The Scientific Theory of Queues
The scientific theory of queueing probability, commonly known as queue theory, gained popularity during World War II. It emerged from the need to better manage landing operations at English airfields for large squadrons of bombers returning from missions over Nazi Germany. On a relatively small number of airfields, formations of hundreds of aircraft were landing simultaneously. Many of the bombers were damaged, some had injured crew members, and fuel supplies were running low. Almost all of the bombers required immediate landing.
This situation presented a significant organizational problem that needed immediate resolution. This led to the standardization of a set of statistical tools known as queue theory. The mathematical concept of this theory was first published in 1917 by Danish mathematician Agner Krarup Erlang.
Queue theory is used to estimate the probability of bottlenecks occurring. This tool enabled the calculation of the likelihood of catastrophes at English airfields in advance, allowing for actions that could save the lives of Allied pilots.
Queue theory assumes that the number of elements entering the queue, denoted as λ (e.g., the number of incoming bombers per minute), is less than the number of elements that can be handled by the process, denoted as μ (e.g., the number of bombers that the airfield can accommodate per minute). Another factor, r, represents the number of operators (e.g., the number of airfields).
Meeting the condition
does not eliminate the possibility of bottlenecks. Queue theory provides the probability of queue formation and a range of its parameters.
Operational Issues in Milling Rollers
A milling roller manufacturer had recently faced quality issues with its products, receiving frequent complaints every few months. The complaints were related to vibrations occurring during roller operation. Engineers identified that the vibrations were caused by improperly seated roller bearings. The roller manufacturer purchased these components from a major bearing supplier.
The manufacturer, bearing significant costs due to defective production, demanded an explanation from the supplier. The bearing producer formed a special committee to investigate the causes of the quality issues. It was noted that similar issues, although to a lesser extent, were present among other buyers of the defective bearings.
The committee determined that the quality issues primarily affected two facilities: ZPAP Kotowiec and Wola. In other facilities, defects were almost nonexistent. Additionally, it was observed that the quality issues with the bearings occurred exclusively in winter months. A microscopic analysis of the connection between the bearing and sleeve revealed a break in the steel fiber structure, an important clue.
According to production technology, the sleeve was supposed to be inserted into a heated bearing collar. As the collar cooled, it would clamp around the sleeve, creating a permanent bond. Analysis showed that in defective bearings, the collar was too cold to accept the sleeve properly. Pressing the sleeve into an undersized opening caused structural damage.
The question arose: why were the components not sufficiently heated to ensure proper bonding?
A thorough investigation led to the conclusion that the heated bearing collars waited too long before bonding with the sleeve. It was determined that when components spent more than six minutes on the production floor after being removed from the furnace, their temperature dropped to a level unsuitable for further processing.
The assembly line manufacturer could not explain the delays in the automated feeders guiding the hot components to the bonding presses.
The committee then used statistical formulas from queue theory. They started by analyzing how many components exited the furnaces per hour (denoted as λ) and how many bonding operations were performed per hour on a single press (denoted as μ). It was found that in all facilities, the number of components leaving the furnaces (λ) was less than the product of press efficiency (μ) and the number of presses (r) performing bonding operations.
The ratio of incoming components (λ) to the product of
is known as the traffic intensity indicator.
Calculation of Waiting Probability
To calculate the probability that there is no congestion at a given moment on the assembly line, the following formula is used:
Where
represents the absence of items in the queue.
To calculate the probability that there is a specific number of items (n) waiting in the queue, use the following formula:
Different formulas are chosen depending on the number of items in the queue and the number of presses performing the bonding operation.
To determine the average number of items waiting to be introduced to the presses, the following formula is used:
For the production technologists supervising the process, the critical factor was the probability of a hot component waiting longer than six minutes, after which it would cool to a level that would cause defective bonds. Each defective bearing incurs significant losses due to the high cost of raw materials.
To calculate the probability of a wait time exceeding six minutes (for
), the following formula is used:
All formulas were entered into a spreadsheet and calculated.
The following table provides an assessment of the probability of production bottlenecks in the furnaces at the facilities producing the combined bearings. The values of λ and μ represent the number of components per hour.
The most critical information in the table is the probability of a component waiting more than six minutes. Remember, a component that waits longer than six minutes cools, resulting in a defective bond.
Analysis revealed that at ZPAP Kotowiec and Wola, there is a 2-3% probability of a wait time exceeding six minutes, which correlates with the complaint rates at these facilities during winter months.
The main cause of production bottlenecks at the two facilities was an insufficient number of presses for bonding operations. Increasing the number of presses will undoubtedly solve the quality issues with the combined bearings.
Conclusion
Queue theory is widely applied wherever bottlenecks can lead to quality loss in processes. This includes operations such as receiving and loading goods, processing payments at toll booths, and practically every industry. An excellent example is a bank’s data center, where administrators may expand servers when transaction capacity issues arise. However, transaction throughput issues can be very costly for a bank. Therefore, it is advantageous to anticipate such risks.
Author Bio
Wojciech Moszczyński – Graduate of the Department of Econometrics and Statistics at Nicolaus Copernicus University in Toruń, specializing in econometrics, data science, and managerial accounting. He focuses on optimizing production and logistics processes and conducts research in AI development and application. He has been engaged in promoting econometrics and data science in business environments for years.