Analysis of an M/M/1 Queue with Customer Interjection
Bibliographic record
Abstract
In our daily life, we often experience waiting in a queue to receive some kind of service. Some customers do not join the queue at the end like other normal customers, and try to cut in the queue hoping to have a shorter waiting time and a higher level of satisfaction. This behaviour is called customer interjection. Some of these customers only try to cut in queue, while some others try to find excuses for interjection. For instance, the first-come-first-served (FCFS) service discipline is usually assumed in public places like restaurants, banks, airports, and supermarkets. However, customer interjections can still be seen in these places. In telecommunications networks, to test the efficiency of transmission, artificial packages are inserted into the normal traffic in a random manner. These interjections can affect the waiting time of other customers in queue. Such interjections may reduce the waiting time of interjecting customers, but increase the waiting time and dissatisfaction of others. In this work, an M/M/1 queueing system with customer interjection is investigated. The arrival of customers to the system is assumed to be a Poisson process with arrival rate . The service times for customers are independent and identically distributed random variables with an exponential distribution with rate . Customers are dispersed into normal customers and interjecting customers. A normal customer joins the queue at the end, and an interjecting customer tries to cut in the queue and occupy a position as close to the head of the queue as possible. Two parameters are introduced to describe the interjection behaviour: the percentage of customers interjecting and the tolerance level of interjection by individual customers who are already waiting in the queue. Using matrix-analytic methods and stochastic comparison methods, the waiting times of normal customers and interjecting customers are being studied. The impacts of the two parameters on the waiting times are analyzed in detail, and the implications of the results are discussed with numerical examples. It is found that the waiting times are sensitive to the tolerance level of interjection by individual customers. It is also found that eliminating customer interjection would be always beneficial to normal customers and arbitrary customers though it would not always be so for interjecting customers.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".