Queueing Theory and Special Consideration to Emergency Care

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Waiting lines may occur in the case of even underloaded systems because of the existing variability in the rates of services and/or arrivals. For instance, a fast food restaurant may have the capacity of handling 200 orders per hour, yet clients will still experience waiting lines when there is an average 150 orders fulfilled. The average component is important because arrival times can range, and some orders may take more time to get fulfilled compared to others. This means that in order to reduce the occurrence or shorten waiting lines overall, businesses should cut the rates of service and arrival. Despite the expectation that customer arrival rates will be stationary, most systems experience its non-stationarity, which ranges from one time to the next. Service delivery rates also can be affected by variability, especially in cases when specialized services are needed. Importantly, because systems do not operate continuously, there may be closing and opening volatility at different points of a day. Thus, even in contexts when systems are underloaded, waiting lines will occur if service and arrival rates are non-stationary.

Conducting a queuing analysis, there are several measures of system performance. The average number (i.e. of customers) standing in line is among the most crucial values because it can help determining other valuable measures (Stevenson, 2018). They include “the average time waiting in line, the average time in the system, and the average number in the system” (Stevenson, 2018, p. 814). Because of this, the average number waiting in line is the first measure of system performance calculated in the queuing analysis. Also, identifying the rates of arrivals and service can allow analysts to determine the effectiveness of existing systems.

Case Analysis

Schedule management and hiring an appropriate amount of personnel is not enough for ensuring the efficiency of systems and leaving customers satisfied. In the context of operations management, the waiting times theory plays a crucial role because of the ability to predict queue lengths and waiting times. Initially developed by Agner Krarup Erlang, the theory is now applied to a variety of contexts (Keesling, n.d.). To receive a service or product, customers arrive at the location randomly, with the expectation of being served immediately. At times of maximum capacity, customers may not be served immediately, which means that they will wait in a line. Since many services require personalization, it is hard to predict how much time will be necessary to serve each customer. Thus, the main objective of the waiting times theory is reaching an “economic equilibrium between the service cost and the patients’ wasted time while waiting in the queue to be served” (Cho, Kim, Chae, & Song, 2017, p. 36). To reach customer satisfaction and ensure system efficiency, the waiting times theory should be applied by operations managers.

In the Big Bank case study, an operations manager of a soon-to-open bank branch is faced with the dilemma associated with optimizing system operations and avoid long waits about which customers will complain. The problem lies in the differences in demographics from one location to another, which means that a solution implemented for one system may not be as effective in the new location. In order to come with a conclusion on which system will work best for the new branch of the bank, several aspects should be considered. These aspects include the following:

There is an average of 80 customers processed during noon hours;
The average time for a single transaction is 90 seconds (1.5 minutes);
The average time for customers for several transactions is 4 minutes;
60% of customers usually have several transactions;
The noon hour on Friday is usually the busiest.

Given the determinants above, the operations manager should make a decision of either creating a single line for all customers and make the person in line closest to the frony go to the teller that is available or have two separate lines: one for clients with one transaction handled by one teller and another for clients with multiple transactions handled by four tellers. The first step in the case analysis is calculating the processing rate and waiting time in line for each of the mentioned options:

Total arrival rate: 80 customers per hour;
Arrival rate for several transactions: 48 customers per hour (60% of 80 in total);
Arrival rate for a single transaction: 32 customers per hour (40% of 80 in total);

Option 1: One waiting line for all customers, managed by the newly available tellers.

Arrival rate: 80 customers per hour

Processing rate = 60/ ((1.5 * 32+4 * 48)/80) = 20 customers per hour

Waiting time = 60/ (no. of servers * service rate-arrival rate) = 60/ (5+20-80) = 3 minutes

Option 2: Two waiting lines: one for one transaction and one for multiple transactions.

Processing rate for single transaction = 60/1.5 = 40 customers per hour

Time waiting in line (a single transaction) = (arrival rate/ service rate * (service rate – arrival rate) * 60 = (32/ (40* (40-32)) * 60 = 6 minutes

Processing rate for multiple transactions = 60/4 = 15 customers per hour

Time waiting in line (multiple transactions) = 60 / (4 * 15-48) = 5 minutes

By comparing the results of the calculations between the two options, the solution with a single line seems more effective because the waiting time is only 3 minutes. However, the single line will only be able to process twenty customers per hour. If the manager implements the option with two lines, the one teller for a single transaction will process forty customers per hour while the four other tellers will process fifteen customers completing multiple transactions. Thus, the total number of customers processed with the help of two lines will be fifty-five. Given the fact that there are eighty customers in total that arrive each hour, the second option also seems feasible.

There is an exponential difference between the results for the two operations management solutions because of the variability in services that customers come to receive in the branch of the bank. On the one hand, a single line can be a solution because customers will not have to stand in a line for long and will be referred to tellers that become available after serving customers. However, because of variation, fewer customers in total will be served. This can lead to subsequent dissatisfaction of customers. On the other hand, waiting two to three minutes longer may not be a problem if more customers are served per hour. Because of this, as a manager of the department, I would select the second option despite the seeming ineffectiveness.

It should be mentioned that there were some concerns of solutions having a different impact on the soon-to-open branch because of the geographic characteristics. Because of this, it is recommended for the manager to test the effectiveness of both options once the branch opens. For instance, the one-line solution may be tested for the first two weeks of a month while the two-line option may be tested for the last two weeks. A short survey can be conducted to involve customers in order to determine whether they were satisfied with the provided services. Thus, the option that will be preferred by customers should be implemented long-term.

In the analysis of the case, several assumptions were made. First, the implemented model assumed that all tellers worked at the same rate and thus were considered identical. If they were assumed non-identical, this would have contributed to the increased complexity of calculations and measurements. Second, the waiting line priority utilized the first-come, first served policy for ensuring fairness and customer satisfaction (Hernandez-Maskivker & Ryan, 2016). Of course, it is possible that some customers may be served first if other customers in line agree. Examples can include the elderly and pregnant women. Third, the model assumed that the customers waiting in line were patient and did not balk as well as came from an ‘infinite population.’ Fourth, the distribution of clients arriving at the branch was assumed to be Poisson (a known constant rate and independent of the time when the last event occurred). Lastly, the model used in the analysis assumed exponential service times. To conclude, the case analysis revealed some complications in operations management measurements and showed that waiting lines times can be addressed through the integration of models that account for the variability of customers’ needs.

References

Cho, K. W., Kim, S. M., Chae, Y. M., & Song, Y. U. (2017). Application of Queueing Theory to the Analysis of Changes in Outpatients’ Waiting Times in Hospitals Introducing EMR. Healthcare Informatics Research, 23(1), 35-42.

Hernandez-Maskivker, G., & Ryan, G. (2016). Priority systems at theme parks from the perspective of managers and customers. Technology Innovation Management Review, 6(11), 40-47.

Keesling, J. (n.d.). Queueing theory with applications and special consideration to emergency care.Web.

Stevenson, W. J. (2018). Operations management (13th ed.). New York, NY: McGraw-Hill Education.

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Queueing Theory and Special Consideration to Emergency Care

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