Louis R. Moore

An exhaustive analysis of multiplicative congruential random number generators with modulus 2 31 -1

This paper presents the results of an exhaustive search to find optimal full period multipliers for the multiplicative congruential random number generator with prime modulus

A Statistical Evaluation of Multiplicative Congruential Random Number Generators with Modulus 231 — 1

2^{31} - 1

Sampling from a Discrete Distribution While Preserving Monotonicity

. Here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which k-tuples lie does not exceed the minimal achievable distance by more than 25 percent for

A Comparison of Reliability Growth Models

k = 2, \cdots ,6

Estimating the Mean of a Correlated Binary Sequence with an Application to Discrete Event Simulation

. This criterion is considerably more stringent than prevailing standards of acceptability and leads to a total of only 414 multipliers among the more than 534 million candidate multipliers.Section 1 reviews the basic properties of linear congruential generators and § 2 describes worst case performance measures. These include the maximal distance between adjacent parallel hyperplanes, the minimal number of parallel hyperplanes, the minimal distance between k-tuples, the lattice ratio and the discrepancy. Section 3 presents the five best multipliers and compares their performances with those of three commonly employed multipliers for all measures but the lattice test. Comparisons using packing measures in the space of k-tuples and in the dual space are also made. Section 4 presents the results of applying a battery of statistical tests to the best five to detect local departures from randomness. None were found. The Appendix contains a list of all optimal multipliers.

In Search of Correlation in Multiplicative Congruential Generators with Modulus 231 -1

Abstract This article presents the results of empirically testing 16 alternative multipliers for a multiplicative congruential random number generator with modulus 231 — 1. Two of the multipliers are in common use, six are the best of 50 candidate multipliers according to the theoretical spectral and lattice tests, and eight are the worst, with regard to 2-tuples, among the 50. The test results raise serious doubts about several of the multipliers, including one in common use. The tests were also applied to a well-known theoretically poor generator, RANDU, and gave strong empirical evidence of its inadequacy. Since comparison of the results of the First eight multipliers with those for the eight worst multipliers failed to show any apparent gross differences, one may want to relax the currently employed stringent criteria for acceptable performance on the lattice and spectral tests.

Starting Times for Data Collection in a Queueing Simulation I: Experiments with a Single Server Model.

Abstract This article describes a cutpoint sampling method for efficiently sampling from an n-point discrete distribution that preserves the monotone relationship between a uniform deviate and the random variate it generates. This property is useful for developing a sampling plan to reduce variance in a Monte Carlo or simulation study. The expected number of comparisons with this method is derived and shown to be bounded above by (m + n −1)/n, where m denotes the number of cut-points. The alias sampling method, which is regarded as the most efficient table sampling technique, generally lacks the monotone property and requires 2n storage locations, whereas the proposed cutpoint sampling method requires m + n storage locations. The article describes two modifications for cases in which n is large and possibly infinite. It is shown that circumstances arise in which the cutpoint method requires fewer comparisons on average than the alias method does for exactly the same space requirement. The article also des...

Monotonic generation of positive random variables

A reliability growth model is an analytic tool that accounts for changes in reliability due to design modifications and other corrective actions taken during the develpment, production, and use of a new piece of equipment. This paper describes a simulation study, and its conclusions, comparing four general reliability growth models that have been proposed in the reliability literature. Details of the simulation results are available in a separate Supplement.

Empirical testing of multiplicative congruential generators with modulus 2 31 −1

This paper discusses a procedure for interval estimation of the mean @@@@ of a correlated binary (0,1) sequence. The method assumes that the sequence is strictly stationary and that a particular string of m binary digits is a recurrent event in the sequence, where m≥l is unknown. Of the 2m choices for the possible recurrent events, the strings of all zeros and of all ones are examined. For each m&equil;1,2,... the sequence is demarcated by entrance to the recurrent event. The subsequences between the demarcation points thus form independent epochs by assumption. Classical techniques then yield variance estimates for the number of ones and zeros in the epoch as well as an estimate of the covariance of the ones and zeros. A quadratic equation in [?] is solved to obtain an interval estimate.

An exhaustive search for optimal multipliers

This paper describes an empirical search for correlation in sample sequences produced by 16 multiplicative congruential random number generators with modulus 231 - 1. Each generator has a distinct multiplier. One multiplier is in common use in the LLRANDOM and IMSL random generation packages as well as in APL and SIMPL/1. A second is used in SIMSCRIPT II. Six multipliers were taken from a recent study that showed them to have the best spectral and lattice test properties among 50 multipliers considered. The last eight multipliers had the poorest spectral and lattice test properties for 2-tupes among the 50. A well known poor generator, RANDU, with modulus 231, was also tested to provide a benchmark for evaluating the empirical testing procedure.