Report Simulation

Question: Simulation of queuing system The management of a company providing financial service is concerned with the existing single channel queuing system adopted by the customer service unit. The management is considering of adding another skilled worker hoping that it will make the system more efficient and beneficial to their customers. The estimated mean interarrival time is 4 minutes and the estimated mean service time is 10 minutes. The estimated cost of waiting in queue is RM 15 per hour. The hourly salary of each worker is RM 12 per hour. Required: (a). Using an appropriate method, generate 300 random numbers. Let the last three digits of your ID as the seed. (b). Are the random numbers generated in (a) uniform and independent to each other? Show your proof. (c). Based on the probability distributions of your choice, develop a random variate generate for generating interarrival and service times. Then, use random numbers produced in (a) to generate interarrival and service times. Based on your results above, should the management maintain the current queuing system or hire another worker? Answer: [a].Random number generation: It is noted that the financial service company used a single channel queuing system for servicing its customer. In this context, the given information shows that the estimated mean interarrival time is 4 minutes and the estimated mean service time is 10 minutes. The estimated cost of waiting in queue is RM 15 per hour. The hourly salary of each worker is RM 12 per hour. Since, the management is concerned about its single queue system; a simulation study has been executed here, to deduce a meaningful conclusion. 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 3rd, 8th, 13th, and so on numbers are auto correlated using = 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 =0.038964 Therefore, = 0.401983186 Now, the critical value z0.025 = 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.