Pierre L'Ecuyer

TestU01: A C library for empirical testing of random number generators

We introduce TestU01, a software library implemented in the ANSI C language, and offering a collection of utilities for the empirical statistical testing of uniform random number generators (RNGs). It provides general implementations of the classical statistical tests for RNGs, as well as several others tests proposed in the literature, and some original ones. Predefined tests suites for sequences of uniform random numbers over the interval (0, 1) and for bit sequences are available. Tools are also offered to perform systematic studies of the interaction between a specific test and the structure of the point sets produced by a given family of RNGs. That is, for a given kind of test and a given class of RNGs, to determine how large should be the sample size of the test, as a function of the generators period length, before the generator starts to fail the test systematically. Finally, the library provides various types of generators implemented in generic form, as well as many specific generators proposed in the literature or found in widely used software. The tests can be applied to instances of the generators predefined in the library, or to user-defined generators, or to streams of random numbers produced by any kind of device or stored in files. Besides introducing TestU01, the article provides a survey and a classification of statistical tests for RNGs. It also applies batteries of tests to a long list of widely used RNGs.

Efficient and portable combined random number generators

In this paper we present an efficient way to combine two or more Multiplicative Linear Congruential Generators (MLCGs) and propose several new generators. The individual MLCGs, making up the proposed combined generators, satisfy stringent theoretical criteria for the quality of the sequence they produce (based on the Spectral Test) and are easy to implement in a portable way. The proposed simple combination method is new and produces a generator whose period is the least common multiple of the individual periods. Each proposed generator has been submitted to a comprehensive battery of statistical tests. We also describe portable implementations, using 16-bit or 32-bit integer arithmetic. The proposed generators have most of the beneficial properties of MLCGs. For example, each generator can be split into many independent generators and it is easy to skip a long subsequence of numbers without doing the work of generating them all.

Random number generation

The fields of probability and statistics are built over the abstract concepts of probability space and random variable. This has given rise to elegant and powerful mathematical theory, but exact implementation of these concepts on conventional computers seems impossible. In practice, random variables and other random objects are simulated by deterministic algorithms. The purpose of these algorithms is to produce sequences of numbers or objects whose behavior is very hard to distinguish from that of their “truly random” counterparts, at least for the application of interest. Key requirements may differ depending on the context.For Monte Carlo methods, the main goal is to reproduce the statistical properties on which these methods are based, so that the Monte Carlo estimators behave as expected, whereas for gambling machines and cryptology, observing the sequence of output values for some time should provide no practical advantage for predicting the forthcoming numbers better than by just guessing at random.

Uniform random number generators

In typical stochastic simulations, randomness is produced by generating a sequence of independent uniform variates (usually real-valued between 0 and 1, or integer-valued in some interval) and transforming them in an appropriate way. In this paper, we examine practical ways of generating (deterministic approximations to) such uniform variates on a computer. We compare them in terms of ease of implementation, efficiency, theoretical support, and statistical robustness. We look in particular at several classes of generators, such as linear congruential, multiple recursive, digital multistep, Tausworthe, lagged-Fibonacci, generalized feedback shift register, matrix, linear congruential over fields of formal series, and combined generators, and show how all of them can be analyzed in terms of their lattice structure. We also mention other classes of generators, like non-linear generators, discuss other kinds of theoretical and empirical statistical tests, and give a bibliographic survey of recent papers on the subject.

Random numbers for simulation

In the mind of the average computer user, the problem of generating uniform variates by computer has been solved long ago. After all, every computer :system offers one or more function(s) to do so. Many software products, like compilers, spreadsheets, statistical or numerical packages, etc. also offer their own. These functions supposedly return numbers that could be used, for all practical purposes, as if they were the values taken by independent random variables, with a uniform distribution between 0 and 1. Many people use them with faith and feel happy with the results. So, why bother? Other (less naive) people do not feel happy with the results and with good reasons. Despite renewed crusades, blatantly bad generators still abound, especially on microcomputers [55, 69, 85, 90, 100]. Other generators widely used on medium-sized computers are perhaps not so spectacularly bad, but still fail some theoretical and/or empirical statistical tests, and/or generate easily detectable regular patterns [56, 65]. Fortunately, many applications appear quite robust to these defects. But with the rapid increase in desktop computing power, increasingly sophisticated simulation studies are being performed that require more and more “random” numbers and whose results are more sensitive to the quality of the underlying generator [28, 40, 65, 90]. Sometimes, using a not-so-good generator can give totally misleading results. Perhaps this happens rarely, but can be disastrous in some cases. For that reason, researchers are still actively investigating ways of building generators. The main goal is to design more robust generators without having to pay too much in terms of portability, flexibility, and efficiency. In the following sections, we give a quick overview of the ongoing research. We focus mainly on efficient and recently proposed techniques for generating uniform pseudorandom numbers. Stochastic simulations typically transform such numbers to generate variates according to more complex distributions [13, 25]. Here, “uniform pseudorandom” means that the numbers behave from the outside as if they were the values of i.i.d. random variables, uniformly distributed over some finite set of symbols. This set of symbols is often a set of integers of the form {0, . . . , m - 1} and the symbols are usually transformed by some function into values between 0 and 1, to approximate the U(0, 1) distribution. Other tutorial-like references on uniform variate generation include [13, 23, 52, 54, 65, 84, 89].

Maximally equidistributed combined Tausworthe generators

Tausworthe random number generators based on a primitive trinomial allow an easy and fast implementation when their parameters obey certain restrictions. However, such generators, with those restrictions, have bad statistical properties unless we combine them. A generator is called maximally equidistributed if its vectors of successive values have the best possible equidistribution in all dimensions. This paper shows how to find maximally equidistributed combinations in an efficient manner, and gives a list of generators with that property. Such generators have a strong theoretical support and lend themselves to very fast software implementations.

Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators

Combining parallel multiple recursive sequences provides an efficient way of implementing random number generators with long periods and good structural properties. Such generators are statistically more robust than simple linear congruential generators that fit into a computer word. We made extensive computer searches for good parameter sets, with respect to the spectral test, for combined multiple recursive generators of different sizes. We also compare different implementations and give a specific code in C that is faster than previous implementations of similar generators.

Modeling Daily Arrivals to a Telephone Call Center

We develop stochastic models of time-dependent arrivals, with focus on the application to call centers. Our models reproduce three essential features of call center arrivals observed in recent empirical studies: a variance larger than the mean for the number of arrivals in any given time interval, a time-varying arrival intensity over the course of a day, and nonzero correlation between the arrival counts in different periods within the same day. For each of the new models, we characterize the joint distribution of the vector of arrival counts, with particular focus on characterizing how the new models are more flexible than standard or previously proposed models. We report empirical results from a study on arrival data from a real-life call center, including the essential features of the arrival process, the goodness of fit of the estimated models, and the sensitivity of various simulated performance measures of the call center to the choice of arrival process model.

Improved long-period generators based on linear recurrences modulo 2

Fast uniform random number generators with extremely long periods have been defined and implemented based on linear recurrences modulo 2. The twisted GFSR and the Mersenne twister are famous recent examples. Besides the period length, the statistical quality of these generators is usually assessed via their equidistribution properties. The huge-period generators proposed so far are not quite optimal in this respect. In this article, we propose new generators of that form with better equidistribution and “bit-mixing” properties for equivalent period length and speed. The state of our new generators evolves in a more chaotic way than for the Mersenne twister. We illustrate how this can reduce the impact of persistent dependencies among successive output values, which can be observed in certain parts of the period of gigantic generators such as the Mersenne twister.

Tables of linear congruential generators of different sizes and good lattice structure

We provide sets of parameters for multiplicative linear congruential generators (MLCGs) of different sizes and good performance with respect to the spectral test. For ` = 8, 9, . . . , 64, 127, 128, we take as a modulus m the largest prime smaller than 2`, and provide a list of multipliers a such that the MLCG with modulus m and multiplier a has a good lattice structure in dimensions 2 to 32. We provide similar lists for power-of-two moduli m = 2`, for multiplicative and non-multiplicative LCGs.