No justified complaints: On fair sharing of multiple resources

Fair allocation has been studied intensively in both economics and computer science. This subject has aroused renewed interest with the advent of virtualization and cloud computing. Prior work has typically focused on mechanisms for fair sharing of a single resource. We consider a variant where each user is entitled to a certain fraction of the system's resources, and has a fixed usage profile describing how much he would want from each resource. We provide a new definition for the simultaneous fair allocation of multiple continuously-divisible resources that we call bottleneck-based fairness (BBF). Roughly speaking, an allocation of resources is considered fair if every user either gets all the resources he wishes for, or else gets at least his entitlement on some bottleneck resource, and therefore cannot complain about not receiving more. We show that BBF has several desirable properties such as providing an incentive for sharing, and also promotes high overall utilization of resources; we also compare BBF carefully to another notion of fairness proposed recently, dominant resource fairness. Our main technical result is that a fair allocation can be found for every combination of user requests and entitlements. The allocation profile of each user is proportionate to the user's profile of requests. The main problem is that the bottleneck resources are not known in advance, and indeed one can find instances that allow different solutions with different sets of bottlenecks. Therefore known techniques such as linear programming do not seem to work. Our proof uses tools from the theory of ordinary differential equations, showing the existence of a sequence of points that converge upon a solution. It is constructive and provides a practical method to compute the allocations numerically.