Markov Decision Processes in Practice pp 487-503 | Cite as

# Flexible Staffing for Call Centers with Non-stationary Arrival Rates

## Abstract

We consider a multi-period staffing problem of a single-skill call center. The call center is modeled as a multi-server queue in which the staffing levels can be changed only at specific moments in time. The objective is to set the staffing levels such that a service level constraint is met in the presence of time-varying arrival rates. We develop a Markov decision model to obtain time-dependent staffing levels for both the case where the arrival rate function is known as well as unknown. The characteristics of the optimal policies associated to the two cases are illustrated through a numerical study based on real-life data. We show that the optimal policies provide a good balance between staffing costs and the penalty probability for not meeting the service level.

## Key words

Call centers Markov decision processes Staffing Time-varying arrival rates## References

- 1.O.Z. Akşin, M. Armony, V. Mehrotra, The modern call center: a multi-disciplinary perspective on operations management research. Prod. Oper. Manag.
**16**(6), 665–688 (2007)Google Scholar - 2.S. Aldor-Noiman, P.D. Feigin, A. Mandelbaum, Workload forecasting for a call center: methodology and a case study. Ann. Appl. Stat.
**3**(4), 1403–1447 (2009)CrossRefGoogle Scholar - 3.S. Asmussen, P.W. Glynn,
*Stochastic Simulation: Algorithms and Analysis*(Springer, New York, 2007)Google Scholar - 4.A.N. Avramidis, A. Deslauriers, P. L’Ecuyer, Modeling daily arrivals to a telephone call center. Manag. Sci.
**50**(7), 896–908 (2004)CrossRefGoogle Scholar - 5.J. Bard, H. Purnomo, Short-term nurse scheduling in response to daily fluctuations in supply and demand. Health Care Manag. Sci.
**8**(4), 315–324 (2005)CrossRefGoogle Scholar - 6.R. Batta, O. Berman, Q. Wang, Balancing staffing and switching costs in a service center with flexible servers. Eur. J. Oper. Res.
**177**(2), 924–938 (2007)CrossRefGoogle Scholar - 7.O. Berman, R.C. Larson, A queueing control model for retail services having back room operations and cross-trained workers. Comput. Oper. Res.
**31**(2), 201–222 (2004)CrossRefGoogle Scholar - 8.L. Brown, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn, and L. Zhao. Multifactor Poisson and gamma-Poisson models for call center arrival times. Working Paper, 2004Google Scholar
- 9.L.D. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn, L. Zhao, Statistical analysis of a telephone call center: a queueing-science perspective. J. Am. Stat. Assoc.
**100**(469), 36–50 (2005)CrossRefGoogle Scholar - 10.F.F. Easton, J.C. Goodale, Schedule recovery: unplanned absences in service operations. Decis. Sci.
**36**(3), 459–488 (2005)CrossRefGoogle Scholar - 11.Z. Feldman, A. Mandelbaum, W.A. Massey, W. Whitt, Staffing of time-varying queues to achieve time-stable performance. Manag. Sci.
**54**(2), 324–338 (2008)CrossRefGoogle Scholar - 12.N. Gans, G.M. Koole, A. Mandelbaum, Telephone call centers: tutorial, review, and research prospects. Manuf. Serv. Oper. Manag.
**5**(2), 79–141 (2003)Google Scholar - 13.L.V. Green, P.J. Kolesar, The pointwise stationary approximation for queues with nonstationary arrivals. Manag. Sci.
**37**(1), 84–97 (1991)CrossRefGoogle Scholar - 14.L.V. Green, P.J. Kolesar, J. Soares, Improving the SIPP approach for staffing service systems that have cyclic demands. Oper. Res.
**49**(4), 549–564 (2001)CrossRefGoogle Scholar - 15.L.V. Green, P.J. Kolesar, J. Soares, An improved heuristic for staffing telephone call centers with limited operating hours. Prod. Oper. Manag.
**12**(1), 46–61 (2003)CrossRefGoogle Scholar - 16.L.V. Green, P.J. Kolesar, W. Whitt, Coping with time-varying demand when setting staffing requirements for a service system. Prod. Oper. Manag.
**16**(1), 13–39 (2007)CrossRefGoogle Scholar - 17.J. Harrison, A. Zeevi, A method for staffing large call centers based on stochastic fluid models. Manuf. Serv. Oper. Manag.
**7**(1), 20–36 (2005)Google Scholar - 18.A. Ingolfsson, E. Akhmetshina, S. Budge, Y. Li, X. Wu, A survey and experimental comparison of service-level-approximation methods for nonstationary
*M*(*t*)∕*M*∕*s*(*t*) queueing systems with exhaustive discipline. INFORMS J. Comput.**19**(2), 201–214 (2007)CrossRefGoogle Scholar - 19.T. Jiménez, G.M. Koole, Scaling and comparison of fluid limits of queues applied to call centers with time-varying parameters. OR Spectr.
**26**(3), 413–422 (2004)CrossRefGoogle Scholar - 20.G. Jongbloed, G.M. Koole, Managing uncertainty in call centers using Poisson mixtures. Appl. Stoch. Model. Bus. Ind.
**17**(4), 307–318 (2001)CrossRefGoogle Scholar - 21.S. Liao, G.M. Koole, C. van Delft, O. Jouini, Staffing a call center with uncertain non-stationary arrival rate and flexibility. OR Spectr.
**34**, 1–31 (2012)CrossRefGoogle Scholar - 22.V. Mehrotra, O. Ozlük, R. Saltzman, Intelligent procedures for intra-day updating of call center agent schedules. Prod. Oper. Manag.
**19**(3), 353–367 (2010)CrossRefGoogle Scholar - 23.E.J. Pinker, R.C. Larson, Optimizing the use of contingent labor when demand is uncertain. Eur. J. Oper. Res.
**144**(1), 39–55 (2003)CrossRefGoogle Scholar - 24.T. Robbins, Managing service capacity under uncertainty. Ph.D. Thesis, Penn State University, 2007Google Scholar
- 25.A. Roubos, G.M. Koole, R. Stolletz, Service level variability of inbound call centers. Manuf. Serv. Oper. Manag.
**14**(3), 402–413 (2012)Google Scholar - 26.H. Shen, J.Z. Huang, Forecasting time series of inhomogeneous Poisson processes with application to call center workforce management. Ann. Appl. Stat.
**2**(2), 601–623 (2008)CrossRefGoogle Scholar - 27.H. Shen, J.Z. Huang, Interday forecasting and intraday updating of call center arrivals. Manuf. Serv. Oper. Manag.
**10**(3), 391–410 (2008)Google Scholar - 28.S.G. Steckley, S.G. Henderson, V. Mehrotra, Service system planning in the presence of a random arrival rate. Working Paper, 2004Google Scholar
- 29.J.W. Taylor, A comparison of univariate time series methods for forecasting intraday arrivals at call a center. Manag. Sci.
**54**(2), 253–265 (2008)CrossRefGoogle Scholar - 30.J. Weinberg, L.D. Brown, J.R. Stroud, Bayesian forecasting of an inhomogeneous Poisson process with applications to call center data. J. Am. Stat. Assoc.
**102**(480), 1186–1199 (2007)CrossRefGoogle Scholar - 31.W. Whitt, Staffing a call center with uncertain arrival rate and absenteeism. Prod. Oper. Manag.
**15**(1), 88–102 (2006)Google Scholar - 32.J. Yoo, Queueing models for staffing service operations. Ph.D. Thesis, University of Maryland, 1996Google Scholar