Effects of node processing time on optimal load balancing in tree hierarchy network configurations

In this paper, optimal static load balancing in a tree hierarchy network that consists of a set of heterogeneous host computers is considered. It is formulated as a nonlinear optimization problem. By parametric analysis, we study the effects of the node processing time on the optimal link flow rate (i.e. the rate at which a node forwards jobs to other nodes for remote processing), the optimal node load (i.e. the rate at which jobs are processed at a node), and the optimal mean response time. We show that the entire network can be divided into several independent sub-tree networks with respect to the link flow rates and node loads. We find that the processing time of a node affects only the link flow rates and the loads of nodes which are in the same sub-tree network. Generally, an increase in the processing time of an arbitrary node causes an increase in the link flow rates of its ancestor nodes and itself, but causes a decrease in the link flow rates of its descendant nodes and its collateral nodes in the same sub-tree network. It also causes a decrease in the load of the node itself, but causes an increase in the loads of other nodes in the same sub-tree network. Furthermore, it causes an increase in the mean response time. By conducting numerical experiments, we find that the node processing time possesses a large influence on the system performance measures. Knowledge of the effects of node processing time is useful in designing networks or making a parametric adjustment to improve the system performance.