Consistent queueing rules

In a "queueing problem ", a group of agents are waiting for a service. Each agent incurs a cost of waiting that is proportional to the time they wait. Monetary transfers can take place. We study the subsolutions of the no-envy solution that are anonymous, consistent, conversely consistent, and continuous. We show that there are infinitely many proper consistent subsolutions from the no-envy solution and characterize a class of these solutions on the basis of basic requirements of continuity, anonymity, monotonicity with respect to the budget available, consistency, and the converse of consistency.