Concept explainers
A priority waiting system assigns arriving customers to one of four classes. Arrival rates (Poisson) of the classes are shown in the following table. Five servers process the customers, and each can handle three customers per hour.
a. What is the system utilization?
b. What is the average wait for service by customers in the various classes? How many are waiting in each class, on average?
c. If the arrival rate of the second priority class could be reduced to three units per hour by shifting some arrivals into the third priority class, how would your answers to part b change?
d. What observations can you make based on your answers to part c?
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.
Answer to Problem 16P
Explanation of Solution
Given Information:
It is given that the processing time is 3 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 0.7333.
b)
To determine: The average customer waiting for service for each class and waiting in each class on average.
Answer to Problem 16P
Explanation of Solution
Given Information:
It is given that the processing time is 3 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 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 0.0333 hours, class 2 is 0.0555 hours, class 3 is 0.1202 hours and class 4 is 0.2705 hours. The waiting in each class on average for class 1 is 0.0666 customers, class 2 is 0.2220 customers, class 3 is 0.3607 customers and class 4 is 0.5411 customers.
c)
To determine: The average customer waiting for service for each class and waiting in each class on average.
Answer to Problem 16P
Explanation of Solution
Given Information:
It is given that the processing time is 3 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 0.0333 hours, class 2 is 0.0499 hours, class 3 is 0.1082 hours and class 4 is 0.2705 hours. The waiting in each class on average for class 1 is 0.0666 customers, class 2 is 0.1498 customers, class 3 is 0.4329 customers and class 4 is 0.5411 customers.
d)
To determine: The observations based on the results from part c.
Answer to Problem 16P
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
STEVENSON: OPERATIONS MANAGEMENT(LL)
- In the library, students arrive randomly at the help desk, which is staffed by 4 employees (all of them answer inquiries from students). The time required for inquiry varies from student to student. Arrivals have been found to follow a Poisson process (i.e., exponentially distributed inter-arrival times), and the service times follow an exponential distribution. The average arrival rate is 12 students per hour, and the average service rate is 13 students per hour. What is the utilization of the help desk? Note: 1. Provide your answer as a percentage, but do not include the % mark. 2. Keep two decimal places. For example, if your result is 35.126%, enter 35.13.arrow_forwardDetermining the Number of ServersIn the service department of the Glenn-Mark Auto Agency, mechanics requiring parts for auto repair or service present their request forms at the parts department counter. The parts clerk fills a request while the mechanic waits. Mechanics arrive in a random (Poisson) fashion at the rate of 40 per hour, and a clerk can fill requests at the rate of 20 per hour (exponential).If the cost for a parts clerk is $30 per hour and the cost for a mechanic is $60 per hour, determine the optimum number of clerks to staff the counter. (Because of the high arrival rate, an infinite source may be assumed.)arrow_forwardat the order fulfillment center of a major mail-order firm, customer orders, already packaged for shipment, arrive at the sorting machines to be sorted for loading onto the appropriate truck for the parcel address. the arrival rate at the sorting machines is at the rate of 100 per hour following Poisson distribution. the machine sorts at the constant rate of 150 per hour.a. what is the utilization rate of system? b. what is the average number of packages waiting to be sorted? c. what is the average number of packages in the sorting system? d. how long must the average package wait until it gets sorted? e. what would Lq and wq be if the service rate were exponential, not constant?arrow_forward
- Practical Management ScienceOperations ManagementISBN:9781337406659Author:WINSTON, Wayne L.Publisher:Cengage,