Repetition of states of a random-number generator under multiple access

In performing computer calculations which require multiple access to a random-number generator, there is the danger of using one and the same or matching segments of pseudorandom numbers. This phenomenon may cause a lack of agreement between the results of calculations and the respective theoretical predictions. This paper considers a simple mathematical model corresponding to the generators with one cycle of internal states and estimates the sizes of data arrays produced by such generators within which the probability of repeated use of one and the same element of pseudorandom sequences is practically negligible. The main results are expressed by inequalities which allow us to get numerical estimates for the appearance of matchings and for the distributions of some random variables related with this phenomenon.