Two-parameter heavy-traffic limits for infinite-server queues

Queueing Systems

Volumn 65, Issue 4, 2010, Pages 325-364

  Pang, Guodong ^a   Whitt, Ward ^a  

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

Author keywords

Functional central limit theorems; Heavy traffic limits for queues; Infinite server queues; Invariance principles; Kiefer process; Markov approximations; Martingales; Measure valued processes; Time varying arrivals; Two parameter processes

Indexed keywords

STOCHASTIC SYSTEMS;

FUNCTIONAL CENTRAL LIMIT THEOREM; HEAVY TRAFFIC LIMIT; INFINITE SERVER QUEUES; INVARIANCE PRINCIPLE; MARKOV APPROXIMATION; MARTINGALES; MEASURE VALUED PROCESS; TIME VARYING; TWO PARAMETER;

QUEUEING THEORY;

EID: 77955096229     PISSN: 02570130     EISSN: 15729443     Source Type: Journal    
DOI: 10.1007/s11134-010-9184-z     Document Type: Article

References (53)

1. Bickel, P. J., Wichura, M. J.: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42, 1656-1670 (1971).
2. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) (2nd edn., 1999).
3. Borovkov, A. A.: On limit laws for service processes in multi-channel systems. Sib. Math J. 8, 746-763 (1967) (in Russian).
4. Brémaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, Berlin (1981).
5. Cairoli, R.: Sur une equation differentielle stochastique. C. R. Acad. Sci. Paris Ser. A 274, 1739-1742 (1972).
6. Cairoli, R., Walsh, J. B.: Stochastic integrals in the plane. Acta Math. 134, 111-183 (1975).
7. Csörgö, M., Révész, P.: Strong Approximations in Probability and Statistics. Wiley, New York (1981).
8. Decreusefond, L., Moyal, P.: A functional central limit theorem for the M/GI/∞ queue. Ann. Appl. Probab. 18(6), 2156-2178 (2008).
9. Duffield, N. G., Whitt, W.: Control and recovery from rare congestion events in a large multi-server system. Queueing Syst. 26, 69-104 (1997).
10. t/G/∞ queue. Oper. Res. 41, 731-742 (1993).
11. Ethier, S. N., Kurtz, T. G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986).
12. Gaenssler, P., Stute, W.: Empirical processes: a survey of results for independent and identically distributed random variables. Ann. Probab. 7, 193-243 (1979).
13. Glynn, P. W.: On the Markov property of the GI/G/∞ Gaussian limit. Adv. Appl. Probab. 14, 191-194 (1982).
14. Glynn, P. W., Whitt, W.: A new view of the heavy-traffic limit theorem for the infinite-server queue. Adv. Appl. Probab. 23, 188-209 (1991).
15. t/GI/∞ queue with a terminating arrival process. Queueing Syst. 58, 77-104 (2008).
16. Halfin, S., Whitt, W.: Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29(3), 567-587 (1981).
17. Harrison, J. M., Reiman, M. I.: Reflected Brownian motion in an orthant. Ann. Probab. 9, 302-308 (1981).
18. Iglehart, D. L.: Limit diffusion approximations for the many server queue and the repairman problem. J. Appl. Probab. 2, 429-441 (1965).
19. Iglehart, D. L.: Weak convergence of compound stochastic processes. Stoch. Process. Appl. 1, 11-31 (1973).
20. Iglehart, D. L., Whitt, W.: Multichannel queues in heavy traffic, I. Adv. Appl. Probab. 2, 150-177 (1970).
21. Iglehart, D. L., Whitt, W.: Multichannel queues in heavy traffic, II: sequences, networks and batches. Adv. Appl. Probab. 2, 355-369 (1970).
22. Jacod, J., Shiryayev, A. N.: Limit Theorems for Stochastic Processes. Springer, Berlin (1987).
23. Kang, W., Ramanan, K.: Fluid limits of many-server queues with reneging. Ann. Appl. Probab. (2010, to appear).
24. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1991).
25. Kaspi, H., Ramanan, K.: Law of large numbers limits for many-server queues. Ann. Appl. Probab. (2010, to appear).
26. Khoshnevisan, D.: Multiparameter Processes: An Introduction to Random Fields. Springer, Berlin (2002).
27. Krichagina, E. V., Puhalskii, A. A.: A heavy-traffic analysis of a closed queueing system with a GI/∞ service center. Queueing Syst. 25, 235-280 (1997).
28. Kurtz, T. G., Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19, 1035-1070 (1991).
29. Kurtz, T. G., Protter, P.: Weak convergence of stochastic integrals and differential equations II: Infinite dimensional case. Lect. Notes Math. 1627, 197-285 (1996).
30. Louchard, G.: Large finite population queuing systems. Part I: the infinite server model. Stoch. Models 4, 373-505 (1988).
31. Mamatov, K. M.: Weak convergence of stochastic integrals with respect to semimartingales. Russ. Math. Surv. 41(5), 155-156 (1986).
32. Mandelbaum, A., Momcilovic, P.: Queues with many servers and impatient customers. EECS Department, University of Michigan (2009).
33. Mandelbaum, A., Massey, W. A., Reiman, M. I.: Strong approximations for Markovian service networks. Queueing Syst. 30, 149-201 (1998).
34. Massey, W. A., Whitt, W.: Networks of infinite-server queues with nonstationary Poisson input. Queueing Syst. 13, 183-250 (1993).
35. Neuhaus, G.: On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Stat. 42, 1285-1295 (1971).
36. Pang, G., Whitt, W.: Two-parameter heavy-traffic limits for infinite-server queues: longer version. Columbia University (2009). Available at: http://www. columbia. edu/~ww2040.
37. Pang, G., Talreja, R., Whitt, W.: Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4, 193-267 (2007).
38. Puhalskii, A. A., Reed, J. E.: On many-server queues in heavy traffic. Ann. Appl. Probab. 20(1), 129-195 (2010).
39. Reed, J. E.: The G/GI/N queue in the Halfin-Whitt regime I: infinite-server queue system equations. Ann. Appl. Probab. 19(6), 2211-2269 (2009).
40. Reed, J. E.: The G/GI/N queue in the Halfin-Whitt regime II: idle-time system equations. Working paper, The Stern School, NYU (2007).
41. Reed, J., Talreja, R.: Distribution-valued heavy-traffic limits for GI/GI/∞ queues. Preprint (2009).
42. Skorohod, A. V.: Limit theorems for stochastic processes. Probab. Theory Appl. 1, 261-290 (1956).
43. Straf, M. L.: Weak convergence of stochastic processes with several parameters. Proc. Sixth Berkeley Symp. Math. Stat. Probab. 2, 187-221 (1971).
44. Talreja, R., Whitt, W.: Heavy-traffic limits for waiting times in many-server queues with abandonments. Ann. Appl. Probab. 19(6), 2137-2175 (2009).
45. van Der Vaart, A. W., Wellner, J.: Weak Convergence and Empirical Processes. Springer, Berlin (1996).
46. Walsh, J. B.: Martingales with a multidimensional parameter and stochastic integrals in the plane. In: Lectures in Probability and Statistics, pp. 329-491. Springer, Berlin (1986).
47. Whitt, W.: On the heavy-traffic limit theorem for GI/G/∞ queues. Adv. Appl. Probab. 14, 171-190 (1982).
48. Whitt, W.: Stochastic-Process Limits. Springer, Berlin (2002).
49. Whitt, W.: Fluid models for multiserver queues with abandonments. Oper. Res. 54, 37-54 (2006).
50. Whitt, W.: Proofs of the martingale FCLT: a review. Probab. Surv. 4, 268-302 (2007).
51. Wong, E., Zakai, M.: Martingales and stochastic integrals for processes with a multidimensional parameter. Z. Wahrscheinlichkeitstheor. Verw. Geb. 29, 109-122 (1974).
52. Wong, E., Zakai, M.: An extension of stochastic integrals in the plane. Ann. Probab. 5, 770-778 (1977).
53. Zhang, J.: Fluid models of multi-server queues with abandonment. Preprint (2009).