The Poisson function, r -C P(r) = C * e / r! where C = the distribution average,

Question

The Poisson function,

              r     -C

P(r) = C * e   / r! where C = the distribution average, is useful for modeling a distribution when the probability of any one occurrence of P(r) is extremely small and r is the number of occurrences of the event. Applying this average to message volumes and interarrival rates, C = * T where = message volume and T = interarrival rate, would imply:

                            r      (- T)

        P(r) = ( T) * e          / r!

Assume the special case r = 0 (no messages arrive for a specific length of time) -- ie

              0      (- T)            (- T)

        (T) * e          / 0! = e

                          t=T

Let us say that      f(t)dt represents the probability of "one arrival within T units of

                        t=0

time after prior arrival." Therefore the probability of no messages

          (- T)            t=T                   t=T                  (- T)

        e        = 1 -        f(f)dt and      f(t)dt = 1 - e

                            t=0                   t=0

For purposes of simulation let the integral be represented by a random number RN such that 0 <= RN < 1.

                (- T)                               (-T)

Thus 1 - e         = RN, (1 - RN) = e         , LN(1 - RN) = - T and

-1/ * LN(1 - RN) = T = interarrival time.

Develop a simulation program to accomplish the following:

1) Ask for and accept message volume per hour, Lambda.

2) Convert this volume to message volume per second, Lambda/3600.

3) Ask for and accept the number of simulations desired.

4) Ask for and accept a random number generator seed.

5) Perform the simulation loop to determine and print interarrival

     times for Lambda where RN is determined by a random number

     generator function.

6. Compute the average message interarrival time = sum of all interarrival times / count.

Use volumes of 3600 and 7200 with 100 simulations each.

Explanation / Answer

#include <iostream>
#include <stdlib.h> /* srand, rand */
#include <math.h> /* log */

/*-1/ L * LN(1 - RN) = T = interarrival time.

Develop a simulation program to accomplish the following:
1) Ask for and accept message volume per hour, Lambda.
2) Convert this volume to message volume per second, Lambda/3600.
3) Ask for and accept the number of simulations desired.
4) Ask for and accept a random number generator seed.
5) Perform the simulation loop to determine and print interarrival
times for Lambda where RN is determined by a random number
generator function.
6. Compute the average message interarrival time = sum of all interarrival times / count.
Use volumes of 3600 and 7200 with 100 simulations each.
*/

int main(int argc, char** argv) {
   //*****************************************************
   // Declaration of variables
   int Lambda, simulation, seed,i;
   float lambda_per_second, T, RN, avg=0;
  
  
   //******************************************************
   // 1) Ask for and accept message volume per hour, Lambda.
   printf ("Please insert the number of message volume per hour, Lambda ");
   scanf ("%d",&Lambda);
  
   //*******************************************************
   //2) Convert this volume to message volume per second, Lambda/3600.
   lambda_per_second=Lambda/3600.00;
   // printf ("%f", lambda_per_second);
  
   //*******************************************************
   // 3) Ask for and accept the number of simulations desired.
   printf (" Please insert the number of simulations desired, simulation ");
   scanf ("%d",&simulation);
  
   //********************************************************
   // 4) Ask for and accept a random number generator seed.
   printf (" Please insert a random number generator seed, seed ");
   scanf ("%d",&seed);
   srand(seed);  
  
   //********************************************************
   /* 5) Perform the simulation loop to determine and print interarrival
times for Lambda where RN is determined by a random number
generator function.
   -1/ L * LN(1 - RN) = T = interarrival time
   *******************************************************/
for(i=1; i<=simulation; i++)
{
    RN=rand() % 10000 + 0;
       RN=RN/10000;
       // printf("random number is %f ",RN);
    T=-log(1-RN)/lambda_per_second;
   printf ("interarrival time for Lambda %d is %f ",i,T);
/***********************************************************
   6. Compute the average message interarrival time = sum of all interarrival times / count.
   ***********************************************************/
       avg=T+avg;
          
   }
   avg=avg/simulation;
   printf("The average time is %f",avg);
   return 0;
}

/*********************************************************************************************

********************************************************************************************

with 3600 The average time is 1.082516

with 7200 The average time is 0.590793

if you have any question about it please leave me a comment

*************************************************************************/

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