Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic

Fleming, W. H. and Soner, H. M. (1993). <i>Controlled Markov Processes and Viscosity Solutions</i>. Springer, New York.

Mandelbaum, A. and Stolyar, A. L. (2004). Scheduling flexible servers with convex delay costs: Heavy-traffic optimality of the generalized $c\mu$-rule. <i>Oper. Res.</i> To appear.

Billingsley, P. (1999). <i>Convergence of Probability Measures</i>, 2nd ed. Wiley, New York.

Karatzas, I. and Shreve, S. E. (1991). <i>Brownian Motion and Stochastic Calculus</i>, 2nd ed. Springer, New York.

Armony, M. and Maglaras, C. (2004). Customer contact centers with multiple service channels. <i>Oper. Res.</i> To appear.

Atar, R., Mandelbaum, A. and Reiman, M. (2004). A Brownian control problem for a simple queueing system in the Halfin–Whitt regime. <i>Systems Control Lett</i>. <b>51</b> 269–275.

Bell, S. L. and Williams, R. J. (2001). Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: Asymptotic optimality of a threshold policy. <i>Ann. Appl. Probab.</i> <b>11</b> 608–649.

Birkhoff, G. and Rota, G.-C. (1962). <i>Ordinary Differential Equations.</i> Ginn and Company, Boston.

Borkar, V. S. (1989). <i>Optimal Control of Diffusion Processes</i>. Longman, Harlow.

Borst, S., Mandelbaum, A. and Reiman, M. (2004). Dimensioning large call centers. <i>Oper. Res.</i> To appear.

Borst, S. C. and Seri, P. (2000). Robust algorithms for sharing agents with multiple skills. Preprint.

Durrett, R. (1996). <i>Probability</i>: <i>Theory and Examples</i>, 2nd ed. Duxbury Press, Pacific Grover, CA.

Ethier, S. N. and Kurtz, T. G. (1986). <i>Markov Processes. Characterization and Convergence.</i> Wiley, New York.

Fleming, P. J., Stolyar, A. and Simon, B. (1994). Heavy traffic limits for a mobile phone system loss model. In <i>Proceedings of 2nd International Conference on Telecommunication Systems Modelling and Analysis</i>.

Gans, N., Koole, G. and Mandelbaum, A. (2003). Telephone call centers: Tutorial, review and research prospects. <i>Manufacturing and Service Operations Management</i> <b>5</b> 79–141.

Garnett, O. and Mandelbaum, A. (2000). An introduction to skills-based routing and its operational complexities. Teaching notes.

Garnett, O., Mandelbaum, A. and Reiman, M. (2002). Designing a call center with impatient customers. <i>Manufacturing and Service Operations Management</i> <b>4</b> 208–227.

Gilbarg, D. and Trudinger, N. S. (1983). <i>Elliptic Partial Differential Equations of Second Order</i>, 2nd ed. Springer, Berlin.

Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. <i>Oper. Res.</i> <b>29</b> 567–588.

Harrison, J. M. (1988). Brownian models of queueing networks with heterogeneous customer populations. <i>Math. Appl.</i> <b>10</b> 147–186.

Harrison, J. M. and Van Mieghem, J. A. (1997). Dynamic control of Brownian networks: State space collapse and equivalent workload formulations. <i>Ann. Appl. Probab.</i> <b>7</b> 747–771.

Harrison, J. M. and Wein, L. M. (1990). Scheduling networks of queues: Heavy traffic analysis of a two-station closed network. <i>Oper. Res.</i> <b>38</b> 1052–1064.

Harrison, J. M. and Zeevi, A. (2004). Dynamic scheduling of a multiclass queue in the Halfin–Whitt heavy traffic regime. <i>Oper. Res.</i> To appear.

Iglehart, D. L. and Whitt, W. (1971). The equivalence of functional central limit theorems for counting processes and associated partial sums. <i>Ann. Math. Statist.</i> <b>42</b> 1372–1378.

Ishii, H. (1989). On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. <i>Comm. Pure Appl. Math.</i> <b>42</b> 15–45.

Jagerman, D. L. (1974). Some properties of the Erlang loss function. <i>Bell Syst. Tech. J.</i> <b>53</b> 525–551.

Krichagina, E. V. and Taksar, M. I. (1992). Diffusion approximation for $GI/G/1$ controlled queues. <i>Queueing Systems Theory Appl.</i> <b>12</b> 333–367.

Kumar, S. (2000). Two-server closed networks in heavy traffic: Diffusion limits and asymptotic optimality. <i>Ann. Appl. Probab.</i> <b>10</b> 930–961.

Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. <i>Ann. Probab.</i> <b>19</b> 1035–1070.

Kushner, H. J. and Chen, Y. N. (2000). Optimal control of assignment of jobs to processors under heavy traffic. <i>Stochastics Stochastics Rep.</i> <b>68</b> 177–228.

Martins, L. F., Shreve, S. E. and Soner, H. M. (1996). Heavy traffic convergence of a controlled, multiclass queueing system. <i>SIAM J. Control Optim.</i> <b>34</b> 2133–2171.

Plambeck, E., Kumar, S. and Harrison, J. M. (2001). A multiclass queue in heavy traffic with throughput time constraints: Asymptotically optimal dynamic controls. <i>Queueing Systems Theory Appl.</i> <b>39</b> 23–54.

Protter, P. (1990). <i>Stochastic Integration and Differential Equations. A New Approach.</i> Springer, Berlin.

Puhalskii, A. A. and Reiman, M. I. (2000). The multiclass $\mathit{GI}/\mathit{PH}/N$ queue in the Halfin–Whitt regime. <i>Adv. in Appl. Probab.</i> <b>32</b> 564–595.

Reiman, M. I. (1984). Some diffusion approximations with state space collapse. <i>Proc. of the Internat. Seminar on Modeling and Performance Evaluation Methodology</i>. <i>Lecture Notes in Control and Informational Science</i> 209–240. Springer, New York.

Sze, D. Y. (1984). A queueing model for telephone operator staffing. <i>Oper. Res.</i> <b>32</b> 229–249.

Van Mieghem, J. A. (1995). Dynamic scheduling with convex delay costs: The generalized $c\mu$ rule. <i>Ann. Appl. Probab.</i> <b>5</b> 809–833.

Zohar, E., Mandelbaum, A. and Shimkin, N. (2002). Adaptive behavior of impatient customers in tele-queues: Theory and empirical support. <i>Management Sci.</i> <b>48</b> 566–583.

Williams, R. J. (2000). On dynamic scheduling of a parallel server system with complete resource pooling. In <i>Analysis of Communication Networks<i>:</i> Call Centres<i>,</i> Traffic and Performance</i> (D. R. McDonald and S. R. E. Turner, eds.) 49–71. Amer. Math. Soc., Providence, RI.