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On first-come, first-served queues with three classes of impatient customers

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Abstract

In this article, we study queuing systems with three classes of impatient customers which differ across the classes in their distribution of service times and patience times. The customers are served on a first-come, first-served (FCFS) policy independent of their classes. Such systems are common in customer call centers, which often segment their arrivals into classes of callers whose requests differ in complexity and criticality. First of all, we consider an \(M/G/1 + M\) queue and then analyze the \(M/M/m + M\) system. Using the virtual waiting time process, we obtain performance measures such as the percentage of customers receiving service in each class, the expected waiting times of customers in each class, and the average number of customers waiting in the queue. We use our characterization to perform a numerical analysis of the \(M/M/m + M\) system. Finally, we compare the performance of a system based on numerical solution with the steady-state performance measures of a comparable \(M/M/m + M\) system.

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References

  1. Adan, I., Boon, M., Weiss, G.: Design and evaluation of overloaded service systems with skill based routing, under fcfs policies. Perform Evalu 70(10), 873–888 (2013)

    Article  Google Scholar 

  2. Adan, I., Hathaway, B., Kulkarni, V.: On first come, first served queues with two classes of impatient customers. Queue Sys 91, 113–142 (2019)

    Article  MathSciNet  Google Scholar 

  3. Baccelli, F., Boyer, P., Hebuterne, G.: Single-server queues with impatient customers. Adv Appl Probab 16(4), 887–905 (1984)

    Article  MathSciNet  Google Scholar 

  4. Benes, V. E.: General stochastic processes in the theory of queues. Addison-Wesley Publishing Company, Inc., (1963)

  5. Brandt, M., Brandt, A.: On the two-class m/m/1 system under preemptive resume and impatience of the prioritized customers. Queueing Syst 47, 147–168 (2004)

    Article  MathSciNet  Google Scholar 

  6. Brandt, M., Brandt, A.: Workload and busy period for M/GI/1 with a general impatience mechanism. Queueing Syst 75, 189–209 (2013)

    Article  MathSciNet  Google Scholar 

  7. Brill, M.J., Posner, P.: Level crossings in point processes applied to queues: single-server case. Oper. Res. 25, 662–674 (1977)

    Article  MathSciNet  Google Scholar 

  8. Brill, M.J., Posner, P.: The system point method in exponential queues: a level crossing approach. Math. Oper. Res. 6, 31–49 (1981)

    Article  MathSciNet  Google Scholar 

  9. Choi, B., Kim, B.D., Chung, J.: M/M/1 queue with impatient customers of higher priority. Queueing Syst. 38, 49–66 (2001)

    Article  MathSciNet  Google Scholar 

  10. Daley, D.: General customer impatience in the queue GI/G/1. J. Appl. Probab. 2, 186–205 (1965)

    Article  MathSciNet  Google Scholar 

  11. Garnett, A., Mandelbaum, O., Reiman, M.: Designing a call center with impatient customers. Manuf. Serv. Oper. Manag. 4, 208–227 (2002)

    Article  Google Scholar 

  12. Houdt, B.: Analysis of the adaptive mmap[k]/ph[k]/1 queue: a multi-type queue with adaptive arrivals and general impatience. Eur J Operat Res 220, 695–704 (2012)

    Article  MathSciNet  Google Scholar 

  13. Iravani, B., Balcıogğlu, F.: Approximations for the M/GI/n+GI type call center. Queueing Syst 58, 137–153 (2008)

    Article  MathSciNet  Google Scholar 

  14. Iravani, B., Balcıogğlu, F.: On priority queues with impatient customers. Queueing Syst. 58, 239–260 (2008)

    Article  MathSciNet  Google Scholar 

  15. Sakuma, T., Takine, Y.: Multi-class m/ph/1 queues with deterministic impatience times. Stoch. Models 33, 1–33 (2017)

    Article  MathSciNet  Google Scholar 

  16. Stanford, R.E.: Reneging phenomena in single channel queues. Math. Oper. Res. 4, 162–178 (1979)

    Article  MathSciNet  Google Scholar 

  17. Takàcs, M.: Introduction to the Theory of Queues. Oxford University Press, (1962)

  18. Zeltyn, A., Mandelbaum, S.: Call centers with impatient customers: many-server asymptotics of the M/M/n+G queue. Queueing Syst. 51, 361–402 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The second author is thankful for the funding by IIT Madras from the IoE project: SB20210848MAMHRD008558.

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  1. Department of Mathematics, Indian Institute of Technology, Madras, Chennai, 600036, India

    Vinay Kumar & Neelesh S. Upadhye

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  1. Vinay Kumar
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  2. Neelesh S. Upadhye
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Correspondence to Neelesh S. Upadhye.

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Kumar, V., Upadhye, N.S. On first-come, first-served queues with three classes of impatient customers. Int J Adv Eng Sci Appl Math 13, 368–382 (2021). https://doi.org/10.1007/s12572-022-00313-4

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  • DOI: https://doi.org/10.1007/s12572-022-00313-4

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