Flow control is the dominant technique currently used in communication networks for preventing excess traffic from flooding the network, and for handling congestion. In rate-based flow control, transmission rates of sessions are adjusted in an end-to-end manner through a sequence of operations. In this work, we present a theory of max-min fair, rate-based flow control sensitive to priorities of different sessions, as a significant extension of the classical theory of max-min fair, rate-based flow control to networks supporting applications with diverse requirements on network resources. Each individual session bears a priority function, which maps the session's priority to a transmission rate; the priority is a working abstraction of the session's priority to bandwidth access. Priority functions enable the specification of requirements on bandwidth access by distributed applications, and the formal handling of such requirements. We present priority max-min fairness, as a novel and well motivated fairness condition which requires that assigned rates correspond, through the priority functions, to priorities comprising a max-min vector. We also introduce priority bottleneck algorithms gradually update a session's rate until when its priority is restricted on a priority bottleneck edge of the network. We establish a collection of interesting combinatorial properties of priority bottleneck algorithms. Most significantly, we show that they can only converge to priority max-min fairness. As an application of our general theory, we embed priority bottleneck algorithms in the more realistic optimistic framework for rate-based flow control. The optimistic framework allows for both decreases and increases of session rates. We exploit these additionally provided semantics to prove further combinatorial properties for the termination of priority bottleneck algorithms in the optimistic framework. We use these properties to conclude the first optimistic algorithms for efficient, max-min fair, rate-based flow control sensitive to priorities.
