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The concept of random data through a simulation example, focusing on self-similarity and exponential arrivals. It discusses the differences in results for various processes and compares them with theoretical distributions. The document also covers data generation and the importance of inverse functions.
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Random Data
Iterations Proc 0 100,000,000 - 200,000,000 - 300,000,000 5141 400,000,000 3197
500,000,000 - 600,000,000 - 700,000,000 - 800,000,000 - 900,000,000 - 1,000,000,000 -
After 400 million iterations, there were 3192 more “heads” than “tails”.
Random Data
Iterations Proc 0 Proc 1 Proc 2 Proc 3 Proc 4 Proc 5 Proc 6 Proc 7 100,000,000 -10299 -9319 -1063 6743 8633 -4421 8123 - 200,000,000 -4245 -10227 3657 -23059 24885 -26655 25865 - 300,000,000 5141 -6819 255 -20175 14469 -33389 27077 -
400,000,000 3197 -8155 -5379 -6633 27387 -50509 24531 2339 500,000,000 -1313 -10547 -153 -14679 29335 -51963 23097 - 600,000,000 -25941 -29847 -26371 5027 32857 -49505 27089 - 700,000,000 -24093 -26331 -43401 13153 24471 -26899 4561 -
800,000,000 -24661 -35315 -31233 41 20425 -11861 13837 - 900,000,000 -27123 -33049 -44461 -11769 -3283 -12477 15865 -
1,000,000,000 -23997 -15483 -44535 22889 -8447 -13671 15743 6023
Random Data
The Max and Min values in All Runs
0
50000
100000
150000
200000
250000
0 200000000 400000000 600000000 800000000 1000000000 1200000000 Iterations
Max/Min + 125000
Min of all runs Max of all runs
Random Data
Secs Proc 0 Proc 1 Proc 2 Proc 3 Proc 4 Proc 5 Proc 6 Proc 7 Proc 8 Proc 9
1000 21660 21650 21810 21800 21790 21720 21850 21740 21640 21730
2000 43000 42960 43080 43120 43220 42960 43190 43110 42900 43080
3000 64790 64650 64850 64930 65060 64680 64900 64860 64770 64940
4000 86610 86450 86620 86680 86750 86530 86640 86660 86560 86690
5000 108450 108280 108370 108450 108520 108410 108480 108380 108400 108580
6000 130010 129860 129990 129950 129980 130050 130090 130010 129910 130080
7000 151730 151600 151710 151730 151730 151770 151750 151820 151750 151800
8000 173340 173340 173400 173640 173480 173400 173520 173660 173470 173500
9000 194950 195050 195010 195300 195090 195000 195230 195440 195130 195150
10000 216760 216880 216780 217140 216860 216740 216990 217240 216880 216960
Random Data
Disk Access Rates With Time - It is (Max Access - Min Access)
0
100
200
300
400
500
600
0 2000 4000 6000 8000 10000 12000 Time (Seconds)
Difference In Accesses
Random Data
Process Writes - How they deviate from the minimum value
0
100
200
300
400
500
600
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Tim e (Seconds)
Process Writes compared to minimum process
Proc 0 Proc 1 Proc 2 Proc 3 Proc 4 Proc 5 Proc 6 Proc 7 Proc 8 Proc 9
Another Numerical Example
Another Numerical Example
// ////////////////////////////////////////////////////////////////////// // We're trying to solve the following problem. // Given two circles, how close should the centers of the circles be such // that the area subtended by the arcs of the two circles is exactly one // half the total area of the circle. // // See example 2.3.8 in Leemis & Park. // We use the book's definition for Uniform - see 2.3. // Here's how this works. Try a number of different distances between // the two circle centers. Then for the ones that are most successful, // zoom in to do them in more detail. // ////////////////////////////////////////////////////////////////////// #include <math.h> #include <stdlib.h> #define PI 3. #define TRUE 1 #define FALSE 0
// Prototypes double GetRandomNumber( void ); void InitializeRandomNumber( ); double ModelTwoCircles( double, int ); double Uniform( double min, double max) { return( min + (max - min)*GetRandomNumber() ); }
double ModelTwoCircles ( double Distance, int NumberOfSamples ) { double HitsInOneCircle = 0, HitsInTwoCircles = 0; double x, y, SecondDistance; int Samples; for ( Samples = 0; Samples < NumberOfSamples; Samples++ ) { do { x = Uniform( -1, 1 ); y = Uniform( -1, 1 ); } while ( (x * x) + (y * y) >= 1 ); // Loop until value in circle
HitsInOneCircle++; SecondDistance = sqrt( ( x - Distance ) * (x - Distance ) + (y * y) ); if ( SecondDistance < 1.0 ) { HitsInTwoCircles++; // printf( "Samples: Second Distance = %8.6f\n", SecondDistance ); } } // End of for return( HitsInTwoCircles / HitsInOneCircle ); }
Network Arrivals
What Did Leland et. al Measure?
Millions of packets from many workstations, as recorded on Bellcore internal networks.
What Did Leland et.al Measure?