This exercise examines the basic principles and concepts of discrete event simul

Question

This exercise examines the basic principles and concepts of discrete event simulation. Let's start with a single channel queue. Assume a small grocery store has only one checkout counter. Customers arrive at this checkout counter at random intervals from 1 to 8 minutes apart with each possible value of integer interarnval time having the same probability of occurrence. That is, customer arrival times are uniformly distributed over the eight integer values 1 through 8. Assume that customer service times vary from 1 to 6 minutes with the following probabilities: The essence of manual simulation is a simulation table. The table construction is designed for the problem at hand and structured to answer any posed questions. For this simple queue we may want to analyse and answer the following question: What is the average customer waiting time? What is the probability a customer had to wait? What portion of the time is the server idle? What is the average service time? What is the average time between arrivals? What is the average waiting time for those customers that had to wait? What is the average time a customer spends in the system?

Explanation / Answer

We start with random digit assignment as shown in the table below (on the basis of cumulative frequency):

Next we will generate random variables and then determine time between arrivals for 20 customers (i have taken 20 customers to solve the problem). Random digit for customer 2 = 913. The figure 0.913 falls in the range of cumulative probabilities of 0.876 and 1.000. This will result in 8 minutes as the generated time.

Now we will determine the service time of each customer. The 1st customer's service time = 4 as the random digit 84 (0.84) falls in the in the cumulative frequency of 0.61 to 0.85.

Now we will make the simulation table for the problem. The 1st customer will arrive at time 0. Service time is 4. So he was in the queue for 4 minutes. 2nd customer arrives at 8th minute. Thus idle time = 8-4 = 4.

1. average wait time = total wait time/no. of customers = 56/20 = 2.8 minutes

2. probability = no. of customers who have to wait/total customers = 13/20 = 0.65

3. portion of idle time = total idle time/total run time of simulation = 18/86 = 0.21

4. average service time = total service time/total customers = 68/20 = 3.4 minutes

5. average time between arrivals = total of all times between arrivals/(number of arrivals-1) = 82/19 = 4.3 minutes

6. average waiting time for those who waited = total wait time/total customers who waited = 56/13 = 4.3 minutes

7. average time in the system = total time spend in the system/total customers = 124/20 = 6.2 minutes

Service time Probability Cumulative frequency Random digit assignment 1 0.1 0.1 "1-10" 2 0.2 0.3 "11-30" 3 0.3 0.6 "31-60" 4 0.25 0.85 "61-85" 5 0.1 0.95 "86-95" 6 0.05 1 "96-00"

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