Martingale proofs of many-server heavy-traffic limits for Markovian queues

Probability Surveys

Volumn 4, Issue 1, 2007, Pages 193-267

  Pang, Guodong ^a   Talreja, Rishi ^a   Whitt, Ward ^a  

^a Department of Earth and Environmental Sciences   (United States)

Author keywords

Diffusion approximations; Functional central limit theorems; Many server heavy traffic limits for queues; Martingales; Multiple server queues

Indexed keywords

EID: 52949126577     PISSN: None     EISSN: 15495787     Source Type: Journal    
DOI: 10.1214/06-PS091     Document Type: Article

References (74)

1. Armony, M. 2005. Dynamic routing in large-scale service systems with heterogeneous servers. Queueing Systems 51, 287-329.
2. Armony M., I., Gurvich and C. Maglaras. 2006. Cross-selling in a call center with a heterogeneous customer population. Working Paper, New York University, New York, NY, and Columbia University, New York, NY.
3. Armony M., I. Gurvich and A. Mandelbaum. 2006. Service level differentiation in call Centers with fully flexible servers. Management Science, forthcoming.
4. Atar R. 2005. Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic. Ann. Appl. Probab. 15, 2606-2650.
5. Atar R., A. Mandelbaum and M. I. Reiman. 2004a. Scheduling a multi-class queue with many exponential servers. Ann. Appl. Probab. 14, 1084-1134.
6. Atar R., A. Mandelbaum and M. I. Reiman. 2004b. Brownian control problems for queueing systems in the Halfin-Whitt regime. Ann. Appl. Probab. 14, 1084-1134.
7. Bickel P.J. and M.J. Wichura. 1971. Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42, 1656-1670.
8. Billingsley, P. 1968. Convergence of Probability Measures, Wiley (second edition, 1999).
9. Borovkov, A. A. 1967. On limit laws for service processes in multi-channel systems (in Russian). Siberian Math J. 8, 746-763.
10. Borovkov, A. A. 1984. Asymptotic Methods in Queueing Theory, Wiley, New York.
11. Borst, S., A. Mandelbaum and M. I. Reiman. 2004. Dimensioning large call centers. Oper. Res. 52, 17-34.
12. Bŕemaud, P. 1981. Point Processes and Queues: Martingale Dynamics, Springer.
13. Cs̈orǵo M. and P. Ŕev́ez. 1981. Strong Approximations in Probability and Statistics. Akademiai Kiado.
14. Dai J. G. and T. Tezcan. 2005. State space collapse in many server diffusion limits of parallel server systems. Working Paper, Georgia Institute of Technology, Atlanta, GA.
15. Dai J. G. and T. Tezcan. 2006. Dynamic control of N systems with many servers: Asymptotic optimality of a static priority policy in heavy traffic. Working Paper, Georgia Institute of Technology, Atlanta, GA.
16. Dai J. G. and T. Tezcan. 2007. Optimal control of parallel server systems with many servers in heavy traffic. Working Paper, Georgia Institute of Technology, Atlanta, GA.
17. Daley, D. J. and D. Vere-Jones. 2003. An Introduction to the Theory of Point Processes, second ed., Springer.
18. Dupuis, P. and H. Ishii. 1991. On when the solution to the Skorohod problem is Lipschitz continuous with applications. Stochastics 35, 31-62.
19. Ethier, S. N. and T. G. Kurtz. 1986. Markov Processes; Characterization and Convergence, Wiley.
20. Gans, N., G. Koole and A. Mandelbaum. 2003. Telephone call centers: Tutorial, review and research prospects. Manufacturing Service Oper. Man- agement 5(2), 79-141.
21. Garnett, O., A. Mandelbaum and M. I. Reiman. 2002. Designing a call center with impatient customers. Manufacturing Service Oper. Management 4, 208-227.
22. Glynn, P. W. and W. Whitt. 1991. A new view of the heavy-traffic limit theorem for the infinite-server queue. Adv. Appl. Prob. 23, 188-209.
23. Gurvich, I. and W. Whitt. 2007a. Queue-and-idleness-ratio controls in many-server servbice systems. working paper, Columbia University. Available at: http://www.columbia.edu/~ww2040
24. Gurvich, I. and W. Whitt. 2007b. Service-level differentiation in many-server service systems: A solution based on fixed-queueratio routing. working paper, Columbia University. Available at: http://www.columbia.edu/~ww2040
25. Gurvich, I. and W. Whitt. 2007c. Scheduling Flexible Servers with convex delay costs in many-server service systems. Manufacturing and Service Operations Management, forthcoming. Available at: http://www.columbia.edu/~ww2040
26. Halfin, S. and W. Whitt. 1981. Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29, 567-588.
27. Harrison, J. M. and A. Zeevi. 2005. Dynamic scheduling of a multiclass queue in the Halfin and Whitt heavy traffic regime. Oper. Res. 52, 243- 257.
28. Helms, L. L. 1974. Ergodic properties of several interacting Poisson particles. Advances in Math. 12, 32-57.
29. Iglehart, D. L. 1965. Limit diffusion approximations for the many server queue and the repairman problem. J. Appl. Prob. 2, 429-441.
30. Jacod, J. and A. N. Shiryayev. 1987. Limit Theorems for Stochastic Pro- cesses, Springer.
31. Jelenkovíc, P., A. Mandelbaum and P. Momcilovíc. 2004. Heavy traffic limits for queues with many deterministic servers. Queueing Systems 47, 53-69.
32. Kallenberg, O. 2002. Foundations of Modern Probability, second edition, Springer.
33. Karatzas, I, and S. Shreve. 1988. Brownian Motion and Stochastic Calculus, Springer.
34. Kaspi, H. and K. Ramanan. 2007. Law of large numbers limit for manyserver queues. Working paper. The Technion and Carnegie Mellon University.
35. Khoshnevisan D. 2002. Multiparameter Processes: An Introduction to Ran- dom Fields, Springer.
36. Kogan, Y., R. Sh. Liptser and A. V. Smorodinskii. 1986. Gaussian diffusion approximations of closed Markov models of computer networks. Problems. Inform. Transmission 22, 38-51.
37. Krichagina, E. V. and A. A. Puhalskii. 1997. A heavy-traffic analysis of a closed queueing system with a GI/8 service center. Queueing Systems 25, 235-280.
38. Kunita, H. and S. Watanabe. 1967. On square-integrble martingales. Nagoya Math. J. 30, 209-245.
39. Kurtz, T. 1978. Strong approximation theorems for density dependent Markov chains. Stoch. Process Appl. 6, 223-240.
40. Kurtz, T. 1980. Representations of Markov processes as multiparameter time changes. Ann. Probability 8, 682-715.
41. Kurtz, T. 2001. Lectures on Stochastic Analysis, Department of Mathematics and Statistics, University of Wisconsin, Madison, WI 53706-1388.
42. Ĺevy, P. 1948. Processus Stochastiques et Mouvement Borownien, Gauthiers-Villars, Paris.
43. Lions, P. and A. Sznitman. 1984. Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37, 511-537.
44. Liptser, R. Sh. and A. N. Shiryayev. 1989. Theory of Martingales, Kluwer (English translation of 1986 Russian edition).
45. Louchard, G. 1988. Large finite population queueing systems. Part I: The infinite server model. Comm. Statist. Stochastic Models 4(3), 473-505.
46. Mandelbaum, A., W. A. Massey and M. I. Reiman. 1998. Strong approximations for Markovian service networks. Queueing Systems 30, 149-201.
47. Mandelbaum, A. and G. Pats. 1995. State-dependent queues: approximations and applications. In Stochastic Networks, F. P. Kelly, R. J. Williams (eds.), Institute for Mathematics and its Applications, Vol. 71, Springer, 239-282.
48. Mandelbaum, A. and G. Pats. 1998. State-dependent stochastic networks. Part I: Approximations and Applications with continuous diffusion limits. Ann. Appl. Prob. 8, 569-646.
49. Mandelbaum, A. and S. Zeltyn. 2005. The Erlang-A/Palm queue, with applications to call centers. Working paper, The Technion, Haifa, Israel. Available at: http://iew3.technion.ac.il/serveng/ References/references.html
50. Massey, W. A. and W.Whitt. 1993. Networks of infinite-server queues with nonstationary Poisson input. Queueing Systems 13, 183-250.
51. Parthasarathy, K. R. 1967. Probability Measures on Metric Spaces, Academic Press.
52. Puhalskii, A. A. 1994. On the invariance principle for the first passage time. Math. Oper. Res. 19, 946-954.
53. Puhalskii, A. A. and M. I. Reiman. 2000. The mutliclass GI/PH/N queue in the Halfin-Whitt regime. Adv. Appl. Prob. 32, 564-595.
54. Rebolledo, R. 1980. Central limit theorems for local martingales. Zeitschrift Wahrscheinlichkeitstheorie verw. Gebiete 51, 269-286.
55. Reed, J. 2005. The G/GI/N queue in the Halfin-Whitt regime. working paper, Georgia Institute of Technology.
56. Reed, J. and A. R. Ward. 2007. Approximating the GI/GI/1 + GI queue with a nonlinear drift diffusion: Hazard-rate scaling in heavy traffic. working paper, Georgia Institute of Technology.
57. Robert, P. 2003. Stochastic Networks and Queues, Springer.
58. Rogers, L. C. G. and D. Williams. 1987. Diffusions, Markov Processes and Martingales, Volume 2: Ito Calculus, Wiley.
59. Rogers, L. C. G. and D. Williams. 2000. Diffusions, Markov Processes and Martingales, Volume 1: Foundations, Cambridge University Press.
60. Skorohod, A. V. 1956. Limit theorems for stochastic processes. Prob. Theory Appl. 1, 261-290.
61. Srikant, R. and W. Whitt. 1996. Simulation run lengths to estimate blocking probabilities. ACM Trans. Modeling Computer Simulations 6, 7-52.
62. Stone, C. 1963. Limit theorems for random walks, birth and death processes and diffusion processes. Illinois J. Math. 4, 638-660.
63. Tezcan T. 2006. Optimal control of distributed parallel server systems under the Halfin and Whitt regime. Working Paper, University of Illinois at Urbana-Champaign. Available at: https://netfiles.uiuc.edu/ ttezcan/www/TolgaTezcansubmittedMOR121906.pdf
64. van der Vaart, A. W. 2006. Martingales, Diffusions and Financial Mathematics Lecture Notes, Available at: http://www.math.vu.nl/sto/onderwijs/mdfm/
65. Ward, A. R. and P. W. Glynn. 2003a. A diffusion approximation for a Markovian queue with reneging. Queueing Systems 43, 103-128.
66. Ward, A. R. and P. W. Glynn. 2003b. Properties of the reflected Ornstein- Uhlenbeck process. Queueing Systems 44, 109-123.
67. Ward, A. R. and P. W. Glynn. 2005. A diffusion approximation for a GI/GI/1 queue with balking or reneging. Queueing Systems 50, 371-400.
68. Whitt, W. 1982. On the heavy-traffic limit theorem for GI/G/8 queues. Adv. Appl. Prob. 14, 171-190.
69. Whitt, W. 2002. Stochastic-Process Limits, Springer.
70. Whitt, W. 2002a. Internet Supplement to Stochastic-Process Limits, Available at: http://www.columbia.edu/~ww2040/supplement.html
71. Whitt, W. 2004. Efficiency-driven heavy-traffic approximations for manyserver queues with abandonments. Management Science 50, 1449-1461.
72. Whitt,W. 2005. Heavy-traffic limits for the G/H* 2/n/m queue. Math. Oper. Res. 30, 1-27.
73. Whitt, W. 2007. Proofs of the martingale functional central limit theorem. Probability Surveys, forthcoming.
74. Zeltyn S. and A.Mandelbaum. 2005. Call centers with impatient customers: Many-server asymptotics of the M/M/n+G queue. Queueing Systems 51, 361-402.