The Generation of Multiple Independent Sequences of Pseudorandom Numbers

The recent availability of highly parallel computers and programming languages has increased the interest in the problem of generating multiple independent sequences of pseudorandom numbers. This has always been a matter of concern to statisticians, especially in such areas as the simulation of queues, but the problem is now affecting a wider group of people. It is easy to generate either multiplicative congruential or shift register sequences in parallel, and thus to obtain the computational efficiency, but it is difficult to choose generators that have good statistical properties both within and between sequences. The most common approach is to use a high quality generator with proven properties within a single sequence, and to validate it with empirical tests when it is used for multiple sequences. The shift register technique (Tausworthe, 1965) has known equidistribution properties in up to m dimensions, where m is the degree of its generating polynomial; m can be divided up arbitrarily as a product of the number of sequences, the length of sequences and the precision of the numbers produced. Even a high quality generator, like the 4113-bit generator used on the ICL DAP at Queen Mary College (Smith et al., 1985), cannot guarantee the quality of all three properties simultaneously to more than a limited extent. This paper describes a technique for generating pseudorandom numbers that can give proven statistical properties, in the sense that the spectral test defines, both within and between sequences simultaneously. It is based on the Wichmann-Hill (Wichmann and Hill, 1982) variation of the multiplicative congruential (Lehmer, 1949) algorithm.