Random number generators for MIMD parallel processors

We discuss and analyze issues related to the design of pseudorandom number generators (prn's) for MIMD (multiple instruction stream/multiple data stream) parallel processors, which are very well suited to Monte Carlo calculations. We are concerned with ensuring reproducibility of runs, providing very long sequences, and assuring an adequate degree of independence of the parallel streams. We consider the class of linear congruential generators x n+1,i ≡ ax n,1 + b i mod m and analyze the effect that different choices of b i have on the correlation properties between such streams. We derive a spectral test ν τ for t parallel linear congruential generators, a modification of Knuth's Algorithm S. From this, we prove a good lower bound for v 2 = min all pairs(i,j) ν 2( i, j) for certain choices of b i 's. The set of the largest r primes p i , i = 1, …, r, satisfying p i < √ m 2 , where m is the period length of each generator, gives a lower bound O(m 1 2 ) to the correlation between a pair of corresponding elements in any two streams. An alternative choice, b i = d i mod m for d = M 1 2 + 1 gives a bound O(m 1 2 (t − 1)) which will be satisfactory for small numbers of streams. Finally, we construct a spectral test for correlations between x n, i and x n+ k, i+ l , but derive no analytic prescriptions from it.