Quiz 13: Queuing Analysis 

Business

Queueing system is a system that has any type of waiting lines(queues). Various models can be built from the queueing system to calculate the parameters like wait time, length of the queue, etc. Poisson process is a counting process in which events occur on a continuous basis with mean rate ?. It is represented as { X(t); t = 0}. a. In a single-server waiting line model, calculate the average waiting time in the queue using the following formula: img Since a victim arrives after every 7 minutes, calculate img as follows: img Since a victim is serviced after every 4.5 minutes, calculate img as follows: img Calculate the waiting time as follows: img Thus, the average waiting time for victims will be img . Now, when the distribution of service times is undefined, img = 9 as aforementioned and calculate img as follows: img Compute the average time using the following formula: img Here, img = expected number of victims in the queue Now, compute img using the following formula: img Obtain the values of img and img as follows: img img Thus, the average waiting time for victims when the distribution of service times is undefined will be img . b. Here, a multi-server model is considered with img = 9 as aforementioned and calculate img as follows: img Calculate the expected number of customers in the system using the following formula: img Here, compute img using the following formula: img Here, img Calculate img and img as shown below: img img Calculate the average waiting time for a victim in the system as follows: img Calculate the average waiting time for a victim in the queue as follows: img img c. Here, img = 18 (number of victims) Now, calculate the average waiting time for a victim in the system as follows: img Calculate the average number of victims in the queue as follows: img Calculate the average waiting time for a victim in the queue as follows: img d. Here, s = 2 (number of servers) Now, calculate img and img as follows: img img Calculate img and img as shown below: img img Now, calculate the average waiting time for a victim in the system as follows: img Calculate the average number of victims in the queue as follows: img Calculate the average waiting time for a victim in the queue as follows: img e. This is a multiple server model with s = 5 (number of physicians). Since a victim arrives after every 8.5 minutes, calculate img as follows: img Calculate img as follows: img Now, compute img and img as shown below: img img img Calculate the average waiting time for a victim in the system as follows: img Calculate the average waiting time for a victim in the queue as follows: img img f. Here, img = 23 (number of victims) Now, calculate the average waiting time for a victim in the system as follows: img Calculate the average number of victims in the queue as follows: img Calculate the average waiting time for a victim in the queue as follows: img g. The most useful model in analyzing the disaster situation will be model e (multi-server model with five physicians in hospital M) since it has the least waiting time for the victims in the queue and the system. Also, the length of the waiting line is the least in the model. To use other models together to analyze the disaster situation , the pros and cons of each model needs to be adjusted so that the overall waiting time and the length of the waiting queue is minimum. This can be done by incorporating more physicians in model b or more emergency vehicles in model c. Other type of waiting line models that might be useful in analyzing the disaster situation will be as follows: • Jockeying - This means the movement between the queues is possible which occurs when there are more than one servers which in turn is preceded by queue. • The service time of the victim is not exponentially distributed or undefined. • Service is provided to the victims not in first-come first basis but on the basis of alphabetical order. • There can also be a model in which victims can be shifted to another hospital if the queue is very long in the hospital to which the patient is first taken to.

BEI airport has a problem in staffing and security checking. In the peak season it is having the problem of long waiting lines and waiting times. BEI decided to forecast the airport passenger arrival in order to find out the number of check points that are needed to avoid the rush and waiting time for 2 hour time segment. Service times are exponentially distributed with a mean of 11.6 seconds. The operational characteristics for the above problem can be calculated using the single server model. The probability that no customers are in the queuing system (either in the queue or being served) img Calculation of the security check points for two hour time segment- By applying the above formulas we can now calculate the values of Po, Pw, L, Lq, W and Wq. • 4:00 A.M to 6:00 A.M img The number of check points required is img . • 6:00 A.M to 8:00 A.M img The number of check points required is img . • 8:00 A.M to 10:00 AM img The number of check points required is img • 10:00 A.M to NOON img The number of check points required is img • NOON- 2:00 PM img The number of check points required is img • 2:00 P.M TO 4:00 P.M img The number of check points required is img • 4:00 P.M TO 6:00 P.M img img The number of check points required is img . • 6:00 P.M to 8:00 P.M img The number of check points required is img • 8:00 P.M to 10:00 P.M img The number of check points required is img . The number of security checkpoints is sufficient to keep passengers move freely through the security systems without excessive delays.

Determine the order service configuration to achieve their set goals of meeting the targets in the form of Excel Spreadsheet. As per the stated case, the current system of achieving the company goals is not feasible and the management of NB does not think if they work 8hours, 5 days per week, they would be able to serve the customers effectively. Calculate the second alternative that is 7 days and 8 hours service in the Excel spreadsheet. The exhibit shows the average time and number taken in the queue and the system. Besides that, the spreadsheet throws light on the total cost that was incurred in meeting the calls. img Insert the formulas in the following way: img The terms for 9 days service are also shown below: img img Calculate the third alternative that is 5 days and 16 hours service in the Excel spreadsheet. The exhibit shows the average time and number taken in the queue and the system. Besides that, the spreadsheet throws light on the total cost that was incurred in meeting the calls. img Insert the formulas in the following way: img The terms for 7 days service are also shown below: img img Calculate the fourth alternative that is 7 days and 16 hours service in the Excel spreadsheet. The exhibit shows the average time and number taken in the queue and the system. Besides that, the spreadsheet throws light on the total cost that was incurred in meeting the calls. img The terms for 5 days service are also shown below: img img Among all the alternatives, second alternative would be the most feasible option for the company P. The cost incurred in meeting the targets is the least that is $22,600 among all the other alternatives.