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STAFFING A SERVICE SYSTEM WITH NON-POISSON NON-STATIONARY ARRIVALS

Published online by Cambridge University Press:  13 June 2016

Beixiang He ,
Yunan Liu  and
Ward Whitt
Beixiang He
Affiliation:
Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695, USA E-mail: [email protected]; [email protected]
Yunan Liu
Affiliation:
Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695, USA E-mail: [email protected]; [email protected]
Ward Whitt
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA E-mail: [email protected]
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Abstract

Motivated by non-Poisson stochastic variability found in service system arrival data, we extend established service system staffing algorithms using the square-root staffing formula to allow for non-Poisson arrival processes. We develop a general model of the non-Poisson non-stationary arrival process that includes as a special case the non-stationary Cox process (a modification of a Poisson process in which the rate itself is a non-stationary stochastic process), which has been advocated in the literature. We characterize the impact of the non-Poisson stochastic variability upon the staffing through the heavy-traffic limit of the peakedness (ratio of the variance to the mean in an associated stationary infinite-server queueing model), which depends on the arrival process through its central limit theorem behavior. We provide simple formulas to quantify the performance impact of the non-Poisson arrivals upon the staffing decisions, in order to achieve the desired service level. We conduct simulation experiments with non-stationary Markov-modulated Poisson arrival processes with sinusoidal arrival rate functions to demonstrate that the staffing algorithm is effective in stabilizing the time-varying probability of delay at designated targets.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 4 , October 2016 , pp. 593 - 621
Copyright
Copyright © Cambridge University Press 2016 

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References

1. Aksin, O.Z., Armony, M., & Mehrotra, V. (2007). The modern call center: a multi-disciplinary perspective on operations management research. Production and Operations Management 16: 665–688.CrossRefGoogle Scholar
2. Armony, M., Israelit, S., Mandelbaum, A., Marmor, Y., Tseytlin, Y., & Yom-Tov, G. (2015). Patient flow in hospitals: a data-based queueing-science perspective. Stochastic Systems 5(1): 146–194.CrossRefGoogle Scholar
3. Asmussen, S. (2003). Applied probability and queues, 2nd ed. New York: Springer.Google Scholar
4. Avramidis, A.N., Deslauriers, A., & L'Ecuyer, P. (2004). Modeling daily arrivals to a telephone call center. Management Science 50: 896–908.CrossRefGoogle Scholar
5. Bassamboo, A. & Zeevi, A. (2009). On a data-driven method for staffing large call centers. Operations Research 57(3): 714–726.CrossRefGoogle Scholar
6. Brown, L., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S., & Zhao, L. (2005). Statistical analysis of a telephone call center: a queueing-science perspective. Journal of the American Statistical Association 100: 36–50.CrossRefGoogle Scholar
7. Choudhury, G.L. & Whitt, W. (1994). Heavy-traffic asymptotic expansions for the asymptotic decay rates in the BMAP/G/1 queue. Stochastic Models 10(2): 453–498.CrossRefGoogle Scholar
8. Cox, D.R. & Lewis, P.A.W. (1966). The Statistical Analysis of Series of Events. London: Methuen.CrossRefGoogle Scholar
9. Defraeye, M. & van Nieuwenhuyse, I. (2013). Controlling excessive waiting times in small service systems with time-varying demand: an extension of the ISA algorithm. Decision Support Systems 54(4): 1558–1567.CrossRefGoogle Scholar
10. Defraeye, M. & van Nieuwenhuyse, I. (2015). Staffing and scheduling under nonstationary demand for service: a literature review. Omega 58: 4–25.CrossRefGoogle Scholar
11. Eick, S.G., Massey, W.A., & Whitt, W. (1993). M t/G/∞ queues with sinusoidal arrival rates. Management Science 39: 241–252.CrossRefGoogle Scholar
12. Feldman, Z., Mandelbaum, A., Massey, W.A., & Whitt, W. (2008). Staffing of time-varying queues to achieve time-stable performance. Management Science 54(2): 324–338.CrossRefGoogle Scholar
13. Fendick, K.W. & Whitt, W. (1989). Measurements and approximations to describe the offered traffic and predict the average workload in a single-server queue. Proceedings of the IEEE 71(1): 171–194.CrossRefGoogle Scholar
14. Fischer, W. & Meier-Hellstern, K. (1992). The Markov-modulated Poisson process (MMPP) cookbook. Performance Evaluation 18: 149–171.CrossRefGoogle Scholar
15. Garnett, O., Mandelbaum, A., & Reiman, M.I. (2002). Designing a call center with impatient customers. Manufacturing and Service Operations Management 4(3): 208–227.CrossRefGoogle Scholar
16. Gerhardt, I. & Nelson, B.L. (2009). Transforming renewal processes for simulation of nonstationary arrival processes. INFORMS Journal on Computing 21: 630–640.CrossRefGoogle Scholar
17. Glynn, P.W. & Whitt, W. (1993). Limit theorems for cumulative processes. Stochastic Processes and their Applications 47: 299–314.CrossRefGoogle Scholar
18. Green, L.V., Kolesar, P.J., & Whitt, W. (2007). Coping with time-varying demand when setting staffing requirements for a service system. Production and Operations Management 16: 13–29.CrossRefGoogle Scholar
19. Halfin, S. & Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Operations Research 29(3): 567–588.CrossRefGoogle Scholar
20. Ibrahim, R., L'Ecuyer, P., Regnard, N., & Shen, H. (2012). On the modeling and forecasting of call center arrivals. Proceedings of the 2012 Winter Simulation Conference 2012: pp. 256–267.Google Scholar
21. Jennings, O.B., Mandelbaum, A., Massey, W.A., & Whitt, W. (1996). Server staffing to meet time-varying demand. Management Science 42: 1383–1394.CrossRefGoogle Scholar
22. Jongbloed, G. & Koole, G. (2001). Managing uncertainty in call centers using Poisson mixtures. Applied Stochastic Models in Business and Industry 17: 307–318.CrossRefGoogle Scholar
23. Kim, S., Vel, P., Whitt, W., & Cha, W.C. (2015). Poisson and non-Poisson properties of appointment-generated arrival processes: the case of an endocrinology clinic. Operations Research Letters 43: 247–253.CrossRefGoogle Scholar
24. Kim, S.-H. & Whitt, W. (2014). Are call center and hospital arrivals well modeled by nonhomogeneous Poisson processes? Manufacturing and Service Operations Management 16(3): 464–480.CrossRefGoogle Scholar
25. Kim, S.-H. & Whitt, W. (2014). Choosing arrival process models for service systems: tests of a nonhomogeneous Poisson process. Naval Research Logistics 61(1): 66–90.CrossRefGoogle Scholar
26. Kuczura, A. (1973). The interrupted Poisson process as an overflow process. Bell System Technical Journal 52(3): 437–448.CrossRefGoogle Scholar
27. Li, A. & Whitt, W. (2014). Approximate blocking probabilities for loss models with independence and distribution assumptions relaxed. Performance Evaluation 80: 82–101.CrossRefGoogle Scholar
28. Li, A., Whitt, W., & Zhao, J. (2016). Staffing to stabilize blocking in loss models with time-varying arrival rates. Probability in the Engineering and Informational Sciences 22(2): 185–211.CrossRefGoogle Scholar
29. Liu, R., Liu, Y., & Wilson, J.R. (2014). Modeling and simulation of nonstationary non-Poisson processes. Working paper. Raleigh, NC: North Carolina State University.Google Scholar
30. Liu, Y. & Whitt, W. (2012). Stabilizing customer abandonment in many-server queues with time-varying arrivals. Operations Research 60(6): 1551–1564.CrossRefGoogle Scholar
31. Liu, Y. & Whitt, W. (2014). Many-server heavy-traffic limits for queues with time-varying parameters. Annals of Applied Probability 24(1): 378–421.CrossRefGoogle Scholar
32. Liu, Y. & Whitt, W. (2014). Stabilizing performance in networks of queues with time-varying arrival rates. Probability in the Engineering and Informational Sciences 28(4): 419–449.CrossRefGoogle Scholar
33. Liu, Y. & Whitt, W. (2016). Stabilizing performance in a service system with time-varying arrivals and customer feedback. Columbia University. Available from http://www.columbia.edu/~ww2040/allpapers.html Google Scholar
34. Ma, N. & Whitt, W. (2015). Efficient simulation of non-Poisson non-stationary point processes to study queueing approximations. Statistics and Probability Letters 102: 202–207.Google Scholar
35. Maman, S. (2009). Uncertainty in demand for service: the case of call centers and emergency departments. PhD dissertation, The Technion, Israel. Available from http://iew3.technion.ac.il/serveng Google Scholar
36. Mandelbaum, A. & Zeltyn, S. (2009). Staffing many-server queues with impatient customers: constraint satisfaction in call centers. Operations Research 57(9): 1189–1205.CrossRefGoogle Scholar
37. Massey, W.A. & Whitt, W. (1994). Unstable asymptotics for nonstationary queues. Mathematics of Operation Research 19: 267–291.CrossRefGoogle Scholar
38. Nelson, B.L. & Gerhardt, I. (2011). Modeling and simulating renewal nonstationary arrival processes to facilitate analysis. Journal of Simulation 5: 3–8.CrossRefGoogle Scholar
39. Neuts, M.F. (1989). Structured Stochastic Matrices of M/G/1 type and Their Applications. New York: Marcel Dekker.Google Scholar
40. Pang, G., Talreja, R., & Whitt, W. (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probability Surveys 4: 193–267.CrossRefGoogle Scholar
41. Pang, G. & Whitt, W. (2010). Two-parameter heavy-traffic limits for infinite-server queues. Queueing Systems 65: 325–364.CrossRefGoogle Scholar
42. Pang, G. & Whitt, W. (2012). The impact of dependent service times on large-scale service systems. Manufacturing and Service Operations Management 14(2): 262–278.CrossRefGoogle Scholar
43. Pang, G. & Whitt, W. (2013). Two-parameter heavy-traffic limits for infinite-server queues with dependent service times. Queueing Systems 73(2): 119–146.CrossRefGoogle Scholar
44. Shen, H. & Huang, J.Z. (2008). Interday forecasting ind intraday updating of call center arrivals. Manufacturing and Service Operations Management 10(3): 391–410.CrossRefGoogle Scholar
45. Sriram, K. & Whitt, W. (1986). Characterizing superposition arrival processes in packet multiplexers for voice and data. IEEE Journal on Selected Areas in Communications SAC-4(6): 833–846.CrossRefGoogle Scholar
46. Stolletz, R. (2008). Approximation of the nonstationary M(t)/M(t)/c(t)-queue using stationary models: the stationary backlog-carryover approach. European Journal of Operational Research 190(2): 478–493.CrossRefGoogle Scholar
47. Whitt, W. (1982). Approximating a point process by a renewal process: two basic methods. Operations Research 30: 125–147.CrossRefGoogle Scholar
48. Whitt, W. (1992). Asymptotic formulas for Markov processes with applications to simulation. Operations Research 40(2): 279–291.CrossRefGoogle Scholar
49. Whitt, W. (1992). Understanding the efficiency of multi-server service systems. Management Science 38(5): 708–723.CrossRefGoogle Scholar
50. Whitt, W. (2002). Stochastic-process Limits. New York: Springer.CrossRefGoogle Scholar
51. Whitt, W. (2004). A diffusion approximation for the G/GI/n/m queue. Operations Research 52(6): 922–941.CrossRefGoogle Scholar
52. Whitt, W. (2006). Staffing a call center with uncertain arrival rate and absenteeism. Production and Operations Management 15(1): 88–102.CrossRefGoogle Scholar
53. Whitt, W. (2015). Stabilizing performance in a single-server queue with time-varying arrival rate. Queueing Systems 81: 341–378.CrossRefGoogle Scholar
54. Wolfe, R.W. (1977). The effect of service time regularity on system performance. InChandy, K.M. & Reiser, M. (eds.), Computer Performance. Amsterdam: Elsevier, pp. 297–304.Google Scholar
55. Yom-Tov, G. & Mandelbaum, A. (2014). Erlang R: a time-varying queue with reentrant customers, in support of healthcare staffing. Manufacturing and Service Operations Management 16(2): 283–299.CrossRefGoogle Scholar
56. Zeltyn, S. & Mandelbaum, A. (2005). Call centers with impatient customers: many-server asymptotics of the M/M/n+G queue. Queueing Systems 51(5): 361–402.CrossRefGoogle Scholar
57. Zhang, X., Hong, L.J., & Glynn, P.W. (2014). Timescales in modeling call center arrivals. working paper, Department of Industrial Engineering and Logistics Management, The Hong Kong University of Science and Technology.Google Scholar