Abstract
In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We consider the random variables Q e(t,y) and Q r(t,y) representing the number of customers in the system at time t that have elapsed service times less than or equal to time y, or residual service times strictly greater than y. We also consider W r(t,y) representing the total amount of work in service time remaining to be done at time t+y for customers in the system at time t. The two-parameter stochastic-process limits in the space D([0,∞),D) of D-valued functions in D draw on, and extend, previous heavy-traffic limits by Glynn and Whitt (Adv. Appl. Probab. 23, 188–209, 1991), where the case of discrete service-time distributions was treated, and Krichagina and Puhalskii (Queueing Syst. 25, 235–280, 1997), where it was shown that the variability of service times is captured by the Kiefer process with second argument set equal to the service-time c.d.f.
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References
Bickel, P.J., Wichura, M.J.: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42, 1656–1670 (1971)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) (2nd edn., 1999)
Borovkov, A.A.: On limit laws for service processes in multi-channel systems. Sib. Math J. 8, 746–763 (1967) (in Russian)
Brémaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, Berlin (1981)
Cairoli, R.: Sur une equation differentielle stochastique. C. R. Acad. Sci. Paris Ser. A 274, 1739–1742 (1972)
Cairoli, R., Walsh, J.B.: Stochastic integrals in the plane. Acta Math. 134, 111–183 (1975)
Csörgö, M., Révész, P.: Strong Approximations in Probability and Statistics. Wiley, New York (1981)
Decreusefond, L., Moyal, P.: A functional central limit theorem for the M/GI/∞ queue. Ann. Appl. Probab. 18(6), 2156–2178 (2008)
Duffield, N.G., Whitt, W.: Control and recovery from rare congestion events in a large multi-server system. Queueing Syst. 26, 69–104 (1997)
Eick, S.G., Massey, W.A., Whitt, W.: The physics of the M t /G/∞ queue. Oper. Res. 41, 731–742 (1993)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)
Gaenssler, P., Stute, W.: Empirical processes: a survey of results for independent and identically distributed random variables. Ann. Probab. 7, 193–243 (1979)
Glynn, P.W.: On the Markov property of the GI/G/∞ Gaussian limit. Adv. Appl. Probab. 14, 191–194 (1982)
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)
Goldberg, D.A., Whitt, W.: The last departure time from an M t /GI/∞ queue with a terminating arrival process. Queueing Syst. 58, 77–104 (2008)
Halfin, S., Whitt, W.: Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29(3), 567–587 (1981)
Harrison, J.M., Reiman, M.I.: Reflected Brownian motion in an orthant. Ann. Probab. 9, 302–308 (1981)
Iglehart, D.L.: Limit diffusion approximations for the many server queue and the repairman problem. J. Appl. Probab. 2, 429–441 (1965)
Iglehart, D.L.: Weak convergence of compound stochastic processes. Stoch. Process. Appl. 1, 11–31 (1973)
Iglehart, D.L., Whitt, W.: Multichannel queues in heavy traffic, I. Adv. Appl. Probab. 2, 150–177 (1970)
Iglehart, D.L., Whitt, W.: Multichannel queues in heavy traffic, II: sequences, networks and batches. Adv. Appl. Probab. 2, 355–369 (1970)
Jacod, J., Shiryayev, A.N.: Limit Theorems for Stochastic Processes. Springer, Berlin (1987)
Kang, W., Ramanan, K.: Fluid limits of many-server queues with reneging. Ann. Appl. Probab. (2010, to appear)
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1991)
Kaspi, H., Ramanan, K.: Law of large numbers limits for many-server queues. Ann. Appl. Probab. (2010, to appear)
Khoshnevisan, D.: Multiparameter Processes: An Introduction to Random Fields. Springer, Berlin (2002)
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)
Kurtz, T.G., Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19, 1035–1070 (1991)
Kurtz, T.G., Protter, P.: Weak convergence of stochastic integrals and differential equations II: Infinite dimensional case. Lect. Notes Math. 1627, 197–285 (1996)
Louchard, G.: Large finite population queuing systems. Part I: the infinite server model. Stoch. Models 4, 373–505 (1988)
Mamatov, K.M.: Weak convergence of stochastic integrals with respect to semimartingales. Russ. Math. Surv. 41(5), 155–156 (1986)
Mandelbaum, A., Momcilovic, P.: Queues with many servers and impatient customers. EECS Department, University of Michigan (2009)
Mandelbaum, A., Massey, W.A., Reiman, M.I.: Strong approximations for Markovian service networks. Queueing Syst. 30, 149–201 (1998)
Massey, W.A., Whitt, W.: Networks of infinite-server queues with nonstationary Poisson input. Queueing Syst. 13, 183–250 (1993)
Neuhaus, G.: On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Stat. 42, 1285–1295 (1971)
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
Pang, G., Talreja, R., Whitt, W.: Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4, 193–267 (2007)
Puhalskii, A.A., Reed, J.E.: On many-server queues in heavy traffic. Ann. Appl. Probab. 20(1), 129–195 (2010)
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)
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)
Reed, J., Talreja, R.: Distribution-valued heavy-traffic limits for GI/GI/∞ queues. Preprint (2009)
Skorohod, A.V.: Limit theorems for stochastic processes. Probab. Theory Appl. 1, 261–290 (1956)
Straf, M.L.: Weak convergence of stochastic processes with several parameters. Proc. Sixth Berkeley Symp. Math. Stat. Probab. 2, 187–221 (1971)
Talreja, R., Whitt, W.: Heavy-traffic limits for waiting times in many-server queues with abandonments. Ann. Appl. Probab. 19(6), 2137–2175 (2009)
van Der Vaart, A.W., Wellner, J.: Weak Convergence and Empirical Processes. Springer, Berlin (1996)
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)
Whitt, W.: On the heavy-traffic limit theorem for GI/G/∞ queues. Adv. Appl. Probab. 14, 171–190 (1982)
Whitt, W.: Stochastic-Process Limits. Springer, Berlin (2002)
Whitt, W.: Fluid models for multiserver queues with abandonments. Oper. Res. 54, 37–54 (2006)
Whitt, W.: Proofs of the martingale FCLT: a review. Probab. Surv. 4, 268–302 (2007)
Wong, E., Zakai, M.: Martingales and stochastic integrals for processes with a multidimensional parameter. Z. Wahrscheinlichkeitstheor. Verw. Geb. 29, 109–122 (1974)
Wong, E., Zakai, M.: An extension of stochastic integrals in the plane. Ann. Probab. 5, 770–778 (1977)
Zhang, J.: Fluid models of multi-server queues with abandonment. Preprint (2009)
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Pang, G., Whitt, W. Two-parameter heavy-traffic limits for infinite-server queues. Queueing Syst 65, 325–364 (2010). https://doi.org/10.1007/s11134-010-9184-z
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DOI: https://doi.org/10.1007/s11134-010-9184-z
Keywords
- Infinite-server queues
- Heavy-traffic limits for queues
- Markov approximations
- Two-parameter processes
- Measure-valued processes
- Time-varying arrivals
- Martingales
- Functional central limit theorems
- Invariance principles
- Kiefer process