The Annals of Applied Probability

Distribution-valued heavy-traffic limits for the $\mathbf{{G/\mathit{GI}/\infty}}$ queue

Abstract

We study the $G/\mathit{GI}/\infty$ queue in heavy-traffic using tempered distribution-valued processes which track the age and residual service time of each customer in the system. In both cases, we use the continuous mapping theorem together with functional central limit theorem results in order to obtain fluid and diffusion limits for these processes in the space of tempered distribution-valued processes. We find that our diffusion limits are tempered distribution-valued Ornstein–Uhlenbeck processes.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 3 (2015), 1420-1474.

Dates
First available in Project Euclid: 23 March 2015

https://projecteuclid.org/euclid.aoap/1427124133

Digital Object Identifier
doi:10.1214/14-AAP1027

Mathematical Reviews number (MathSciNet)
MR3325278

Zentralblatt MATH identifier
1315.60102

Citation

Reed, Josh; Talreja, Rishi. Distribution-valued heavy-traffic limits for the $\mathbf{{G/\mathit{GI}/\infty}}$ queue. Ann. Appl. Probab. 25 (2015), no. 3, 1420--1474. doi:10.1214/14-AAP1027. https://projecteuclid.org/euclid.aoap/1427124133

References

• [1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
• [2] Bojdecki, T. and Gorostiza, L. G. (1986). Langevin equations for $\mathcal{S}'$-valued Gaussian processes and fluctuation limits of infinite particle systems. Probab. Theory Related Fields 73 227–244.
• [3] Bojdecki, T. and Gorostiza, L. G. (1991). Gaussian and non-Gaussian distribution-valued Ornstein–Uhlenbeck processes. Canad. J. Math. 43 1136–1149.
• [4] Bojdecki, T., Gorostiza, L. G. and Ramaswamy, S. (1986). Convergence of $\mathcal{S}'$-valued processes and space–time random fields. J. Funct. Anal. 66 21–41.
• [5] Borovkov, A. A. (1967). On limit laws for service processes in multi-channel systems. Siberian Journal of Mathematics 8 983–1004.
• [6] Chung, K. L. (2001). A Course in Probability Theory, 3rd ed. Academic Press, San Diego, CA.
• [7] Decreusefond, L. and Moyal, P. (2008). Fluid limit of a heavily loaded EDF queue with impatient customers. Markov Process. Related Fields 14 131–158.
• [8] Decreusefond, L. and Moyal, P. (2008). A functional central limit theorem for the $M/\mathit{GI}/\infty$ queue. Ann. Appl. Probab. 18 2156–2178.
• [9] Down, D. G., Gromoll, H. C. and Puha, A. L. (2009). Fluid limits for shortest remaining processing time queues. Math. Oper. Res. 34 880–911.
• [10] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
• [11] Glynn, P. W. and Whitt, W. (1991). A new view of the heavy-traffic limit theorem for infinite-server queues. Adv. in Appl. Probab. 23 188–209.
• [12] Gromoll, H. C. (2004). Diffusion approximation for a processor sharing queue in heavy traffic. Ann. Appl. Probab. 14 555–611.
• [13] Gromoll, H. C., Puha, A. L. and Williams, R. J. (2002). The fluid limit of a heavily loaded processor sharing queue. Ann. Appl. Probab. 12 797–859.
• [14] Gromoll, H. C., Robert, P. and Zwart, B. (2008). Fluid limits for processor-sharing queues with impatience. Math. Oper. Res. 33 375–402.
• [15] Hitsuda, M. and Mitoma, I. (1986). Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions. J. Multivariate Anal. 19 311–328.
• [16] Holley, R. A. and Stroock, D. W. (1978). Generalized Ornstein–Uhlenbeck processes and infinite particle branching Brownian motions. Publ. Res. Inst. Math. Sci. 14 741–788.
• [17] Iglehart, D. L. (1965). Limiting diffusion approximations for the many server queue and the repairman problem. J. Appl. Probab. 2 429–441.
• [18] Kallianpur, G. (1986). Stochastic differential equations in duals of nuclear spaces with some applications. In IMA Preprint Series. Institute for Mathematics and Its Applications, Univ. Minnesota.
• [19] Kallianpur, G. and Pérez-Abreu, V. (1988). Stochastic evolution equations driven by nuclear-space-valued martingales. Appl. Math. Optim. 17 237–272.
• [20] Kallianpur, G. and Pérez-Abreu, V. (1989). Weak convergence of solutions of stochastic evolution equations on nuclear spaces. In Stochastic Partial Differential Equations and Applications, II (Trento, 1988) (G. Da Prato and L. Tubaro, eds.). Lecture Notes in Math. 1390 119–131. Springer, Berlin.
• [21] Kallianpur, G. and Xiong, J. (1995). Stochastic Differential Equations in Infinite-Dimensional Spaces. Institute of Mathematical Statistics Lecture Notes—Monograph Series 26. IMS, Hayward, CA.
• [22] Karatzas, I. and Shreve, S. E. (1999). Brownian Motion and Stochastic Calculus, 5th ed. Springer, New York.
• [23] Kaspi, H. and Ramanan, K. (2011). Law of large numbers limits for many-server queues. Ann. Appl. Probab. 21 33–114.
• [24] Kaspi, H. and Ramanan, K. (2013). SPDE limits of many-server queues. Ann. Appl. Probab. 23 145–229.
• [25] Krichagina, E. V. and Puhalskii, A. A. (1997). A heavy-traffic analysis of a closed queueing system with a $\mathit{GI}/\infty$ service center. Queueing Systems Theory Appl. 25 235–280.
• [26] Mitoma, I. (1983). Tightness of probabilities on $C([0,1];\mathcal{S}')$ and $D([0,1];\mathcal{S}')$. Ann. Probab. 11 989–999.
• [27] Mitoma, I. (1985). An $\infty$-dimensional inhomogeneous Langevin’s equation. J. Funct. Anal. 61 342–359.
• [28] Mitoma, I. (1987). Generalized Ornstein–Uhlenbeck process having a characteristic operator with polynomial coefficients. Probab. Theory Related Fields 76 533–555.
• [29] Pang, G., Talreja, R. and Whitt, W. (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4 193–267.
• [30] Pang, G. and Whitt, W. (2010). Two-parameter heavy-traffic limits for infinite-server queues. Queueing Syst. 65 325–364.
• [31] Puhalskii, A. A. and Reed, J. E. (2010). On many-server queues in heavy traffic. Ann. Appl. Probab. 20 129–195.
• [32] Whitt, W. (1982). On the heavy-traffic limit theorem for $\mathit{GI}/G/\infty$ queues. Adv. in Appl. Probab. 14 171–190.
• [33] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.
• [34] Whitt, W. (2007). Proofs of the martingale FCLT. Probab. Surv. 4 268–302.