On randomness of random number generators

On randomness of random number generators

Citation for published version (APA):

Verhaegh, W. F. J. (1989). On randomness of random number generators. (Memorandum COSOR; Vol. 8920). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1989

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

Memorandum COSOR 89-20 On randomness of random

number generators W.F.I. Verbaegh

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513

5600 MB Eindhoven The Netherlands

Eindhoven, August 1989 The Netherlands

On randomness of random number generators

W.FJ. Verhaegh

Eindhoven University of

Technology

17-8-1989

Abstract

Many computer simulations use random number generators and since most

computer languages have a built-in generator it is very easy just to use

that

one.

However these random number generators can be very non-random.

1.

Introduction

This paper is a warning for them who use random number generators that are already

imple-mented on a computer system. These generators can be very bad for simulations. At least in our

case the random number generator we used had a very bad perfonnance. We were simulating a

queueing network and we wanted to detennine some average waiting time. Among the situations

we simulated were some situations for which we could detennine the average waiting time

analytically, by means of the theory of M

I

Mil-queues [1]. So we were able

to

compare the

results of the simulations to the theoretical values.

We simulated a long run and split this run in ten subruns. Each subrun contained 10,000 arrivals

so the dependency between two subruns was small enough. Then for each subrun we detennined

the experimental average waiting time.

and

we used these values of subrun 2 - 10 to detennine an

estimation for the average waiting time and the standard deviation of this estimation [2].

The results showed something

strange~

For example in a situation for which the occupancy-level

was 80

% and

for which

the

waiting time could be detennined like in a M

I

M II-queue, the results

were:

-2-subrun average waiting time

2 4.192 3 4.363 4 4.653 5 4.211 6 4.383 7 4.564 8 3.953 9 4.319 10 4.690 estimation 4.370 standard deviation 0.079

But the theoretical average waiting time is 4. so the results are very unlikely. Therefore we doub-lechecked our simulation algorithm. by tracing the events. but we didn't find any error. The only thing left which could have caused the bad results was the random number generator. So we decided to do the simulation with other sequences of random numbers. We did this by changing the initial seed of the random number generator (we will come back to this soon). The simulation above was done with initial seed Xo = I, the default value. We performed some other simulations and the results were:

subrun average waiting time

Xo= 1234 Xo=236 Xo =528194 2 4.282 2.627 3.692 3 4.281 2.624 3.767 4 4.474 2.625 3.662 5 4.331 2.618 3.506 6 4.286 2.633 3.343 7 4.465 2.603 2.817 8 4.059 2.633 3.281 9 4.417 2.623 3.408 10 4.419 2.619 3.574 estimation 4.331 2.623 3.450 standard deviation 0.044 0.003 0.096

These results are extremely unlikely so we decided to examine the random number generator thoroughly.

2. The random number generator

First we tried to find out what random number generator we had used. This generator was the

standard random number generator of the PASCAL-compiler of the VAX-UNIX (Ultrix V2.2) at

-3-congruential generator, i.e.

with

a

=

62605,

c

=

113218009 and

m

=

536870912

=

229• However, we discovered that this was

not true. The actual generator was

with

a

= 1103515245 and

m

= 2147483647 = 231_1. Xo is the initial seed, 0 <Xo <

m,

and the

ith random number'i is Xilm. This value of m is prime so the sequence ('i)i~ 1 is periodic, and

for this value of

a

the random sequence has maximal period length,

m

-1, because

a

is

a

primitive

element in GF(m). That means alll

-1 mod m

=

I, and ak mod m :1:1 for k = 1,2, ... ,m-2 [4].

3. A Chi-square test

The first test we perfonned was a Chi-square test of the unifonn independent distribution of the

random numbers between 0 and 1. This intelVal was partitioned in 400 intelVals of equal length

(1/400) and for each intelVal i: [ (i - 1)/400, il400 )

Ii was detennined, where Ii is the number of

,/s, j=I,2, ... ,I00000, contained in intelVal i. The expected value Ii

=

250 and if the ~om

numbers are independent and uniformly distributed then

Ii is binomial distributed. Because Ii is

big enough (»5)

Ii

is about nonnally distributed with average

Ii

=

250 and variance

100000 x 1J400 X 399/400 ::::

t.

Then

- 2

400 (Ji

-Ii)

'1.2 :=

'L---i-I

Ii

is about Chi-square distributed with 399 degrees of freedom [5]. The critical value of '1.;99 of a

1 %0 (I) unreliability test is about 491. The values of the tests were:

Xo 1 1234 236 528194

'1.2 2285 2141 119510 124351

So we can say that the random number generator is very bad. These results are so extremely

unlikely that we decided to look at the values of

Ii for X

0 = 528194 in detail. For this value of X 0

~4-1295

i

972 649 ~ •• ~ • • • • • • • • • • • • • • m . -... ~-... ""'" ... --" ... - .. -~;. .. _ ... - .~ ... - .. ~ ... ,. _ ... '"---- .. _ .. -_ ... _ .... ~ .. + "4 ... .1' .. " ' - " . " _ _ •• - . _ ... - _ • • ~ .. _ _ _ _ _ ... _ _ - .. -.... ~-... _.",. - _ . _ - . - • • • • - - . I 400

So this random number generator is definitely not random. 4. The spectral test

To get more insight in our random number generator, we apply the spectral test on it. According to D.E. Knuth [6] this is a very important test. He says about this test:

... not only do all good generators pass this test, all generators now known to be b~d

actually fail it. Thus it is by far the most powerfull test known, and it deserves

particu-lar attention."

The test detennines how random all points of t consecutive random numbers (rll,rll+l .... ,rll+l-l)

are distributed in [O,l]t. It does this by looking at parallel hyperplanes through all these points

and detennining the distance between these hyperplanes. Then the maximum distance is taken

over all sets of parallel hyperplanes that cover the points. For example, when t

=

2 and the

gen-eratorisXj =7 XXj_l mod 11 andXo

=

I, then

7 5 2 3 10 4 6 9 8 1 7 5

(ri)i~

1

=

11'11'11'11'11'11'11'11'11'11'11'11""

-5-1

...

•••.•• A'

.,-...,

"

,

" , ••••• oIi. ,

"

,~...,'

,

,

....

,

,

,

...

.. .,s....

_,...

"

_'

"

I •.••••• " , ···A ,

,

,-

...

,

"

"

.... -p. ..

...•

_,

_{.. ...'}

" "

_,

,

••• 1!f.

'

"

,

...•

'

" "

... . ..

o

o

1

For t

=

2 the hyperplanes are lines. The dotted lines form one set of parallel lines and the dashed

lines form another set (there are of course many other sets). The distance between the dotted lines is the maximum, so this is taken.

The smaller this maximum distance is, the better the distribution of the points in [0,1], is, and

thus the better the generator is. Let lIVt be that maximum distance.

We apply the spectral test for t

=

2,3,4,5,6, which is sufficient according to Knuth. The values of

the Vt in our case (m

=

2147483647, a

=

1103515245) are the following

t 2 3 4 5 6

Vt 45096.01 1413.21 197.25 67.79 35.50

Now what criterion do we use to determine whether a random number generator passes the spec-tral test or not? Knuth notes that this depends on the application and that it appears that in most

applications Vt ~ 2301t for t

=

2,3,4,5,6 will be quite adequate. The values of 2301t for

t

=

2,3,4,5,6 are:

t 2 3 4 5 6

2301t 32768 1024 181.02 64 32

Now we see that for our random number generator the inequalities hold, so our generator passes the spectral test!

For some purposes we would like a criterion that is relatively independant of m, so we can say

that a particular multiplier a is good or bad with respect to the set of all other multipliers for the

given m, without examining any others. According to Knuth [6, page 101] a reasonable figure of

merit for rating the goodness of a particular multiplier seems to be the volume of the ellipsoid in

-6-(

xlm -XZa - ... -x,a

t-l)l +X1 + ... 2

+Xt ;;::

2.... V, 2

This volume is

where

For our

a

and

m

this gives

t 2

""

2.98

x'12

v:

"" =

(t/2)!

m

3 4 5.51 3.48 5 6 3.51 4.81

Now Knuth says that the multiplier a passes the test if ~ ~ 0.1 for t

=

2.3.4,5.6 and it "passes with flying colors" if ~ ~ 1 for t

=

2,3,4,5.6. So our

a

passes this test very well!

5. Conclusion

We have seen that the global behaviour of our random number generator is very good. but the local behaviour is extremely bad. Therefore we advise always to apply several tests to a random

number generator before using it. Further we advise to do also some simulations of situations for

which one can compute the results analytically, and compare the experimental values to the

theoretical values. This is also a test on the randomness of the generator.

If one wants to take an other random number generator. Park and Miller [7] have some nice suggestions (for example

m

=

231-1 and

a

=

39373 or

a

=

48271).

[1] Kleinrock. L., Queueing Systems, vol. 1: Theory, John Wiley & Sons, New York, London,

Sydney, Toronto, 1975. pp. 94-99.

[2] Bratley, P .• Fox, B.L .• Schrage. L.E., A Guide To Simulation. 2nd Ed, Springer-Verlag inc.,

New York, 1987.

[3] Ultrix-32™ Supplementary Documents, vol. 2: Programmers, Digital Equipment

Corpora-tion, 1984, part 2: Languages, pp. 208.

[4] Biggs, N.L., Discrete Mathematics, Clarendon Press, Oxford, 1985, pp.344-353.

[5] Hogg, RV., Tanis, E.A.. Probability And Statisticalln/erence , Macmillan Publishing Co.

Inc., New York. 1977. pp. 329-358.

[6] Knuth, D.E., The Art Of Computer Programming, 2nd Ed., vol. 2: Seminumerical

Algo-rithms, Addison-Wesley, Reading, Massachusetts, 1981, pp. 89-110.

[7] Park, S.K., Miller, K. W., Random Number Generators: Good Ones Are Hard To Find,