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Performance Approximation for Time-Dependent Queues with Generally Distributed Abandonments

Author

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  • Gregor Selinka

    (Business School, University of Mannheim, 68131 Mannheim, Germany)

  • Raik Stolletz

    (Business School, University of Mannheim, 68131 Mannheim, Germany)

  • Thomas I. Maindl

    (Department of Astrophysics, University of Vienna, 1180 Vienna, Austria; SDB Science-Driven Business Ltd, 6025 Larnaca, Cyprus)

Abstract

Many stochastic systems face a time-dependent demand. Especially in stochastic service systems, for example, in call centers, customers may leave the queue if their waiting time exceeds their personal patience. As discussed in the extant literature, it can be useful to use general distributions to model such customer patience. This paper analyzes the time-dependent performance of a multiserver queue with a nonhomogeneous Poisson arrival process with a time-dependent arrival rate, exponentially distributed processing times, and generally distributed time to abandon. Fast and accurate performance approximations are essential for decision support in such queueing systems, but the extant literature lacks appropriate methods for the setting we consider. To approximate time-dependent performance measures for small- and medium-sized systems, we develop a new stationary backlog-carryover (SBC) approach that allows for the analysis of underloaded and overloaded systems. Abandonments are considered in two steps of the algorithm: (i) in the approximation of the utilization as a reduced arrival stream and (ii) in the approximation of waiting-based performance measures with a stationary model for general abandonments. To improve the approximation quality, we discuss an adjustment to the interval lengths. We present a limit result that indicates convergence of the method for stationary parameters. The numerical study compares the approximation quality of different adjustments to the interval length. The new SBC approach is effective for instances with small numbers of time-dependent servers and gamma-distributed abandonment times with different coefficients of variation and for an empirical distribution of the abandonment times from real-world data obtained from a call center. A discrete-event simulation benchmark confirms that the SBC algorithm approximates the performance of the queueing system with abandonments very well for different parameter configurations. Summary of Contribution: The paper presents a fast and accurate numerical method to approximate the performance measures of a time‐dependent queueing system with generally distributed abandonments. The presented stationary backlog carryover approach with abandonment combines algorithmic ideas with stationary queueing models for generally distributed abandonment times. The reliability of the method is analyzed for transient systems and numerically studied with real‐world data.

Suggested Citation

  • Gregor Selinka & Raik Stolletz & Thomas I. Maindl, 2022. "Performance Approximation for Time-Dependent Queues with Generally Distributed Abandonments," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 20-38, January.
    • Handle: RePEc:inm:orijoc:v:34:y:2022:i:1:p:20-38
    DOI: 10.1287/ijoc.2021.1090
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    References listed on IDEAS

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    1. Michael C. Fu, 2002. "Feature Article: Optimization for simulation: Theory vs. Practice," INFORMS Journal on Computing, INFORMS, vol. 14(3), pages 192-215, August.
    2. Schwarz, Justus Arne & Selinka, Gregor & Stolletz, Raik, 2016. "Performance analysis of time-dependent queueing systems: Survey and classification," Omega, Elsevier, vol. 63(C), pages 170-189.
    3. Ryan Palmer & Martin Utley, 2020. "On the modelling and performance measurement of service networks with heterogeneous customers," Annals of Operations Research, Springer, vol. 293(1), pages 237-268, October.
    4. Jun Kim & Sung Ha, 2012. "Advanced workforce management for effective customer services," Quality & Quantity: International Journal of Methodology, Springer, vol. 46(6), pages 1715-1726, October.
    5. Ger Koole & Avishai Mandelbaum, 2002. "Queueing Models of Call Centers: An Introduction," Annals of Operations Research, Springer, vol. 113(1), pages 41-59, July.
    6. William A. Massey & Jamol Pender, 2018. "Dynamic rate Erlang-A queues," Queueing Systems: Theory and Applications, Springer, vol. 89(1), pages 127-164, June.
    7. Castillo, Ignacio & Joro, Tarja & Li, Yong Yue, 2009. "Workforce scheduling with multiple objectives," European Journal of Operational Research, Elsevier, vol. 196(1), pages 162-170, July.
    8. Dietz, Dennis C., 2011. "Practical scheduling for call center operations," Omega, Elsevier, vol. 39(5), pages 550-557, October.
    9. Benjamin Legros & Oualid Jouini & Ger Koole, 2018. "A Uniformization Approach for the Dynamic Control of Queueing Systems with Abandonments," Operations Research, INFORMS, vol. 66(1), pages 200-209, January.
    10. Mok, Stephen K. & Shanthikumar, J. George, 1987. "A transient queueing model for Business Office with standby servers," European Journal of Operational Research, Elsevier, vol. 28(2), pages 158-174, February.
    11. Bertsimas, Dimitris & Doan, Xuan Vinh, 2010. "Robust and data-driven approaches to call centers," European Journal of Operational Research, Elsevier, vol. 207(2), pages 1072-1085, December.
    12. Lawrence Brown & Noah Gans & Avishai Mandelbaum & Anat Sakov & Haipeng Shen & Sergey Zeltyn & Linda Zhao, 2005. "Statistical Analysis of a Telephone Call Center: A Queueing-Science Perspective," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 36-50, March.
    13. Zohar Feldman & Avishai Mandelbaum & William A. Massey & Ward Whitt, 2008. "Staffing of Time-Varying Queues to Achieve Time-Stable Performance," Management Science, INFORMS, vol. 54(2), pages 324-338, February.
    14. Young Myoung Ko & Natarajan Gautam, 2013. "Critically Loaded Time-Varying Multiserver Queues: Computational Challenges and Approximations," INFORMS Journal on Computing, INFORMS, vol. 25(2), pages 285-301, May.
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