Pseudorandom number

Monte Carlo Method The Monte Carlo method makes use of random numbers. A digital computer can be used to generate pseudorandom numbers in the range from 0 to 1. To describe the use of random numbers, let us consider the frequency distribution cui ve of a particular factor, e.g., sales volume. Each value of the sales volume has a certain probabihty of occurrence. The cumulative probabihty of that value (or less) being realized is a number in the range from 0 to 1. Thus, a random number in the same range can be used to select a random value of the sales volume. 

The Monte Carlo method is especially suited for use on a digital computer, particularly one of the stored-program type. The mathematical model and the distribution function, even if quite complicated, can be expressed on the computer and the necessary calculations are highly repetitive. Also, random numbers (or rather pseudorandom numbers) can be synthesized so that the computer procedure becomes fully automatic and self-contained (M9, S5). 

The variance estimators s20j) of the estimated effects allow unequal response variances and the use of common pseudorandom numbers. This is a well-known technique used in simulation experiments to reduce the variances of the estimated factor effects (Law and Kelton, 2000). This technique uses the same pseudorandom numbers when simulating the system for different factor combinations, thus creating positive correlations between the responses. Consequently, the variances of the estimated effects are reduced. This technique is similar to blocking in real-world experiments see, for example, Dean and Voss (1999, Chapter 10). 

We give only a short description of the three supply chain configurations and their simulation models for details we refer to Persson and Olhager (2002). At the start of our sequential bifurcation, we have three simulation models programmed in the Taylor II simulation software for discrete event simulations see Incontrol (2003). We conduct our sequential bifurcation via Microsoft Excel, using the batch run mode in Taylor II. We store input-output data in Excel worksheets. This set-up facilitates the analysis of the simulation input-output data, but it constrains the setup of the experiment. For instance, we cannot control the pseudorandom numbers in the batch mode of Taylor II. Hence, we cannot apply common pseudorandom numbers nor can we guarantee absence of overlap in the pseudorandom numbers we conjecture that the probability of overlap is negligible in practice. 

Fig. 4. Realization of bond percolation on a 50 X 60 section of the square lattice for four different values of q. The diagrams have been created using the same sequence of pseudorandom numbers, with the result that each graph is a subgraph of the next. Attentive readers may verify that open paths exist joining the left to the right side when = 0.51 but not at q = 0.49. (From Ref. 13, with permission.) continued)...

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The random generator seed completely determines the sequence of the pseudorandom numbers that will be used for random events in simulations. Depending on the platform the sofiware is used on, the random generator seed may be set globally. This has to be taken into account when running simulation simultaneously. 

Compared to the other techniques, simulation is a brute force approach using pseudorandom numbers to reproduce the uncertainty in the model, sample model behavior is generated and analyzed. 

This method of incorporating randomness into computer simulations has a profound impact on the design and analysis of simulation experiments. Most importantly, it means that different simulation runs will be dependent if they employ the same pseudorandom numbers. This can be good, yielding sharper comparisons between alternative systems, or bad, invalidating the assumptions behind statistical procedures that assume independent observations. 

An important feature of most simulation languages is that they permit control of the pseudorandom numbers through random number streams or seeds, which permit the user to access different, and thus apparently independent, portions of the pseudorandom number sequence. Such control allows the user to induce dependence where desired or obtain independence where necessary. 

Observations or realizations of the inputs are obtained by transforming the pseudorandom numbers. One or more pseudorandom numbers may be required to produce each input, depending on what transformation is used. The inputs, X, are functions of the pseudorandom numbers, u, so they are completely determined by the seed or stream, s, say X = X(s). 

Perhaps it seems strange that statistical methods are employed to analyze a completely deterministic process, which a simulation is after the seeds are specified (particularly since many users accept the default seeds or streams). However, if the pseudorandom numbers cannot easily be distinguished from random numbers, then treating functions of these numbers—the inputs, outputs and statistics— as random variables will not be misleading. 

The role of the pseudorandom number streams or seeds is important. This method of representing randomness, and the corresponding control it permits, is the primary difference between simulation experiments and classical statistical experiments. 

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