Gamma-distribution order statistics, maximal multinomial frequency and randomization designs

The paper discusses the relationship between the gamma-distribution order statistics, the maximal cell frequency in multinomial trials and the distribution arising in a commonly used balanced randomization scheme with several treatments. The latter is used in clinical trials, load balancing in computer files storage, parapsychological experiments, etc., where a randomization design starts with the uniform probability assignment of subjects to treatments. The limiting distributions of the waiting time until a treatment receives the given number of subjects are described. The relationship to classical occupancy problems, in particular, to the Banach match-box problem and to the birthday problem is discussed.