The most commonly used techniques to generate a sequence of pseudorandom numbers are those that apply some form of recursive computation. In principle, such recursive formulas are based on calculating the residuals modulo of some integers of a linear transformation. The process of producing a random number sequence is completely deterministic. However, the generated sequence would appear to be uniformly distributed and independent.

Congruential methods for generating n random numbers are based on the fun­damental congruence relationship, which can be expressed as (Lehmer, 1951)

Xi+i = {aXi + c}(mod m) i = 1,2,…, n (6.1)

in which a is the multiplier, c is the increment, and m is an integer-valued modulus. The modulo notation (mod m) in Eq. (6.1) represents that

Xi+1 = aXi + c – mIi (6.2)

with Ii = [(aXi + c)/m] denoting the largest positive integer value in (aXi + c)/m. In other words, Xi+1 determined by Eq. (6.1) is the residual resulting from (aXi + c)/m. Therefore, the values of the number sequence generated by Eq. (6.1) would satisfy Xi < m, for all i = 1, 2,…, n. Random number gener­ators that produce a number sequence according to Eq. (6.1) are called mixed congruential generators.

Applying Eq. (6.1) to generate a random number sequence requires the spec­ification of a, c, and m, along with X0, called the seed. Once the sequence of random number Xs are generated, the random number from the unit interval ui є [0,1] can be obtained as

Ui = — i = 1, 2,…, n (6.3)

m

It should be pointed out that the process of generating uniform random numbers is the building block in Monte Carlo simulation.

Owing to the deterministic nature of the number generation, it is clear that the number sequence produced by Eq. (6.1) is periodic, which will repeat itself in, at most, m steps. This implies that the sequence would contain, at most, m distinct numbers and will have a maximum period of length m — 1 be­yond which the sequence will get into a loop. For example, consider Xi+1 = 2Xi + 3(mod m = 5), with X0 = 3; the number sequence generated would be 4,1,0, 3,4,1,0,….

From the practical application viewpoint, it is desirable that the generated number sequence have a very long periodicity to ensure that sufficiently large amounts of distinct numbers are produced before the cycle occurs. Therefore, one would choose the value ofthe modulus m to be as large as possible. However, the length of the periodicity in a sequence also depends on the values of mul­tiplier a and increment c. Knuth (1981) derived three conditions under which a sequence from Eq. (6.1) has a full period m. Based on the three conditions of Knuth (1981), Rubinstein (1981) showed that for a computer with a binary digit system, using m = 2в, with в being the word length of the computer, along with an odd number for parameter c and a = 2r + 1, r > 2 would produce a full period sequence. The literature (Hull and Dobell, 1964; MacLaren and Marsagalia, 1965; Olmstead, 1946) indicates that good statistical results can be achieved by using m = 235, a = 27 + 1, and c = 1. Table 6.2 lists suggested values for the parameters in Eq. (6.1) for different computers.

A second commonly used generator is called the multiplicative generator-.

Xi+1 = {aXi }(mod m) i = 1,2,…, n (6.4)

which is a special case of the mixed generator with c = 0. Knuth (1981) showed that a maximal period can be achieved for the multiplicative generator in a binary computer system when m = 2e and a = 8r ± 3, with r being any positive integer.

Another type of generator is called the additive congruential generator having the recursive relationship as

Xi+1 = {Xi + Xi-t }(mod m) t = 1,2,…, i – 1 (6.5)

As can be seen, the random numbers generated by the additive congruen – tial generator depend on more than one of its preceding values. When t = 1, Eq. (6.5) would generate a sequence of Fibonacci numbers, which are not satis­factorily random. However, the statistical properties improve as t gets larger.

In summary, to ensure that a sequence of random numbers generated by the congruential methods would have satisfactory statistical properties, Knuth (1981) recommended the following principles to choose the parameters a, c, m, and X0. [9]

As noted previously, the accuracy of the model output statistics and probabil­ity distribution (e. g., probability that a specified safety level will be exceeded) obtained from Monte Carlo simulation is a function of the number of simu­lations performed. For models or problems with a large number of uncertain basic variables and for which low probabilities (<0.1) are of interest, tens of thousands of simulations may be required. Rules for determining the number of simulations required for convergence are not available, and thus replication of the Monte Carlo simulation runs for a given number of simulations is the only way to check convergence (Melching, 1995). Cheng et al. (1982) considered the convergence characteristics of Monte Carlo simulation for a simple case of Z = X3X4 – (X1 + X2), where the distributions and statistics of the variables are listed in Table 6.1. They found that failure probabilities (i. e., probability of Z < 0) down to 0.0025 could be estimated reliably with 32,000 simulations, failure probabilities down to 0.015 could be estimated reliably with 8000 sim­ulations, and failure probabilities down to 0.2 could be estimated reliably with 1000 simulations.

Problems involving more complex system functions Z and more basic vari­ables may require more simulations to obtain similar accuracy. For example, Melching (1992) found that 1000 simulations were adequate to estimate the mean, standard deviation, and quantiles above 0.2 for an application of the HEC-1 (U. S. Army Corps of Engineers, 1990) and RORB (Laurenson and Mein, 1985) rainfall-runoff models and that 10,000 simulations were needed to ac­curately estimate quantiles between 0.01 and 0.2. Brown and Barnwell (1987) reported that for the QUAL2E multiple-constituent (dissolved oxygen, nitrogen cycle, algae, etc.) steady-state surface water-quality model, 2000 simulations were required to obtain accurate estimates of the output standard deviation. With the computational speed of today’s computers, making even 10,000 runs is not prohibitive for simpler models. However, increased computational speed has made possible the use of computational fluid dynamics codes in three di­mensions for hydrosystems design work. When such codes are applied, the variance-reduction techniques described in Sec. 6.7 may be preferred to Monte Carlo simulation.

This chapter focuses on the basic principles and applications of Monte Carlo simulations in the reliability analysis of hydrosystems engineering problems. Section 6.2 describes some basic concepts of generating random numbers, fol­lowed by discussions on the classifications of algorithms for a generation of ran­dom variates in Sec. 6.3. Algorithms for generating univariate random numbers

are described in Sec. 6.4 for several commonly used distribution functions. In Sec. 6.5, attention is given to algorithms that generate multivariate ran­dom numbers. As reliability assessment involves mathematical integration, Sec. 6.6 describes several Monte Carlo simulation techniques for reliability evaluation. Given that Monte Carlo simulations, in essence, are sampling tech­niques, they provide only estimations, which inevitably are subject to certain degrees of errors. To improve the accuracy of the Monte Carlo estimation while reducing excessive computational time, several variance-reduction techniques are discussed in Sec. 6.7. Finally, resampling techniques are described in Sec. 6.8, which allow for assessment of the uncertainty of the quantity of interest based on the available random data without having to make assumptions about the underlying probabilistic structures.