The authors consider the problem of optimally regulating the source traffic in a communication network to simultaneously satisfy a finite number of affine burstiness constraints, They prove that an optimal solution is a series connection of correspondingly dimensioned "leaky buckets," They propose a simple "fork-join" implementation of the optimal solution and study extensions to the problem of optimally shaping the traffic flow to meet a burstiness constraint specified by a concave increasing function. A consequence of their optimality results is that permutations of leaky buckets in a series connection are input-output equivalent.