The concept of the quickest path refers to the path with the minimum transmission time, considering both its length and capacity. We investigate the problem of finding a point on a cycle such that the maximum quickest distance from any vertex to that point is minimized. We refer to this problem as the quickest 1-center problem on cycles. First, we solve the
problem on paths in linear time based on the optimality criterion. Then, we address the problem on cycles in O(n^2) time by leveraging the solution approach on the induced path in each iteration, where n is the number of vertices. We also consider the problem of reducing the quickest distance objective at a predetermined vertex of a cycle as much as possible by augmenting the edge capacities within a given budget. This problem is called the reverse quickest 1-center problem on cycles. We develop a combinatorial algorithm that solves the problem in O(n^2) time by solving each subproblem in linear time.