Balanced Loads in Infinite Networks

A set of nodes and a set of consumers are given, and to each consumer there corresponds a subset of the nodes. Each consumer has a demand, which is a load to be distributed among the nodes corresponding to the consumer. The load at a node is the sum of the loads placed on the node by all consumers. The load is balanced if no single consumer can shift some load from one node to another to reduce the absolute difference between the total loads at the two nodes. The model provides a setting to study the performance of load balancing as an allocation strategy in large systems. The set of possible balanced load vectors is examined for infinite networks with deterministic or random demands. The balanced load vector is shown to be unique for rectangular lattice networks, and a method for computing the load distribution is explored for tree networks. An FKG-type inequality is proved. The concept of load percolation is introduced and is shown to be associated with infinite sets of nodes with identical load.