Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers

Siddharth Mahajan

doi:doi:10.11648/j.ajomis.20200504.11

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Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers

Siddharth Mahajan

Published in American Journal of Operations Management and Information Systems (Volume 5, Issue 4)

Received: 3 October 2020 Accepted: 19 October 2020 Published: 26 October 2020

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Abstract

Both large and small firms maintain call centers to establish contact between themselves and their customers. A call center is staffed by Customer Service Representatives (CSRs). In addition to CSRs, a call center needs computers and telecommunication equipment such as an Automatic Call Distributor (ACD). If calls arrive according to a Poisson arrival process and the service times have an exponential distribution, the call center can be modeled as an M/M/s queue where s is the number of CSRs. Typical calculations include finding the number of CSRs required and finding the number of trunks lines required. However, if call center models ignore the abandonment behavior of customers, they distort information that is relevant to management. Typically, ignoring abandonment would lead to overstaffing, as fewer CSRs are actually needed, because some of the callers abandon the system. Also, nonstationarity of arrivals is highly prevalent in call centers. Green, Kolesar and Whitt plot hourly arrival rates for a financial services call center. There is significant variation in arrivals by time of day. In this paper, we model call centers as multiserver Markovian queues with both nonstationarity and abandonment. Nonstationarity is modeled by having a Poisson arrival stream with time dependent mean, which varies according to a sinusoid. Abandonment is defined in terms of an exponential patience random variable, which is the amount of time the caller would be on hold before abandoning the call. We numerically study the performance of this nonstationary M(t)/M/s+M queue with abandonment and compare its performance measures with those of the stationary M/M/s+M queue. We find that approximating a nonstationary system by a stationary system, even at low levels of nonstationarity, can lead to significant errors. Similar results in a system without abandonment, have been obtained by Green, Kolesar and Svoronos. Additionally, we find that abandonment dampens the effect of nonstationarity. Since abandonment and nonstationarity are both present together in real call centers, a real call center with a high level of abandonment behaves closer to an ideal stationary system.

Published in	American Journal of Operations Management and Information Systems (Volume 5, Issue 4)
DOI	10.11648/j.ajomis.20200504.11
Page(s)	74-83
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), 2024. Published by Science Publishing Group

Keywords

Queueing, Nonstationarity, Abandonment, Markovian Queues, Call Centers

References

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[2]	Reynolds, P. 2003. Call center staffing: The complete practical guide to workforce management. Nashville (Tennessee): Call Center School Press.
[3]	Garnett, O, Mandelbaum, A, Reiman, MI. 2002. Designing a call center with impatient customers. Manufacturing and Service Operations Management. 4 (3): 208-227.
[4]	Halfin, S, Whitt, W. 1981. Heavy-traffic limits for queues with many exponential servers. Operations Research. 29: 567-588.
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[7]	Gans, N, Koole, G, Mandelbaum, A. 2003. Telephone call centers: Tutorial, review and research prospects. Manufacturing and Service Operations Management. 5 (2): 79-141.
[8]	Yunan, L, Whitt, W. 2016. Approximations for heavily loaded G/GI/n+GI queues. Naval Research Logistics. 63 (3): 187-217.
[9]	Batt, R. J, Terwiesch, C. 2015. Waiting patiently: An empirical study of queue abandonment in an emergency department. Management Science. 61 (1): 39-59.
[10]	Green, L, Kolesar, P, Whitt, W. 2007. Coping with time-varying demand when setting staffing requirements for a service system. Production and Operations Management. 16 (1): 13-39.
[11]	Green, L, Kolesar, P, Svoronos, A. 1991. Some effects of nonstationarity on multiserver Markovian queueing systems. Operations Research. 39 (3): 502-511.
[12]	Ross, SM. 1978. Average delay in queues with nonstationary Poisson arrivals. Journal of Applied Probability. 15: 602-609.
[13]	Rolski, T. 1984. Comparison theorems for queues with dependent interarrival times. In: Baccelli F, Fayolle G, editors. Modeling and performance evaluation methodology. p. 42-67.
[14]	Svoronos, A, Green, L. 1988. A convexity result for single server exponential loss systems with nonstationary arrivals. Journal of Applied Probability. 25: 224-227.
[15]	Aksin, ZO, Harker, PT. 2003. Capacity sizing in the presence of a common shared resource: Dimensioning an inbound call center. European Journal of Operational Research. 147 (3): 464-483.
[16]	Tirdad, A, Grassman, WK, Tavakoli, J. 2016. Optimal policies of M(t)/M/c/c queues with two different levels of servers. European Journal of Operational Research. 249 (3): 1124.
[17]	Cho, Y, Ko, YM. 2020. Stabilizing the virtual response time in single-server processor sharing queues with slowly time-varying arrival rates. Annals of Operations Research. 293 (1): 27-55.
[18]	Whitt, W, You, W. 2019. Time-varying robust queueing. Operations Research. 67 (6): 1766-1782.
[19]	Hunt, BR, Lipsman, RL, Osborn, JE, Rosenberg, JM. 2012. Differential equations with MATLAB. 3rd ed. New York: John Wiley.
[20]	Mathews, JH, Fink, KD. 2004. Numerical methods using MATLAB. 4th ed. New Delhi: Pearson Education.
[21]	Abou-El-Ata, MO, Hariri, AMA. 1992. The M/M/c/N queue with balking and reneging. Computers and Operations Research. 19 (8): 713-716.
[22]	Green, L, Kolesar, P. 1991. The pointwise stationary approximation for queues with nonstationary arrivals. Management Science. 37 (1): 84-97.

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APA Style

Siddharth Mahajan. (2020). Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers. American Journal of Operations Management and Information Systems, 5(4), 74-83. https://doi.org/10.11648/j.ajomis.20200504.11

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Siddharth Mahajan. Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers. Am. J. Oper. Manag. Inf. Syst. 2020, 5(4), 74-83. doi: 10.11648/j.ajomis.20200504.11

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AMA Style

Siddharth Mahajan. Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers. Am J Oper Manag Inf Syst. 2020;5(4):74-83. doi: 10.11648/j.ajomis.20200504.11

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@article{10.11648/j.ajomis.20200504.11,
  author = {Siddharth Mahajan},
  title = {Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers},
  journal = {American Journal of Operations Management and Information Systems},
  volume = {5},
  number = {4},
  pages = {74-83},
  doi = {10.11648/j.ajomis.20200504.11},
  url = {https://doi.org/10.11648/j.ajomis.20200504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajomis.20200504.11},
  abstract = {Both large and small firms maintain call centers to establish contact between themselves and their customers. A call center is staffed by Customer Service Representatives (CSRs). In addition to CSRs, a call center needs computers and telecommunication equipment such as an Automatic Call Distributor (ACD). If calls arrive according to a Poisson arrival process and the service times have an exponential distribution, the call center can be modeled as an M/M/s queue where s is the number of CSRs. Typical calculations include finding the number of CSRs required and finding the number of trunks lines required. However, if call center models ignore the abandonment behavior of customers, they distort information that is relevant to management. Typically, ignoring abandonment would lead to overstaffing, as fewer CSRs are actually needed, because some of the callers abandon the system. Also, nonstationarity of arrivals is highly prevalent in call centers. Green, Kolesar and Whitt plot hourly arrival rates for a financial services call center. There is significant variation in arrivals by time of day. In this paper, we model call centers as multiserver Markovian queues with both nonstationarity and abandonment. Nonstationarity is modeled by having a Poisson arrival stream with time dependent mean, which varies according to a sinusoid. Abandonment is defined in terms of an exponential patience random variable, which is the amount of time the caller would be on hold before abandoning the call. We numerically study the performance of this nonstationary M(t)/M/s+M queue with abandonment and compare its performance measures with those of the stationary M/M/s+M queue. We find that approximating a nonstationary system by a stationary system, even at low levels of nonstationarity, can lead to significant errors. Similar results in a system without abandonment, have been obtained by Green, Kolesar and Svoronos. Additionally, we find that abandonment dampens the effect of nonstationarity. Since abandonment and nonstationarity are both present together in real call centers, a real call center with a high level of abandonment behaves closer to an ideal stationary system.},
 year = {2020}
}

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TY - JOUR
T1 - Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers
AU - Siddharth Mahajan
Y1 - 2020/10/26
PY - 2020
N1 - https://doi.org/10.11648/j.ajomis.20200504.11
DO - 10.11648/j.ajomis.20200504.11
T2 - American Journal of Operations Management and Information Systems
JF - American Journal of Operations Management and Information Systems
JO - American Journal of Operations Management and Information Systems
SP - 74
EP - 83
PB - Science Publishing Group
SN - 2578-8310
UR - https://doi.org/10.11648/j.ajomis.20200504.11
AB - Both large and small firms maintain call centers to establish contact between themselves and their customers. A call center is staffed by Customer Service Representatives (CSRs). In addition to CSRs, a call center needs computers and telecommunication equipment such as an Automatic Call Distributor (ACD). If calls arrive according to a Poisson arrival process and the service times have an exponential distribution, the call center can be modeled as an M/M/s queue where s is the number of CSRs. Typical calculations include finding the number of CSRs required and finding the number of trunks lines required. However, if call center models ignore the abandonment behavior of customers, they distort information that is relevant to management. Typically, ignoring abandonment would lead to overstaffing, as fewer CSRs are actually needed, because some of the callers abandon the system. Also, nonstationarity of arrivals is highly prevalent in call centers. Green, Kolesar and Whitt plot hourly arrival rates for a financial services call center. There is significant variation in arrivals by time of day. In this paper, we model call centers as multiserver Markovian queues with both nonstationarity and abandonment. Nonstationarity is modeled by having a Poisson arrival stream with time dependent mean, which varies according to a sinusoid. Abandonment is defined in terms of an exponential patience random variable, which is the amount of time the caller would be on hold before abandoning the call. We numerically study the performance of this nonstationary M(t)/M/s+M queue with abandonment and compare its performance measures with those of the stationary M/M/s+M queue. We find that approximating a nonstationary system by a stationary system, even at low levels of nonstationarity, can lead to significant errors. Similar results in a system without abandonment, have been obtained by Green, Kolesar and Svoronos. Additionally, we find that abandonment dampens the effect of nonstationarity. Since abandonment and nonstationarity are both present together in real call centers, a real call center with a high level of abandonment behaves closer to an ideal stationary system.
VL - 5
IS - 4
ER -

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Siddharth Mahajan

Production and Operations Management Area, Indian Institute of Management, Bangalore, India

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Siddharth Mahajan. (2020). Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers. American Journal of Operations Management and Information Systems, 5(4), 74-83. https://doi.org/10.11648/j.ajomis.20200504.11

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ACS Style

Siddharth Mahajan. Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers. Am. J. Oper. Manag. Inf. Syst. 2020, 5(4), 74-83. doi: 10.11648/j.ajomis.20200504.11

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AMA Style

Siddharth Mahajan. Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers. Am J Oper Manag Inf Syst. 2020;5(4):74-83. doi: 10.11648/j.ajomis.20200504.11

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@article{10.11648/j.ajomis.20200504.11,
  author = {Siddharth Mahajan},
  title = {Nonstationarity and Abandonment in Markovian Queues with Application to Call Centers},
  journal = {American Journal of Operations Management and Information Systems},
  volume = {5},
  number = {4},
  pages = {74-83},
  doi = {10.11648/j.ajomis.20200504.11},
  url = {https://doi.org/10.11648/j.ajomis.20200504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajomis.20200504.11},
  abstract = {Both large and small firms maintain call centers to establish contact between themselves and their customers. A call center is staffed by Customer Service Representatives (CSRs). In addition to CSRs, a call center needs computers and telecommunication equipment such as an Automatic Call Distributor (ACD). If calls arrive according to a Poisson arrival process and the service times have an exponential distribution, the call center can be modeled as an M/M/s queue where s is the number of CSRs. Typical calculations include finding the number of CSRs required and finding the number of trunks lines required. However, if call center models ignore the abandonment behavior of customers, they distort information that is relevant to management. Typically, ignoring abandonment would lead to overstaffing, as fewer CSRs are actually needed, because some of the callers abandon the system. Also, nonstationarity of arrivals is highly prevalent in call centers. Green, Kolesar and Whitt plot hourly arrival rates for a financial services call center. There is significant variation in arrivals by time of day. In this paper, we model call centers as multiserver Markovian queues with both nonstationarity and abandonment. Nonstationarity is modeled by having a Poisson arrival stream with time dependent mean, which varies according to a sinusoid. Abandonment is defined in terms of an exponential patience random variable, which is the amount of time the caller would be on hold before abandoning the call. We numerically study the performance of this nonstationary M(t)/M/s+M queue with abandonment and compare its performance measures with those of the stationary M/M/s+M queue. We find that approximating a nonstationary system by a stationary system, even at low levels of nonstationarity, can lead to significant errors. Similar results in a system without abandonment, have been obtained by Green, Kolesar and Svoronos. Additionally, we find that abandonment dampens the effect of nonstationarity. Since abandonment and nonstationarity are both present together in real call centers, a real call center with a high level of abandonment behaves closer to an ideal stationary system.},
 year = {2020}
}

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