Search

The way engineering is being taught has changed in recent years with the introduction of commercial and educational software thus allowing a better understanding of the subject matter and increased teaching efficiency. This paper presents the educational software package designed to work in PCs and provides an attractive way to introduce the fundamentals of random variables. The software can be used as a complement to theoretical classes or alone as a self-study tool. The package was implemented in MATLAB and uses MATLAB™ tool "makeshow" which allows the students to create interactive slideshows without building their own graphic interface. The software is easy to use and is in accordance with the established standards, mouse operations, icon menus, etc. The demos include Probability Density Function (PDF), and Cumulative Distribution Function (CDF), normal, lognormal, exponential, Rayleigh and Rician random variables. The package also demonstrates how to generate the desired random sequence, how to examine its characteristics and how to estimate its PDF and CDF.

The acceptance/rejection approach is widely used in universal nonuniform random number generators. Its key part is an accurate approximation of a given probability density from above by a hat function. This article uses a piecewise constant hat function, whose values are overestimates of the density on the elements of the partition of the domain. It uses a sawtooth overestimate of Lipschitz continuous densities, and then examines all local maximizers of such an overestimate. The method is applicable to multivariate multimodal distributions. It exhibits relatively short preprocessing time and fast generation of random variates from a very large class of distributions.

We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions. The algorithms can be implemented in a few lines of high-level language code. In opposition to other black-box algorithms hardly any setup step is required, and thus it is superior in the changing-parameter case.

Applying the ratio-of-uniforms method for generating random variates results in very efficient, fast, and easy-to-implement algorithms. However parameters for every particular type of density must be precalculated analytically. In this article we show, that the ratio-of-uniforms method is also useful for the design of a black-box algorithm suitable for a large class of distributions, including all with log-concave densities. Using polygonal envelopes and squeezes results in an algorithm that is extremely fast. In opposition to any other ratio-of-uniforms algorithm the expected number of uniform random numbers is less than two. Furthermore, we show that this method is in some sense equivalent to transformed density rejection.

We suggest an interesting and fast method for generating normal, exponential, t, von Mises, and certain other important random variables used in Monte Carlo studies. The right half of a symmetric density is cut into pieces, then, using simple area-preserving transformations, reassembled into a rectangle from which the x-coordinate—or a linear function of the x-coordinate—of a random point provides the required variate. To illustrate the speed and simplicity of the Monty Python method, we provide a small C program, self-contained, for rapid generation of normal (Gaussian) variables. It is self-contained in the sense that required uniform variates are generated in-line, as pairs of 16-bit integers by means of the remarkable new multiply-with-carry method.

In this article the author describes a procedure for generating gamma variates with shape parameter >1. Given a supply of “good” uniform (0,1) variates, the procedure makes use of composition method, squeeze method, and aliasing to generate gamma variates. Comparison with existing methods shows that the author's method is faster in terms of computer time and uses a smaller number of uniform (0,1) variates. The method is also statistically exact.

Antithetic variates and control variates are two well-known variance reduction techniques. We consider combining antithetic variates and control variates to estimate the mean response in a stochastic simulation experiment. When applying antithetic variates to generate control variates across paired replications, we show that the integrated control-variate estimator is unbiased and yields, under the assumption of common correlations induced for all control variates, a smaller variance than the conventional control-variate estimator without using antithetic variates. We examine the proposed estimator and two alternative integrated control-variate estimators when applying antithetic variates on control variates and show that the proposed estimator is the optimal integrated control-variate estimator We implement these three integrated control-variate estimators and the conventional control-variate estimator in a simulation model of a stochastic network to evaluate the performance of each control-variate estimator Empirical results show that the proposed estimator outperforms the other control-variate estimators.

For discrete distributions a variant of reject from a continuous hat function is presented. The main advantage of the new method, called rejection-inversion , is that no extra uniform random number to decide between acceptance and rejection is required, which means that the expected number of uniform variates required is halved. Using rejection-inversion and a squeeze, a simple universal method for a large class of monotone discrete distributions is developed. It can be used to generate variates from the tails of most standard discrete distributions. Rejection-inversion applied to the Zipf (or zeta) distribution results in algorithms that are short and simple and at least twice as fast as the fastest methods suggested in the literature.

In this paper, we propose three combined Tausworthe random number generators with period length about 1018, whose k-distribution properties are good and which can be implemented in a portable way. These generators are found through an exhaustive search for the combination with the best lattice structure in GF{2, x }~, the k-dimensional vector space over the field of all Laurent series with coefficients in GF(2). We then apply a battery of statistical tests to these generators for the comprehensive investigation of their empirical statistical properties. No apparent defect was found. In the appendix, we give a sample program in C for the generators.

Stochastic models based on matrix exponential structures, like matrix exponential distributions and rational arrival processes, have gained popularity in analytical models recently. However the application of these models in simulation based evaluations is not as widespread yet. One of the possible reasons is the lack of efficient random variates generation methods. In this paper we propose methods for efficient random variates generation for matrix exponential stochastic models based on appropriate representations of the models.

This pedagogical tip presents a physical means for generating a random number - that turns out to be not so random after all.

A suitable square root transformation of a gamma random variable with mean a ≥ 1 yields a probability density close to the standard normal density. A modification of the rejection technique then begins by sampling from the normal distribution, being able to accept and transform the initial normal observation quickly at least 85 percent of the time (95 percent if a ≥ 4). When used with efficient subroutines for sampling from the normal and exponential distributions, the resulting accurate method is significantly faster than competing algorithms.

When the shape parameter, α , is integral, generating gamma random variables with a digital computer is straightforward. There is no simple method for generating gamma random variates with non-integral shape parameters. A common procedure is to approximately generate such random variables by use of the so-called probability switch method. Another procedure, which is exact, is due to Jöhnk. This paper presents a rejection method for exactly generating gamma random variables when α is greater than 1. The efficiency of the rejection method is shown to be better than the efficiency of Jöhnk's method. The paper concludes that when α is non-integral the following mix of procedures yields the best combination of accuracy and efficiency: (1) when α is less than 1, use Jöhnk's method; (2) when 1 is less than α and α is less than 5, use the rejection method; (3) when α is greater than 5, use the probability switch method.

This paper notes several results from statistical distribution theory which are useful in checking random number generators. It illustrates these methods by application to the normal random number generator GGNRF from the IMSL program package. Independence is checked with both time and frequency domain tests. The normal distributional properties are checked with goodness of fit tests and tests of the mean and variance.

A new rejection method is described for generating beta variates. The method is compared with previously published methods both theoretically and through computer timings. It is suggested that the method has advantages in both speed and programming simplicity over previous methods, especially for “difficult” combinations of parameter values.

The study of highly unstable nonlinear dynamical systems—chaotic systems—has emerged recently as an area of major interest and applicability across the mathematical, physical and social sciences. This interest has been triggered by advances in the past decade, particularly in the mathematical understanding of complex systems. An important insight that has become widely recognized in recent years is that deterministic systems can give rise to chaotic behavior. Surprisingly, many of these systems are extremely simple, yet they exhibit complex chaotic behavior.

An algorithm to generate variates having a gamma distribution with shape parameter greater than one is presented in this paper. This algorithm is faster than Schmeiser and Lal's G4PE, which is the fastest one currently available, yet is equally simple and easy to implement.

This paper presents the comparative results of several test runs involving the use of several unit uniform random number generators with an inverse transform to generate Erland random variates with λ=1 and k = 3, 4, 5, and 6. The results show that two commonly used generators (a multiplicative and a mixed one) produce unacceptable results. Two others (an additive one and a multiplicative one) produce acceptable results. The computer system used is an IBM 370/3033.

A method for generating values of continuous symmetric random variables that is relatively fast, requires essentially no computer memory, and is easy to use is developed. The method, which uses a uniform zero-one random number source, is based on the inverse function of the lambda distribution of Tukey. Since it approximates many of the continuous theoretical distributions and empirical distributions frequently used in simulations, the method should be useful to simulation practitioners.

A useful feature found in most programing languages, but missing from Ada, is a standard predefined (pseudo) random number generator. In this article we present an Ada implementation a "universal" (i.e., portable) random number generator developed by George Marsaglia. This implementation generates real numbers with 24-bit fractions, uniform on the interval (0,1). We also show a sample program along with the resulting random numbers that were generated by executing that program on a Sun workstation under UNIX.

The quixotic quest for the perfect random number generator has been an overriding factor for a group of simulation purists. Although we have developed a series of tests for the evaluation of random number generators, only a few of us engaged in modeling and simulation have actually taken the time to analyze the generators we use; most are willing to accept the random number generators available on the system -- just because it is there!

A very fast method for generating exponential random variables in a digital computer is outlined.

Extremely portable subroutines are sometimes needed for which moderate quality and efficiency suffice. Typically, this occurs for library functions (like random number generation and incore sorting) which are not entirely universal or are not used in a standardized way. The literature on random number generators does not seem to contain an algorithm that meets requirements of this sort. An extremely portable 8-line FORTRAN program is provided which is based on an important paper by Coveyou and MacPherson (1967). Using their methods, Fourier analysis is applied to the probability function for the consecutive n -tuples provided by our generator (with n less than or equal to 4). While the small modulus which must be used to maintain portability prevents the quality of the generator from being high, the generator compares well with the bounds established in the above mentioned paper.

The recent flurry of random number routines and your continued plea for material had led me to dust off my favorite trick uniform random number generator. It requires no seed, produces numbers uniformly distributed between 0 and 1, is acyclic and for all practical purposes, is completely unpredictable and irreproducible.