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Analysis of a Markovian queue with customer interjections and finite buffer

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Abstract

This paper studies a finite Markovian queue with customer interjections. Arriving customers are dispersed into normal customers and interjecting customers, the normal customers join the queue at the end and the interjecting customers try to cut in line and occupy a position as close to the head of the queue as possible. The behavior of interjecting customers is described by the percentage of customers interjecting and the tolerance level of interjection by individual customers who are already waiting in the queue. Customers in different positions in the waiting line have different tolerance levels on customer interjection. By using the theory of the exponential distribution, the Laplace–Stieltjes transform and the formula of the total probability, we obtain the mean and variance of waiting time of a customer in position n, a normal customer and an interjecting customer. Further, numerical results show that these means and variances of waiting times increase with increase of the arrival process, and are sensitive to the percentage of customers interjecting and the tolerance level of interjection. It is found that eliminating customer interjection would be beneficial to normal customers.

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Acknowledgements

The research was supported by the Funding of V.C. & V.R. Key Lab of Sichuan Province (SCVCVR2019.05VS), the National Natural Science Foundation of China (No.71571127) and the FSUSE (No. 2017RCL55).

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Correspondence to Wenqing Wu.

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Wu, W., Zhang, Y. Analysis of a Markovian queue with customer interjections and finite buffer. OPSEARCH (2020). https://doi.org/10.1007/s12597-020-00437-w

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Keywords

  • \(M/M/1/m+1\) queueing system
  • Customer interjection
  • Normal customer
  • Interjecting customer
  • Waiting time