Abstract
Motivated by modern call centers, we consider large-scale service systems with multiple server pools and a single customer class. For such systems, we propose a simple routing rule which asymptotically minimizes the steady-state queue length and virtual waiting time. The proposed routing scheme is FSF which assigns customers to the Fastest Servers First. The asymptotic regime considered is the Halfin-Whitt many-server heavy-traffic regime, which we refer to as the Quality and Efficiency Driven (QED) regime; it achieves high levels of both service quality and system efficiency by carefully balancing between the two. Additionally, expressions are provided for system limiting performance measures based on diffusion approximations. Our analysis shows that in the QED regime this heterogeneous server system outperforms its homogeneous server counterpart.
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References
M. Armony and N. Bambos, Queueing dynamics and maximal throughput scheduling in switched processing systems, Queueing Systems 44 (2003) 209–252.
M. Armony and C. Maglaras, On customer contact centers with a call-back option: Customer decisions, routing rules and system design, {Operations Research} {52}(2) (2004) 271–292.
M. Armony and C. Maglaras, Contact centers with a call-back option and real-time delay information, {Operations Research} {52}(4) (2004) 527–545.
M. Armony and A. Mandelbaum, Routing and staffing in large-scale service systems with heterogeneous servers and impatient customers, Preprint (2005).
R. Atar, A diffusion model of scheduling control in queueing systems with many servers, { Ann. Appl. Probab.} { 15}(1B) (2005) 820–852.
R. Atar, Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic, { Ann. Appl. Probab.}, to appear (2005).
R. Atar, A. Mandelbaum and M. Reiman, Scheduling a multi-class queue with many exponential servers: Asymptotic optimality in heavy traffic, { Ann. Appl. Prob} { 14}(3) (2004) 1084–1134.
A. Bassamboo, J.M. Harrison and A. Zeevi, Design and control of a large call center: Asymptotic analysis of an LP-based method, preprint (2004).
A. Bassamboo, J.M. Harrison and A. Zeevi, Dynamic routing and admission control in high-volume service systems: Asymptotic analysis via multi-scale fluid limits, preprint (2004).
S.L. Bell and R.J. Williams, Dynamic scheduling of a system with two parallel servers in heavy traffic with complete resource pooling: Asymptotic optimality of a continuous review threshold policy, {Annals of Applied Probability} {11} (2001) 608–649.
S. Bhulai and G. Koole, A queueing model for call blending in call centers, {IEEE Transactions on Automatic Control} {48} (2003) 1434–1438.
S. Borst, A. Mandelbaum and M. Reiman, Dimensioning large call centers, { Operations Research} {52}(1) (2004) 17–34.
M. Bramson, State space collapse with applications to heavy-traffic limits for multiclass queueing networks, {Queueing Systems} { 30} (1997) 89–148.
A. Brandt and M. Brandt, On a two-queue priority system with impatience and its application to a call center, { Methodology and Computing in Applied Probablity} {1} (1999) 191–210.
S. Browne and W. Whitt, Piecewise-linear diffusion processes, in: {Advances in Queueing. Theory, Methods, and Open Problems}, ed. J.H. Dshalalow, (CRC Press, 1995), Chapter 18, pp. 463–480.
H. Chen and D. Yao, Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization., (Springer, New-York, 2001).
J.G. Dai, Stability of fluid and stochastic processing networks, {MaPhySto}, 9 (1999).
F. de Véricourt and Y.-P. Zhou, A routing problem for call centers with customer callbacks after service failure, { Operations Research}, to appear, (2004).
F. de Véricourt and Y.-P. Zhou, On the incomplete results for the multiple-server slow-server problem, Technical report, Duke University, The Fuqua School of Business (2004).
A.K. Erlang, On the rational determination of the number of circuits, in: The Life and Works of. A.K. Erlang. eds. E. Brockmeyer, H.L. Halstrom, and A. Jensen, (Copenhagen: The Copenhagen Telephone Company, Copenhagen, 1948).
S.N. Ethier and T.G. Kurtz, Markov Processes, Characterization and Convergence. (John Wiley & Sons, 1985).
A. Federgruen and H. Groenevelt, M/G/c. systems with multiple customer classes: Characterization and achievable performance unrder nonpreemptive priority rules, Management Science {34} (1988) 1121–1138.
P. Fleming, A. Stolyar and B. Simon, Heavy traffic limit for a mobile phone system loss model, in: Proceedings of 2nd Int'l Conf. on Telecomm. Syst. Mod. and Analysis., Nashville, TN (1994).
G.J. Foschini, On heavy traffic diffusion analysis and dynamic routing in packet switched networks, in: Computer Performance., eds. K.M. Chandy and M. Reiser (North Holland, 1977).
N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutorial, review and research prospects, {Manufacturing & Service Operations Management} 5(2) (2003) 79–141.
N. Gans and Y.-P. Zhou, A call-routing problem with service-level constraints, {Operations Research} {51}(2) (2003) 255–271.
O. Garnett, A. Mandelbaum and M. Reiman, Designing a call center with impatient customers, { Manufacturing & Service Operations Management} {4}(3) (2002) 208–227.
K. Glazebrook and J. Ni {n}o-Mora, Parallel scheduling of multiclass M./M./m. queues: Approximate and heavy-traffic optimization of achievable performance, { Operations Research} 49(4) (2001) 609–623.
P.W. Glynn, Diffusion approximations, in: Stochastic Models, Handbooks in OR & MS., eds. D. Heyman and M. Sobel (North-Holland, 1990) vol. 2, pp. 145–198.
I. Gurvich, Design and control of the M/M/N. queue with multi-class customers and many servers, {Masters Thesis}, Technion Institute of Technology, Israel (2004).
I. Gurvich, M. Armony and A. Mandelbaum, Staffing and control of large-scale service systems with multiple customer classes and fully flexible servers, preprint, (2004).
S. Halfin and W. Whitt, Heavy-traffic limits for queues with many exponential servers, {Operations Research} {29}(3) (1981) 567–588.
J.M. Harrison, Heavy traffic analysis of a system with parallel servers: Asymptotic analysis of discrete-review policies, { Annals of Applied Probability} {8} (1998) 822–848.
J.M. Harrison and A. Zeevi, Dynamic scheduling of a multiclass queue in the Halfin and Whitt heavy traffic regime, {Operations Research} {52}(2) (2004) 243-257.
D.L. Jagerman, Some properties of the Erlang loss function, {Bell Systems Technical Journal} {53}(3) (1974) 525–551.
P. Jelenkovic, A. Mandelbaum and P. Momcilović, The GI/D/N queue in the {QED} regime, {Queueing Systems} {47} (2004) 53–69.
O. Kella and U. Yechiali, Waiting times in the nonpreemptive priority M/M/c. queue, {Stochastic Models} {1}(2) (1985) 257–262.
F.P. Kelly and C.N. Laws, Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling, {Queueing Systems} {13} (1993) 47–86.
W. Lin and P.R. Kumar, Optimal control of a queueing system with two heterogeneous servers, {IEEE Trans. Automat. Control} {29} (1984) 696–703.
R.Sh. Lipster and A.N. Shiryaev, Theory of Martingales., (Kluwer, Amsterdam, 1989).
H.P. Luh and I. Viniotis, Threshold control policies for heterogeneous server systems, {Math Meth Oper Res} {55} (2002) 121–142.
C. Maglaras and A. Zeevi, Pricing and capacity sizing for systems with shared resources: Scaling relations and approximate solutions, {Management Science} {49}(8) (2003) 1018–1038.
A. Mandelbaum and A.L. Stolyar, Scheduling flexible servers with convex delay costs: Heavy-traffic optimality of the generalized c.μ-rule, Operations Research 52(6) (2004) 836–855.
W.A. Massey and R.B. Wallace, An optimal design of the M/M/C/K. queue for call centers, { Queueing Systems} to appear (2004).
A. Puhalskii, On the invariance principle for the first passage time, { Mathematics of Operations Research} {19}(4) (1994) 946–954.
A.A Puhalskii and M.I. Reiman, The multiclass GI/PH/N. queue in the Halfin-Whitt regime, { Advances in Applied Probability} {32} (2000) 564–595.
M.I. Reiman, Some diffusion approximations with state space collapse, in: Modelling and Performance Evaluation Methodology., eds. F. Baccelli and G. Fayolle (Springer-Verlag, 1984) pp. 209–240.
V.V. Rykov, Monotone control of queueing systems with heterogeneous servers, {Queueing Systems} {37} (2001) 391–403.
V.V. Rykov and D. Efrosinin, Optimal control of queueing systems with heterogeneous servers, { Queueing Systems} {46}, (2004) 389–407.
A.A. Scheller-Wolf, Necessary and sufficient conditions for delay moments in FIFO multiserver queues with an application comparing slow servers with one fast one, { Operations Research} {51} (2003) 748–758.
J.G. Shanthikumar and D.D. Yao, Comparing ordered-entry queues with heterogeneous servers, { Queueing Systems} {2} (1987) 235–244.
R.A. Shumsky, Approximation and analysis of a call center with flexible and Specialized servers, {OR Spectrum} {26}(3) (2004) 307–330.
S. Stolyar, Optimal routing in output-queues flexible server systems, {Probability in the Engineering and Informational Sciences} 19 (2005) 141–189.
D.Y. Sze, A queueing model for telephone operator staffing, {Operations Research} {32} (1984) 229–249.
Y.-Ch. Teh and A.R. Ward, Critical thresholds for dynamic routing in queueing networks, {Queueing Systems} {42} (2002) 297–316.
R.B. Wallace and W. Whitt, A Staffing Algorithm for Call Centers with Skill-Based Routing, working paper (2004).
W. Whitt, Heavy traffic approximations for service systems with blocking, { AT & T Bell Lab. Tech. Journal} {63} (1984) 689–708.
W. Whitt, {Stochastic-process limits: An introduction to stochastic-process limits and their application to queues}, Springer (2002).
W. Whitt, A diffusion approximation for the G/GI/n/m queue, {Operations Research} {52}(6) (2004) 922–941.
W. Whitt, Heavy-traffic limits for the G./H.2 */n./m. queue, {Mathematics of Operations Research} {30}(1) (2005) 1–27.
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AMS subject classification: 60K25, 68M20, 90B22
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Armony, M. Dynamic Routing in Large-Scale Service Systems with Heterogeneous Servers. Queueing Syst 51, 287–329 (2005). https://doi.org/10.1007/s11134-005-3760-7
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DOI: https://doi.org/10.1007/s11134-005-3760-7