We study a discrete problem of scheduling activities of several types under the constraint that at most a single activity can be scheduled to any one period. Applications of such a model are the scheduling of maintenance service to machines and multi-item replenishment of stock. In this paper we assume that the cost associated with any given type of activity increases linearly with the number of periods since the last execution of this type. The problem is to lind an optimal schedule specifying at which periods to execute each of the activity types in order to minimize the long-run average cost per period. We investigate properties of an optimal solution and show that there is always a cyclic optimal policy. We propose a greedy algorithm and report on computational comparison with the optimal. We also provide a heuristic, based on regular cycles for all but one activity type, with a guaranteed worse case bound.