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Determining Trunk Lines in Call Centers with Nonstationary Arrivals and Lognormal Service Times

Received: 26 June 2019     Accepted: 18 July 2019     Published: 5 August 2019
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Abstract

Two important resources in a call center are the number of staff and the number of trunk lines required. In this paper, we focus on the decision of the number of trunk lines that a call center should have. The current practice is to use the Erlang B or the M/M/s/0 queueing model which assumes Poisson arrivals, exponential service times, s servers and no places in queue, i.e. no customers can wait. In this paper, we improve on the state of practice in determining the required number of trunk lines, by including two realistic features present in call centers. The first realistic feature is to consider nonstationarity of arrivals. The second feature is to consider the lognormal service time distribution instead of the exponential distribution. There is extensive empirical evidence for both features. In order to carry out our computations we use the results of a paper by Massey and Whitt, Operations Research, 44(6), 1996. We have two main findings. Firstly, we find numerically that in our nonstationary Erlang loss model, Mt/G/s/0, an insensitivity result holds. The blocking probability of arrivals at the call center depends only on the mean of the lognormal service time distribution and not on its variance. Our second finding is that current practice is quite robust. In particular, we find the number of trunk lines required using a stationary Poisson approximation. This approximation assumes stationary Poisson arrivals with an appropriately chosen arrival rate and exponential service times. The approximation does quite well in predicting the number of trunk lines required.

Published in American Journal of Operations Management and Information Systems (Volume 4, Issue 3)
DOI 10.11648/j.ajomis.20190403.11
Page(s) 71-79
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Queueing, OR in Service Industries, Call Centers, Nonstationary Arrivals, Lognormal Distribution

References
[1] P. Reynolds, Call Center Staffing: The Complete Practical Guide to Workforce Management, 2003, Call Center School Press, Nashville, Tennessee.
[2] D. Gross, J. F. Shortle, J. M. Thompson, and C. M. Harris, Fundamentals of Queueing Theory (4th ed.), 2008, John Wiley, New Delhi, India.
[3] Green, L., P. Kolesar and W. Whitt (2007). Coping with time-varying demand when setting staffing requirements for a service system. Production and Operations Management 16 (1), 13-39.
[4] Brown, L., N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn and L. Zhao (2005). Statistical analysis of a telephone call center: A Queueing-Science perspective. Journal of the American Statistical Association 100 (469), 36-50.
[5] Gans, N., G. Koole and A. Mandelbaum (2003). Telephone call centers: Tutorial, review and research prospects. Manufacturing and Service Operations Management 5 (2), 79-141.
[6] Bolotin, V. A. (1994). Telephone circuit holding time distributions. Proc. 14th International Teletraffic Conference, 125-134.
[7] Chlebus, E.(1997). Empirical validation of call holding time distributions in cellular communication systems. Proc. 15th International Teletraffic Conference, Elsevier, Amsterdam, 1179-1188.
[8] Mandelbaum, A., A. Sakov and S. Zeltyn (2001). Empirical analysis of a call center. Technical Report, Technion, Haifa, Israel.
[9] Massey, W. A. and W. Whitt (1996). Stationary-process approximations for the nonstationary Erlang loss model. Operations Research 44 (6), 976-983.
[10] Davis, J. L., W. A. Massey and W. Whitt (1995). Sensitivity to the service-time distribution in the nonstationary Erlang loss model. Management Science 41 (6), 1107-1116.
[11] Kim, J. W. and S. C. Park (2010). Outsourcing strategy in two-stage call centers. Computers & Operations Research 37 (4), 790-805.
[12] Klincli, T. G. and X. Zhang (2017). Mathematical models and solution approach for cross-training staff scheduling at call centers. Computers and Operations Research, 87, 258-269.
[13] Yu, M., J. Gong, J. Tang and F. Kong (2017). Delay announcements for call centers with hyperexponential patience modelling. Industrial Management and Data Systems, 117 (6), 1037-1057.
[14] Li, G., J. Z. Huang and H. Shen (2018). To wait or not to wait: Two-way functional hazards model for understanding waiting in call centers. Journal of the American Statistical Association, 113, 1503-1514.
[15] Bimpikis, K and G. M. Markakis (2019). Learning and hierarchies in service systems. Management Science, 65 (3), 1268-1285.
[16] J. H. Mathews and K. D. Fink (2004), Numerical Methods Using MATLAB (4th ed.), Pearson Education, India.
Cite This Article
  • APA Style

    Siddharth Mahajan. (2019). Determining Trunk Lines in Call Centers with Nonstationary Arrivals and Lognormal Service Times. American Journal of Operations Management and Information Systems, 4(3), 71-79. https://doi.org/10.11648/j.ajomis.20190403.11

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    ACS Style

    Siddharth Mahajan. Determining Trunk Lines in Call Centers with Nonstationary Arrivals and Lognormal Service Times. Am. J. Oper. Manag. Inf. Syst. 2019, 4(3), 71-79. doi: 10.11648/j.ajomis.20190403.11

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    AMA Style

    Siddharth Mahajan. Determining Trunk Lines in Call Centers with Nonstationary Arrivals and Lognormal Service Times. Am J Oper Manag Inf Syst. 2019;4(3):71-79. doi: 10.11648/j.ajomis.20190403.11

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  • @article{10.11648/j.ajomis.20190403.11,
      author = {Siddharth Mahajan},
      title = {Determining Trunk Lines in Call Centers with Nonstationary Arrivals and Lognormal Service Times},
      journal = {American Journal of Operations Management and Information Systems},
      volume = {4},
      number = {3},
      pages = {71-79},
      doi = {10.11648/j.ajomis.20190403.11},
      url = {https://doi.org/10.11648/j.ajomis.20190403.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajomis.20190403.11},
      abstract = {Two important resources in a call center are the number of staff and the number of trunk lines required. In this paper, we focus on the decision of the number of trunk lines that a call center should have. The current practice is to use the Erlang B or the M/M/s/0 queueing model which assumes Poisson arrivals, exponential service times, s servers and no places in queue, i.e. no customers can wait. In this paper, we improve on the state of practice in determining the required number of trunk lines, by including two realistic features present in call centers. The first realistic feature is to consider nonstationarity of arrivals. The second feature is to consider the lognormal service time distribution instead of the exponential distribution. There is extensive empirical evidence for both features. In order to carry out our computations we use the results of a paper by Massey and Whitt, Operations Research, 44(6), 1996. We have two main findings. Firstly, we find numerically that in our nonstationary Erlang loss model, Mt/G/s/0, an insensitivity result holds. The blocking probability of arrivals at the call center depends only on the mean of the lognormal service time distribution and not on its variance. Our second finding is that current practice is quite robust. In particular, we find the number of trunk lines required using a stationary Poisson approximation. This approximation assumes stationary Poisson arrivals with an appropriately chosen arrival rate and exponential service times. The approximation does quite well in predicting the number of trunk lines required.},
     year = {2019}
    }
    

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    AU  - Siddharth Mahajan
    Y1  - 2019/08/05
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    AB  - Two important resources in a call center are the number of staff and the number of trunk lines required. In this paper, we focus on the decision of the number of trunk lines that a call center should have. The current practice is to use the Erlang B or the M/M/s/0 queueing model which assumes Poisson arrivals, exponential service times, s servers and no places in queue, i.e. no customers can wait. In this paper, we improve on the state of practice in determining the required number of trunk lines, by including two realistic features present in call centers. The first realistic feature is to consider nonstationarity of arrivals. The second feature is to consider the lognormal service time distribution instead of the exponential distribution. There is extensive empirical evidence for both features. In order to carry out our computations we use the results of a paper by Massey and Whitt, Operations Research, 44(6), 1996. We have two main findings. Firstly, we find numerically that in our nonstationary Erlang loss model, Mt/G/s/0, an insensitivity result holds. The blocking probability of arrivals at the call center depends only on the mean of the lognormal service time distribution and not on its variance. Our second finding is that current practice is quite robust. In particular, we find the number of trunk lines required using a stationary Poisson approximation. This approximation assumes stationary Poisson arrivals with an appropriately chosen arrival rate and exponential service times. The approximation does quite well in predicting the number of trunk lines required.
    VL  - 4
    IS  - 3
    ER  - 

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Author Information
  • Production and Operations Management Area, Indian Institute of Management, Bangalore, India

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