Analysis of a Call Center with Impatient Customers and Repairable Server

Analysis of a Call Center with Impatient Customers and Repairable Server

Peishu Chen Yan Chen 

School of Applied Mathematics, Chaohu University Hefei, Anhui 238000, China

Faculty of Science, Jiangsu University Zhenjiang, Jiangsu, 212013, China

Corresponding Author Email:,
15 March 2017
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15 April 2017
| | Citation



This article considers a call center with impatient customers, repairable server and Interactive Voice Response Units (IVRU). Calls enter the center whenever a trunk line is available; otherwise it is lost. After completing the instructions at the IVRU, the call may leave the center or be routed to an available agent. While waiting for the agent, calls may abandon the queue once the waiting time beyond their patience. We consider that each call abandons the queue independently of each other while waiting for the agent after a random amount of time. The failure rate of the server in the idle and busy periods are both exponential variables with different parameters.  We obtain seven performance measures for the stationary distribution of the system. Numerical examples will illustrate the effects of impatience, repair and some other on design parameters.


IVRU, impatient, repairable, call center, performance measures.

1. Introduction
2. Model Description
3. Derivation of the Stationary Distribution
4. Numerical Examples
5. Conclusion

The authors would like to thank the anonymous referees for their constructive comments and valuable suggestions on improving this paper. This research is partially sponsored by NSFA (KJ2014A174, KJ2015A270, 1608085QA03). It is also sponsored by key discipline of Chaohu University.


1. W. Whitt, Dynamic staffing in a telephone call center aiming to immediately answer all calls, Operations Research Letters, 1999, vol.24, pp.205-212.

2. Z.Aksin, M. Armony, V.Mehrotra, The modern call center: a multi-disciplinary perspective on operations management research, 2010, Production & Operations Management, vol.16, no.6, pp.665-688.

3. Jaime Gil, Title of paper, 2006, XXX conference, 2006, Barcelona, Spain, pp 54-56. S.Aguir, F.Karaesmen, O. Z. Akşin, F. Chauvet, The impact of retrials on call center performance, OR Spectrum,2004, vol.26, no. 3, pp.353-376.

4. E. A. Stepanov, B. Favre, F. Alam, et al, Automatic summarization of call-center conversations, 2015, IEEE Automatic Speech Recognition and Understanding Workshop, pp.1-3.

5. S. Dudin, O, Dudina, Call center operation model as a MAP/PH/N/R−N system with impatient customers, 2011, Problems of Information Transmission, vol. 47, pp.364–377. 

6. R. Srinivasan, J. Talim, J. Wang, Performance analysis of a call center with interactive voice response units, 2004,TOP, vol. 12,no.1, pp.91-110. 

7. Jinting Wang, Raj Srinivasan, Staffing a call center with Interactive Voice Response Units and impatient calls, 2008, IEEE, pp. 514–519.

8. Yan Chen, Peishu Chen,Yijuan Zhu, Performance analysis in a call center with calls' abandonment and optional feedback, 2015, Scientific Bulletin of National Mining University, no.4, pp.109-114.

9. Polyna Khudyakov, Paul Feigin, Avishai Mandelbaum, Designing a call center with an IVR, 2010, Queuing systems, vol.66,pp.1-23.

10. A. Brezavšček, A. Baggia, Optimization of a call centre performance using the stochastic queuing models, 2014, Business Systems Research Journal, vol.5, no.5, pp.6-18. 

11. W. K. Grassman, Matrix-geometric solutions in stochastic models (Marcel F. Neuts), 2006, Johns Hopkins University Press.

12. K. Kawanishi, QBD approximations of a call center queueing model with general patience distribution, 2008, Computers & Operations Research, vol.35, no.8, pp. 2463-2481.