The bowl phenomenon provides a way of increasing the throughput of some production line systems with variable processing times by purposely unbalancing the line in a certain manner. However, previously available numerical results for applying the bowl phenomenon have been quite limited. We extend these numerical results here by obtaining the optimal allocation of work for somewhat larger cases than previously considered. We also develop some guidelines and data for extrapolating these results to estimate the optimal allocation of work for even larger production lines that are beyond the reach of exact solution methods. The results cover a broad cross-section of values of both buffer capacities and the coefficientof variation of processing times. I. IntroductionHillier and Boling (1966, 1979) investigated the optimal allocation of work in unpaced production line systems with variable processing times. Their 1966 paper discovered the 'bowl phenomenon', whereby a plot of the optimal allocation vs. station number has the shape of the cross-section of the interior of a bowl. Their 1979 paper obtained numerical results on the optimal allocation for a variety of small cases, and then gave guidelines for extrapolating these results to larger cases. (Also see Rao 1976 for the case where processing times at different stations differ substantially in their variability, Magazine and Silver 1978 for heuristics related to the bowl phenomenon, and Muth and Alkaff 1987 for further analysis of the bowl phenomenon.)Advances in computing power now make it possible to obtain exact results on the optimal allocation for somewhat larger cases than in the 1979 paper. The purposes of the present paper are to present these extended numerical results, and then to use them to refine the 1979 extrapolation guidelines. These results and guidelines are given in Sections 3 and 4, respectively, after first summarizing the model and notation in