On-Line Load Balancing in a Hierarchical Server Topology

Abstract

In a hierarchical server environment jobs are to be assigned in an on-line fashion to a collection of servers which form a hierarchy of capability: each job requests a specific server meeting its needs, but the system is free to assign it either to that server or to any other server higher in the hierarchy. Each job carries a certain load, which it imparts to the server it is assigned to. The goal is to find a competitive assignment in which the maximum total load on a server is minimized. We consider the linear hierarchy in which the servers are totally ordered in terms of their capabilities. We investigate several variants of the problem. In the unweighted (as opposed to weighted) problem all jobs have unit weight. In the fractional (as opposed to integral) model a job may be assigned to several servers, each receiving some fraction of its weight. Finally, temporary (as opposed to permanent) jobs may depart after being active for some finite duration of time. We show an optimal e-competitive algorithm for the unweighted integral permanent model. The same algorithm is (e+1)-competitive in the weighted case. Its fractional version is e-competitive even if temporary jobs are allowed. For the integral model with temporary jobs we show an algorithm which is 4-competitive in the unweighted case and 5-competitive in the weighted case. We show a lower bound of e for the unweighted case (both integral and fractional). This bound is valid even with respect to randomized algorithms. We also show a lower bound of 3 for the unweighted integral model when temporary jobs are allowed. We generalize the problem and consider hierarchies in which the servers form a tree. In the tree hierarchy, any job assignable to a node is also assignable to the node's ancestors. We show a deterministic algorithm which is 4-competitive in the unweighted case and 5-competitive in the weighted case, where only permanent jobs are allowed. Randomizing this algorithm improves its competitiveness to e and e+1, respectively. We also show an $\Omega(\sqrt{n})$ lower bound when temporary jobs are allowed.