Randomized Load Balancing with Non-Uniform Task Lengths

In this paper we consider the classical multiprocessor scheduling problem. In this problem, the goal is to assign a collection of tasks with different execution times to a set of machines so that the total load is balanced as evenly as possible across the machines. This objective is typically expressed as minimization of the makespan, or the load on the most heavily loaded machine. In general, the problem of minimizing makespan is known to be NP-hard [2]. Here we consider the scheduling policy that simply assigns tasks to machines uniformly at random, regardless of the task execution times and previous assignments. We show that for any problem instance, the expected makespan under this policy is no greater than a factor of 2 ln(m)/ ln(ln(m)+1) from optimal, where m is the number of machines. This is an extension of existing results that establish similar bounds for the special case of loading m machines with m unit-length tasks [3, 10]. After considering the case of arbitrary task lengths, we then consider the performance of uniform random assignment when task lengths are generated randomly according to an exponential distribution. We show that if the number of tasks exceeds m ln(m), then the expected makespan under uniform random assignment is no greater than a factor of 4 from optimal. Furthermore, the expected makespan under uniform random assignment approaches the optimal makespan as the number of tasks grows beyond m ln(m).