Competitive analysis of network load balancing

This paper presents a theoretical analysis of the Load Balancing Problem (LBP) in a network of processing units. The performance objective is to minimize the makespan, i.e., the time spent to finish all jobs in a network of processing units. Because of the communication delay that results from the network topology, it is impossible to have a strategy which obtains the exact optimum under all load distributions. Instead, we measure the information efficiency of a load balancing policy by the worst case ratio of the solution (for each load distribution) of a load balancing policy to the optimal solution (for the same load distribution) assuming that processors have complete information about the load distribution over the network. This ratio is called the competitive ratio of this strategy [17, 24, 34]. In particular, a policy is called competitive if this ratio is bounded by a constant. As a first step, we discuss the centralized LBP, where all the processors have complete information of the load distribution over a network. Its solution serves as a benchmark to compare with realistic strategies, both in theoretical analysis, and experimental and simulational studies of distributed algorithms. We show that when jobs have different sizes, even with preemptive scheduling, LBP is NP-complete. When the jobs are of the same size, we give a polynomial algorithm, using network-flow techniques, which extends to approximate solutions for jobs of different sizes. We apply this benchmark solution in order to analyze the competitiveness for three network topologies: completely connected graphs, rings, and hierarchical complete k-ary trees. The constant competitive ratio results for complete network and hierarchical complete k-ary trees are applied to a study on the issues of network designs suitable for the LBP. We further discuss the problem for general networks with jobs of different sizes for slightly weaker results than those for the constant competitive ratio requirement. Finally, we comment on the related issues of job partitioning over parallel/distributed distributed systems. (C) 1997 Academic Press.