Pigeons and other animals soon learn to wait (pause) after food delivery on periodic-food schedules before resuming the food-rewarded response. Under most conditions the steady-state duration of the average waiting time, t, is a linear function of the typical interfood interval. We describe three experiments designed to explore the limits of this process. In all experiments, t was associated with one key color and the subsequent food delay, T, with another. In the first experiment, we compared the relation between t (waiting time) and T (food delay) under two conditions: when T was held constant, and when T was an inverse function of t. The pigeons could maximize the rate of food delivery under the first condition by setting t to a consistently short value; optimal behavior under the second condition required a linear relation with unit slope between t and T. Despite this difference in optimal policy, the pigeons in both cases showed the same linear relation, with slope less than one, between t and T. This result was confirmed in a second parametric experiment that added a third condition, in which T + t was held constant. Linear waiting appears to be an obligatory rule for pigeons. In a third experiment we arranged for a multiplicative relation between t and T (positive feedback), and produced either very short or very long waiting times as predicted by a quasi-dynamic model in which waiting time is strongly determined by the just-preceding food delay.