Report Simulation

on. The Random numbers are a necessary basic ingredient in the simulation of almost all discrete systems. Here, the random numbers are generated using Microsoft excel for the given two variables. Mainly, the linear congruential method is used here. In order to do so, the following assumptions were made:

Base: 333

Multiplier: 312

Summand: 152

Seed: 382

The below figure represents the random number generated here for running the simulation. The details of the random numbers are found in the attached file with this report (Alonso & Schott, 1995).

[b] Case 1: Test for uniformity

To test whether the random numbers are generated here are uniform or not, chi square is followed here. The details of the Chi Square Test are shown below:

Step 1: Calculation

Here, the chi-square test with = 0.05 is employed to test for whether the random number series shown below are uniformly distributed. Here, n = 10 intervals of equal length.

Class	Interval	Oi	Ei	(Oi-Ei)^2/Ei
1	0.00 < R < 0.10	33	30	0.300
2	0.10 < R < 0.20	33	30	0.300
3	0.20 < R < 0.30	66	30	43.200
4	0.30 < R < 0.40	1	30	28.033
5	0.40 < R < 0.50	34	30	0.533
6	0.50 < R < 0.60	0	30	30.000
7	0.60 < R < 0.70	0	30	30.000
8	0.70 < R < 0.80	0	30	30.000
9	0.80 < R < 0.90	67	30	45.633
10	0.90 < R < 1.00	66	30	43.200
		300	300	251.200

Step 2: Hypothesis

H₀: The numbers are uniformly distributed on the interval [0, 1];

H₁: The numbers are non-uniformly distributed;

Step 3: Test statistics

^X2    

Step 4: Critical value

Step 5: Decision

Since, 

(Aritzhaupt.com, 2015)

The null hypothesis is rejected here.

So, there is enough proof to conclude that the distribution of interarrival time is not uniform.

Case 2: Test for independence

Hypothesis: 

In order to test the independency of the random numbers, autocorrelation test is used here.

Here, for the generated random numbers, auto correlation test is used to test whether the 3^rd, 8^th, 13^th, and so on numbers are auto correlated using α = 0.05.

Here,  i = 3

m = 5

N = 300

Therefore,

i + (M + 1) m <=N

or, 3 + (M + 1) 5 <=300

or, 3 + 5M + 5 <= 300

or, 8 +5M <= 300

or, M <= 292 / 5

or, M = 58

Now,

Therefore,  

=0.015663

And

Therefore,

= 0.401983186

Now, the critical value ± z_0.025 = ±1.96.

Since, z₀ > -1.96, null hypothesis should not be rejected here.

Therefore, there is sufficient evidence to conclude that the random number series are independent (Barker & Kelsey, 2006).

[c].

Random variate generation:

Interarrival time:

(Barker & Kelsey, 2007)

Service time:

(Eg.bucknell.edu, 2015)

Simulation:

(Niederreiter, 1992)

Therefore, Average Interarrival time = (sum of Interarrival times) / (number of arrivals – 1)

= 692.0671 / 123 = 5.627 min

Average service time = total service time / number of customers = 1730.1678 /222 = 7.794 min

 (Gentle, 1998)

After running the simulation for several times, it is noted that if the management add another skilled worker, then, the system will become more efficient. Hence, it is recommended that the management should add another worker into the current queuing system.

References

Alonso, L., & Schott, R. (1995). Random generation of trees. Boston: Kluwer Academic.

Aritzhaupt.com,. (2015). Autocorrelation Test Tutorial. Retrieved 19 January 2015, from https://www.aritzhaupt.com/resource/autocorrelation/

Barker, E., & Kelsey, J. (2006). Recommendation for random number generation using deterministic random bit generators. [Gaithersburg, MD]: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology.

Barker, E., & Kelsey, J. (2007). Recommendation for random number generation using deterministic random bit generators (revised). [Gaithersburg, MD]: U.S. Dept. of Commerce, Technology Administration, National Institute of Standards and Technology, Computer Security Division, Information Technology Laboratory.

Eg.bucknell.edu,. (2015). Tests for Auto-correlation. Retrieved 19 January 2015, from https://www.eg.bucknell.edu/~xmeng/Course/CS6337/Note/master/node45.html

Gentle, J. (1998). Random number generation and Monte Carlo methods. New York: Springer.

Niederreiter, H. (1992). Random number generation and quasi-Monte Carlo methods. Philadelphia, Pa.: Society for Industrial and Applied Mathematics.