Concept explainers
a)
To determine: System utilization rate.
Introduction: Poisson distribution is utilized to ascertain the probability of an occasion happening over a specific time period or interval. The interval can be one of time, zone, volume or separation. The probability of an event happening is discovered utilizing the equation in the Poisson distribution.
a)
Answer to Problem 17P
Explanation of Solution
Given Information:
It is given that the processing time is 4 customers per hour and there are 5 servers to process the customers.
Class | Arrivals per Hour |
1 | 2 |
2 | 4 |
3 | 3 |
4 | 2 |
Calculate the system utilization:
It is calculated by adding all the total customer hours for each class and the result is divided with number of servers and customer process per hour.
Here,
M = number of servers
Hence the system utilization is 0.5500.
b)
To determine: The average customer waiting for service for each class and waiting in each class on average.
b)
Answer to Problem 17P
Explanation of Solution
Given Information:
Class | Arrivals per Hour |
1 | 2 |
2 | 4 |
3 | 3 |
4 | 2 |
It is given that the processing time is 4 customers per hour and there are 5 servers to process the customers.
Calculate the average number of customers
It is calculated by dividing the total customers arrive per hour with customer process per hour.
Here,
r = average number of customers
Calculate average number of customers waiting for service (Lq) using infinite-source table values for
The Lq values for
Calculate A using Formula 18-16 from book:
It is calculated by subtracting 1 minus system utilization rate and multiplying the result with Lq, the whole result is divided by total customer arrival rate.
Here,
Lq = average number of customers waiting for service
Calculate B using Formula 18-17 from book for each category:
It is calculated by multiplying number of servers with customer service process rate per hour and the result is divided by total customer arrival rate for each category.
Here,
M = number of servers
Calculate the average waiting time for class 1 and class 2
It is calculated by multiplying A with B0 and B1, the result is divided by 1.
Calculate the average number of customers that are waiting for service for class 1 and class 2
It is calculated by multiplying total customer arrival rate with average waiting time for units in each category.
Excel Spreadsheet:
Excel Workings:
Hence the average wait time for service by customers for class 1 is 0.0099 hours, class 2 is 0.0142 hours, class 3 is 0.0232 hours and class 4 is 0.0361 hours. The waiting in each class on average for class 1 is 0.0199 customers, class 2 is 0.0567 customers, class 3 is 0.0696 customers and class 4 is 0.0722 customers.
c)
To determine: The average customer waiting for service for each class and waiting in each class on average.
c)
Answer to Problem 17P
Explanation of Solution
Given Information:
It is given that the processing time is 4 customers per hour and there are 5 servers to process the customers. The second priority class is reduced to 3 units per hour by shifting some into the third party class. The arrival rate is as follows,
Class | Arrivals per Hour |
1 | 2 |
2 | 3 |
3 | 4 |
4 | 2 |
Calculate the average number of customers
It is calculated by dividing the total customers arrive per hour with customer process per hour.
Here,
r = average number of customers
Calculate average number of customers waiting for service (Lq) using infinite-source table values for
The Lq values for
Calculate A using Formula 18-16 from book
It is calculated by subtracting 1 minus system utilization rate and multiplying the result with Lq, the whole result is divided by total customer arrival rate.
Here,
Lq = average number of customers waiting for service
Calculate B using Formula 18-17 from book for each category
It is calculated by multiplying number of servers with customer service process rate per hour and the result is divided by total customer arrival rate for each category.
Here,
M = number of servers
Calculate the average waiting time for class 1 and class 2
It is calculated by multiplying A with B0 and B1, the result is divided by 1.
Calculate the average number of customers that are waiting for service for class 1 and class 2
It is calculated by multiplying total customer arrival rate with average waiting time for units in each category.
Excel Spreadsheet:
Excel Workings:
Hence the average wait time for service by customers for class 1 is 0.0099 hours, class 2 is 0.0132 hours, class 3 is 0.0217 hours and class 4 is 0.0361 hours. The waiting in each class on average for class 1 is 0.0199 customers, class 2 is 0.0397 customers, class 3 is 0.0867 customers and class 4 is 0.0722 customers.
d)
To determine: The observations based on the results from part c.
d)
Answer to Problem 17P
Explanation of Solution
Calculate the change in average wait time for each class.
It is calculated by subtracting the final answer for average wait time for service by customers from part b with the final answer for average wait time for service by customers from part c.
The above results suggest that there is a decrease in average wait time for class 2 and class 3. Class 1 and 4 remains constant.
Calculate the change in average number waiting for each class.
It is calculated by subtracting the final answer for waiting on average from part b with the final answer for waiting on average from part c.
The above results suggest that there is a decrease in average waiting for class 2 and an increase in class 3. Class 1 and 4 remains constant.
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Chapter 18 Solutions
Operations Management
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- Practical Management ScienceOperations ManagementISBN:9781337406659Author:WINSTON, Wayne L.Publisher:Cengage,