Problem
During the dinner hour, the distribution of the inter-arrival time of customers at a restaurant is estimated to be as shown below. The mode of payment and the service times of the cash and credit card customers are shown in the following tables. Complete the tables and simulate the system for 20 customer arrivals and determine the average time a cash and credit card customer must wait in line before paying the cashier. Use Column A of the given random number table to determine the customer inter-arrival time, Column B to determine whether the customer pays with cash or credit, and Column C to determine the service time.
Inter-arrival time
Inter-arrival Time Probability Cumulative Probability Random Number Interval
30 seconds 0.45
60 seconds 0.25
90 seconds 0.15
120 seconds 0.10
150 seconds 0.05
Mode of Payment
Payment Mode Probability Cumulative Probability Random Number Interval
Cash 0.6
Credit Card 0.4
Cash Service Time
Service Time Probability Cumulative Probability Random Number Interval
20 seconds 0.35
40 seconds 0.30
60 seconds 0.25
80 seconds 0.10
Credit Card Service Time
Service Time Probability Cumulative Probability Random Number Interval
30 seconds 0.20
60 seconds 0.45
90 seconds 0.25
120 seconds 0.10
Random Numbers
(A) (B) (C)
6320 1094 1995
4630 7371 7971
8657 2809 3554
0030 5148 6300
5624 9115 5495
6728 1469 5165
5925 6480 9339
2829 2447 6997
7939 7031 1443
6476 8442 3574
3319 7387 0150
8134 1788 0933
1712 4891 7082
6317 1149 5025
6605 8822 4081
2734 9451 4100
0432 2990 7190
3441 8314 6822
0726 7176 5053
6969 2766 8284
Complete the table below to show the results of the simulation.
Customer Random Number Arrival Time Random Number Mode of Payment Random Number Service Time Service Time Waiting Time
Begins Ends Cash Credit
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Average Waiting Time (Cash)=
Average Waiting Time (Credit)=
To simulate the system and determine the average waiting time for cash and credit card customers, follow these steps:
1. Determine the inter-arrival time for each customer:
- Use the random number in Column A (from the random number table) to find the corresponding inter-arrival time from the given intervals. For example, if the random number is 6320, it falls in the interval of 30 seconds.
- Repeat this process for all 20 customers, filling in the "Inter-arrival Time" column in the table.
2. Determine the mode of payment for each customer:
- Use the random number in Column B to determine whether the customer pays with cash or credit card based on the cumulative probabilities provided. For example, if the random number is 1094 and falls between 0 and 0.6 (cumulative probability for cash), the customer pays with cash.
- Repeat this process for all 20 customers, filling in the "Mode of Payment" column in the table.
3. Determine the service time for each customer based on their mode of payment:
- For cash customers:
- Use the random number in Column C to determine the service time from the cash service time table. For example, if the random number is 1995 and falls in the interval for 40 seconds.
- Repeat this process for all cash customers, filling in the "Service Time" column in the table.
- For credit card customers:
- Use the random number in Column C to determine the service time from the credit card service time table. For example, if the random number is 1995 and falls in the interval for 30 seconds.
- Repeat this process for all credit card customers, filling in the "Service Time" column in the table.
4. Calculate the arrival time, service time begins, service time ends, and waiting time for each customer:
- Start with the first customer. The arrival time for the first customer is the same as their inter-arrival time.
- For subsequent customers, the arrival time is the sum of the previous customer's arrival time and their inter-arrival time.
- The service time begins for each customer is the maximum value between their arrival time and the service time ends of the previous customer.
- The service time ends for each customer is the sum of their service time begins and service time.
- The waiting time for each customer is the difference between their service time begins and arrival time.
- Fill in the respective columns in the table for each customer.
5. Calculate the average waiting time for cash and credit card customers:
- Add up the waiting times for all cash customers and divide by the total number of cash customers to get the average waiting time for cash.
- Add up the waiting times for all credit card customers and divide by the total number of credit card customers to get the average waiting time for credit card.
- Fill in the respective cells in the table for the average waiting times.
Once you complete these steps, you will have filled in all the required information in the table and found the average waiting times for cash and credit card customers.