines such that all machines have approximately the same load. Of course, there is a solution with load at most dmn e on every machine, but that requires central coordination. Without central coordination, the easiest thing you can do is let each job drawn one machine uniformly and independently. Such random assignments are prevalent in different settings. In the general setting, one assumes that balls (in our case jobs) are thrown into bins (in our case machines) at random. We will study the balls-into-bins problems for the case that m = n. We are interested in the number of balls within a single bin. Formally, let Li be the load of bin i. By symmetry reasons, E [Li] = 1 for any fixed i. However, the expected maximum load E [maxi Li] is higher. Just consider the case that n = 2. Then, only with probability 1 2 the maximum load is 1 (the balls fall into different bins), with probability 12 it is 2 (the balls fall into the same bin). In the first lecture, we used a union bound to upper-bound the distribution of the maximum of some random variables. To apply the union bound, we first need to understand the distribution of Li.