An extension of the standard rationing model is introduced. Agents are not only identi fied by their respective claims over some amount of a scarce resource, but also by some payoff thresholds. These thresholds introduce exogenous differences among agents (full or partial priority, past allocations, past debts, ...) that may influence the final distribution. Within this framework we provide generalizations of the constrained equal awards rule and the constrained equal losses rule. We show that these generalized rules are dual from each other. We characterize the generalization of the equal awards rule by using the properties of consistency, path-independence and compensated exemption. Finally, we use the duality between rules to characterize the generalization of the equal losses solution.