Application of an Inhomogeneous Ctmc Model for a Telephone Call Center

Maciej Rafał Burak


In this paper we consider modeling an inbound telephone call center, where the callers may balk or abandon, with an inhomogeneous continuous time Markov chain model. We further discuss the practical application of the uniformization method and compare it to traditional call center modeling methods based on stationary approximations.


operations research; process planning; Markov processes; human resource management; call center; numerical methods

Full Text:



Aksin Z., Armony M., Mehrotra V.: The Modern Call Center: A Multidisciplinary Perspective on Operations Management Research. Production and Operations Management, 16(6), Nov. 2007, p. 665-688, doi: 10.1111/j.1937-5956.2007.tb00288.x.

Arns M., Buchholz P., Panchenko A.: On the Numerical Analysis of Inhomogeneous Continuous-time Markov Chains. INFORMS Journal on Computing, 22(3), Aug. 2010, p. 416-432, doi: 10.1287/ijoc.1090.0357.

Baccelli F., Hebuterne G.: On Queues with Impatient Customers, 1981.

Brown L., Gans N., Mandelbaum A., Sakov A., Shen H., Zeltyn S., Zhao L.: Statistical Analysis of a Telephone Call Center. Journal of the American Statistical Association, 100(469), Mar. 2005, p. 36-50, doi: 10.1198/016214504000001808.

Burak M.: Multi-step Uniformization with Steady-state Detection in Nonstationary M/M/s Queuing Systems. arXiv preprint arXiv:1410.0804, 2014.

Burak M. R.: Inhomogeneous CTMC Model of a Call Center with Balking and Abandonment. Studia Informatica, 36(2), 2015, p. 23-34.

Creemers S., Defraeye M., Van Nieuwenhuyse I.: G-RAND: A Phase-type Approximation for the Nonstationary Queue. Performance Evaluation, 80, 2014, p. 102-123.

Czachórski T., Nycz T., Nycz M., Pekergin F.: Traffic Engineering: Erlang and Engset Models Revisited with Diffusion Approximation. In Information Sciences and Systems, Springer Science – Business Media, 2014, p. 249-256, doi: 10.1007/978-3-319-09465-6_26.

Defraeye M., Nieuwenhuyse I. V.: Staffing and Scheduling Under Nonstationary Demand for Service: A literature review. Omega, Apr. 2015, doi: 10.1016/ 2015.04.002.

Deslauriers, L’Ecuyer P., Pichitlamken J., Ingolfsson A., Avramidis A.N.: Markov Chain Models of a Telephone Call Center with Call Blending. Computers & Operations Research, 34(6), Jun. 2007, p. 1616-1645, doi: 10.1016/j.cor.2005.06.019.

Gans N., Koole G., Mandelbaum A.: Telephone Call Centers: Tutorial, Review, and Research Prospects. Manufacturing & Service Operations Management, 5(2), Apr. 2003, p.79-141, doi: 10.1287/msom.

Grassmann W.K.: Transient Solutions in Markovian Queueing Systems. Computers & Operations Research, 5(2), Jan. 1978, p.161, doi: 10.1016/0305-0548(78)90010-2.

Green L.V., Kolesar P.J., Soares J.: Improving the SIPP Approach for Staffing Service Systems that Have Cyclic Demands. Operations Research, 49 (4), Aug. 2001, p. 549-564, doi: 10.1287/opre.49.4.549.11228.2.

Green L.V., Kolesar P.J., Whitt W.: Coping with Time-Varying Demand when Setting Staffing Requirements for a Service System. Production and Operations Management, 16(1), Jan. 2007, p. 13-39, doi: 10.1111/j.1937-5956.2007.tb00164.x.

Gross D., Miller D.R.: The Randomization Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes. Operations Research, 32(2), Apr. 1984,

p. 343-361, doi: 10.1287/opre.32.2.343.

Helber S., Henken K.: Profit-oriented Shift Scheduling of Inbound Contact Centers with Skills-based Routing, Impatient Customers, and Retrials. OR Spectrum, 32(1), Jul. 2008, p. 109-134, doi: 10.1007/s00291-008-0141-8.

Ingolfsson, Akhmetshina E., Budge S., Li Y., Wu X.: A Survey and Experimental Comparison of Service-level-approximation Methods for Nonstationary M(t)/M/s(t) Queueing Systems with Exhaustive Discipline. INFORMS Journal on Computing, 19(2), May. 2007, p. 201-214, doi: 10.1287/ijoc.1050.0157.

Ingolfsson, Campello F., Wu X., Cabral E.: Combining Integer Programming and the Randomization Method to Schedule Employees. European Journal of Operational Research, 202(1), Apr. 2010, p. 153-163, doi: 10.1016/j.ejor.2009.04.026.

Kendall D.G.: Stochastic Processes Occurring in the Theory of Queues and Their Analysis by the Method of the Imbedded Markov Chain. Ann. Math. Statist., 24(3), Sep. 1953, p. 338-354, doi: 10.1214/aoms/1177728975.

Mehrotra V., Ozlük O., Saltzman R.: Intelligent Procedures for Intraday Updating of Call Center Agent Schedules. Production and Operations Management, 19(3): 2010, p. 353-367.

Jimenez T., Koole G.: Scaling and Comparison of Fluid Limits of Queues Applied to Call Centers with Time-varying Parameters. OR Spectrum, 26(3), Jul. 2004, p. 413-422, doi: 10.1007/s00291-004-0162-x.

Phung-Duc T., Kawanishi K.: Performance Analysis of Call Centers with Abandonment. Retrial and After-call Work, Performance Evaluation, 80, Oct. 2014,

p. 43-62, doi: 10.1016/j.peva.2014.03.001.

Pichitlamken J., Deslauriers A., L`Ecuyer P., Avramidis A.: Modelling and Simulation of a Telephone Call Center. In Proceedings of the 2003 International Conference on Machine Learning and Cybernetics (IEEE Cat. No.03EX693), IEEE, 2003, doi: 10.1109/wsc.2003.1261636.

Reibman, Trivedi K.: Numerical Transient Analysis of Markov Models. Computers & Operations Research, 15(1), Jan. 1988, p. 19-36, doi: 10.1016/0305-0548(88)90026-3.

Schwarz J.A., Selinka G., Stolletz R.: Performance Analysis of Time-dependent Queueing Systems: Survey and Classification. Omega 2015, doi: 10.1016/ .2015.10.013.

Stewart W.J.: Introduction to the Numerical Solution of Markov Chains. Vol. 41, Princeton University Press NJ, 1994.

Stolletz R.: Performance Analysis and Optimization of Inbound Call Centers. Springer, Berlin Heidelberg 2003, doi: 10.1007/978-3-642-55506-0.3

Whitt W.: Fluid Models for Multiserver Queues with Abandonments. Operations Research, 54(1), Feb. 2006, p. 37-54, doi: 10.1287/opre.1050.0227.

Whitt W.: Sensitivity of Performance in the Erlang-A Queueing Model to Changes in the Model Parameters. Operations Research, 54(2), Apr. 2006, p. 247-260, doi: 10.1287/opre.1050.0257.

Zeltyn S., Mandelbaum A.: Call Centers with Impatient Customers: Many-server Asymptotics of the M/M/N + G Queue. Queueing Systems, 51 (3-4), Dec. 2005,

p. 361-402, doi: 10.1007/s11134-005-3699-8.4.