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Second, the value of any number in the sequence must be statistically independent of the other numbers In practice it is dif cult to satisfy these criteria Some commercially available packages include random number generators that rely on the architecture of a particular computer and are, therefore, not portable to different machines Kahaner et al (1989), Press et al (1992) and Fishman (1995) provide detailed programming information or actual Fortran or C code that can be implemented on different machines Some analytical systems, such as MATHCAD, give references for the algorithms used to generate random numbers, but most spreadsheets do not In either case, the user should be aware of some of the considerations in the following paragraphs Most random number generators employ a linear congruential algorithm, in which a sequence of uniformly distributed random integers (I1 , I2 , , Ii , Ii+1 , .

Figure 5.3 Binary tree after second splaying step. The new item has been moved up by another level.

5

, In ) is generated from Ii+1 = a Ii + c (mod m) (1716) In this equation, a and c are constants and (mod m) indicates that the calculations are to be modulo m; that is, the result is divided by m and only the remainder retained Since integer operations are faster than real calculations, the usual procedure is to perform the calculations in integer mode and then divide the results by m to obtain real random numbers Zi between 0 and 1 Equation (1716) implies that each random variable depends only on the previous value in the sequence, if any value recurs, the sequence starts over, the length of the non-repeating sequence cannot exceed m, but it may be a great deal shorter..

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Figure 5.4 Binary tree after third splaying step. The new item has been moved up by yet another level.

Just think how many tacks and gybes you could make in the average race day If you could make each one 1/3 of a boat length better, that would add up to quite a few places when you think how close the eet usually nish..

PUSH EBP MOV EBP,ESP LEA EAX,DWORD PTR [EBP+C] PUSH EAX LEA EAX,DWORD PTR [EBP+8] PUSH EAX PUSH DWORD PTR [EBP+C] PUSH DWORD PTR [EBP+8] CALL ntdll.7C9215DA POP EBP RET 8

Therefore, m is usually taken to be as large as practicable In most applications m is 232 or 231 , or close to it In many applications c = 0, and the algorithm is simpli ed It can be shown that, with the appropriate choice of a and m, the results are as good as they would be with c = 0 However, it is critically important that the rst value of the sequence of random number, called the seed, not be set to zero, even though many users of random numbers are tempted to do so Park and Miller (1969) and Press et al (1992) endorse using a = 75 = 16,807 and m = 231 1 = 2,147,483,647 The most commonly used sequences of random numbers are not random at all but are determined exactly by the computational algorithm They are sometimes called pseudorandom numbers.

Figure 5.5 Binary after splaying process. The new item is now the root node, and the rest of the tree is centered on it.

However, the sequence must be suf ciently long that it does not repeat during the simulation If the parameters of the sequence are chosen with care, the sequence will appear random and can be used as though it were A linear congruential algorithm is very fast if implemented properly One dif culty with pseudo-random number generators of this type is well known in the simulation community but is not so familiar to casual users of simulation software As Press et al (1992) state, [The generator] is not free of sequential correlation on successive calls If k random numbers at a time are used to plot points in k dimensional space (with each coordinate between 0 and 1), then the points will not ll up the k-dimensional space, but rather will lie on (k 1)-dimensional planes There will be at most about m1/k such planes.

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