An Asymptotic Optimality Result for the Multiclass Queue with Finite Buffers in Heavy Traffic

References

• Ata, B., Dynamic control of a multiclass queue with thin arrival streams. Operations Research, 54(5):876–892, 2006. MR2267797Link, Google Scholar
• Ata, B. and Kumar, S., Heavy traffic analysis of open processing networks with complete resource pooling: Asymptotic optimality of discrete review policies. Ann. Appl. Probab., 15(1A):331–391, 2005. MR2115046Google Scholar
• Ata, B. and Olsen, T. L., Near-optimal dynamic lead-time quotation and scheduling under convex-concave customer delay costs. Operations Research, 57(3):753–768, 2009.Link, Google Scholar
• Atar, R., A diffusion regime with nondegenerate slowdown. Oper. Res., 60(2):490–500, 2012. MR2935073Link, Google Scholar
• Atar, R., A diffusion regime with nondegenerate slowdown: Appendix. Electronic companion, downloadable at http://webee.technion.ac.il/people/atar/online-appendix.pdf, 2012. MR2935073Google Scholar
• Atar, R. and Budhiraja, A., Singular control with state constraints on unbounded domain. Ann. Probab., 34(5):1864–1909, 2006. MR2271486Google Scholar
• Atar, R., Budhiraja, A., and Williams, R. J., HJB equations for certain singularly controlled diffusions. Ann. Appl. Probab., 17(5–6):1745–1776, 2007. MR2358640Google Scholar
• Atar, R. and Gurvich, I., Scheduling parallel servers in the non-degenerate slowdown diffusion regime: Asymptotic optimality results. Ann. Appl. Probab., to appear, 2013. MR3178497Google Scholar
• Atar, R., Mandelbaum, A., and Reiman, M. I., Scheduling a multi class queue with many exponential servers: Asymptotic optimality in heavy traffic. Ann. Appl. Probab., 14(3):1084–1134, 2004. MR2071417Google Scholar
• Atar, R. and Solomon, N., Asymptotically optimal interruptible service policies for scheduling jobs in a diffusion regime with nondegenerate slowdown. Queueing Systems Theory Appl., 69(217–235), 2011. MR2886469Google Scholar
• Bell, S. L. and Williams, R. J., Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: Asymptotic optimality of a threshold policy. Ann. Appl. Probab., 11(3):608–649, 2001. MR1865018Google Scholar
• Bennani, M. N. and Menasce, D. A., Resource allocation for autonomic data centers using analytic performance models. In Autonomic Computing, 2005. ICAC 2005. Proceedings. Second International Conference on, pages 229–240. IEEE, 2005.Google Scholar
• Billingsley, P., Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999. ISBN 0-471-19745-9. x+277 pp. A Wiley-Interscience Publication. MR1700749Google Scholar
• Bramson, M., State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Systems Theory Appl., 30(1–2):89–148, 1998. MR1663763Google Scholar
• Budhiraja, A. and Ghosh, A. P., Diffusion approximations for controlled stochastic networks: An asymptotic bound for the value function. Ann. Appl. Probab., 16(4):1962–2006, 2006. MR2288710Google Scholar
• Budhiraja, A. and Ghosh, A. P., Controlled stochastic networks in heavy traffic: Convergence of value functions. Ann. Appl. Probab., 22(2):734–791, 2012. MR2953568Google Scholar
• Buyukkoc, C., Varaiya, P., and Walrand, J., The cμ rule revisited. Adv. in Appl. Probab., 17(1):237–238, 1985. MR0778604Google Scholar
• Cox, D. R. and Smith, W. L., Queues. Methuen’s Monographs on Statistical Subjects. Methuen & Co. Ltd., London; John Wiley & Sons Inc., New York, 1961. xii+180 pp. MR0133178Google Scholar
• Dai, J. G. and Dai, W., A heavy traffic limit theorem for a class of open queueing networks with finite buffers. Queueing Systems Theory Appl., 32(1–3):5–40, 1999. MR1720547Google Scholar
• Dellacherie, C. and Meyer, P.-A., Probabilities and Potential, volume 29 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam-New York, 1978. ISBN 0-7204-0701-X. viii+189 pp. MR0521810Google Scholar
• Durrett, R., Probability: Theory and Examples. Duxbury Press, Belmont, CA, second edition, 1996. ISBN 0-534-24318-5. xiii+503 pp. MR1609153Google Scholar
• Ethier, S. N. and Kurtz, T. G., Markov Processes. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986. ISBN 0-471-08186-8. x+534 pp. Characterization and convergence. MR0838085Google Scholar
• Ghamami, S. and Ward, A. R., Dynamic scheduling of a two-server parallel server system with complete resource pooling and reneging in heavy traffic: Asymptotic optimality of a two-threshold policy. Mathematics of Operations Research, 38(4):761–824, 2013. MR3125918Link, Google Scholar
• Ghosh, A. P. and Weerasinghe, A. P., Optimal buffer size for a stochastic processing network in heavy traffic. Queueing Syst., 55(3):147–159, 2007. MR2319488Google Scholar
• Harrison, J. M., Brownian models of queueing networks with heterogeneous customer populations. In Stochastic Differential Systems, Stochastic Control Theory and Applications (Minneapolis, Minn., 1986), volume 10 of IMA Vol. Math. Appl., pages 147–186. Springer, New York, 1988. MR0934722Google Scholar
• Harrison, J. M., Brownian models of open processing networks: canonical representation of workload. Ann. Appl. Probab., 10(1):75–103, 2000. MR1765204Google Scholar
• Harrison, J. M., A broader view of Brownian networks. Ann. Appl. Probab., 13(3):1119–1150, 2003. MR1994047Google Scholar
• Harrison, J. M. and Taksar, M. I., Instantaneous control of Brownian motion. Math. Oper. Res., 8(3):439–453, 1983. MR0716123Link, Google Scholar
• Harrison, J. M. and Van Mieghem, J. A., Dynamic control of Brownian networks: state space collapse and equivalent workload formulations. Ann. Appl. Probab., 7(3):747–771, 1997. MR1459269Google Scholar
• Harrison, J. M. and Williams, R. J., Workload reduction of a generalized Brownian network. Ann. Appl. Probab., 15(4):2255–2295, 2005. MR2187295Google Scholar
• Jacod, J. and Shiryaev, A. N., Limit Theorems for Stochastic Processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1987. ISBN 3-540-17882-1. xviii+601 pp. MR0959133Google Scholar
• Kruk, L., Lehoczky, J., Ramanan, K., and Shreve, S., An explicit formula for the Skorokhod map on [0, a]. Ann. Probab., 35(5):1740–1768, 2007. MR2349573Google Scholar
• Plambeck, E., Kumar, S., and Harrison, J. M., A multiclass queue in heavy traffic with throughput time constraints: Asymptotically optimal dynamic controls. Queueing Syst., 39(1):23–54, 2001. MR1865457Google Scholar
• Plambeck, E. L., Optimal leadtime differentiation via diffusion approximations. Oper. Res., 52(2):213–228, 2004. MR2066397Link, Google Scholar
• Reiman, M. I., The heavy traffic diffusion approximation for sojourn times in Jackson networks. In Applied Probability—Computer Science: The Interface, Vol. II (Boca Raton, Fla., 1981), volume 3 of Progr. Comput. Sci., pages 409–421. Birkhäuser Boston, Boston, MA, 1982. MR0718529Google Scholar
• Rubino, M. and Ata, B., Dynamic control of a make-to-order, parallel-server system with cancellations. Oper. Res., 57(1):94–108, 2009. MR2555589Link, Google Scholar
• Shifrin, M., Atar, R., and Cidon, I., Optimal scheduling in the hybrid-cloud. In IFIP/IEEE International Symposium on Integrated Network Management. IEEE, 2013.Google Scholar
• Smith, W. E., Various optimizers for single-stage production. Naval Res. Logist. Quart., 3:59–66, 1956. MR0089109Google Scholar
• van Mieghem, J. A., Dynamic scheduling with convex delay costs: the generalized cμ rule. Ann. Appl. Probab., 5(3):809–833, 1995. MR1359830Google Scholar
• Ward, A. R. and Kumar, S., Asymptotically optimal admission control of a queue with impatient customers. Math. Oper. Res., 33(1):167–202, 2008. MR2393546Link, Google Scholar
• Whitt, W., Weak convergence theorems for priority queues: Preemptive-resume discipline. J. Appl. Probability, 8:74–94, 1971. MR0307389Google Scholar
• Williams, R. J., Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Systems Theory Appl., 30(1–2):27–88, 1998. MR1663759Google Scholar