Diffusion Models for Double-Ended Queues with Renewal Arrival Processes

Published Online:https://doi.org/10.1287/13-SSY113

References

  • Amore, P., Asymptotic and exact series representations for the incomplete Gamma function. arXiv: math-ph/0501019, 2005. MR2170316Google Scholar
  • Ancker, C. J. and Gafarian, A., Queueing with impatient customers who leave at random. J. Industr. Engrg., 13:84–90, 1962.Google Scholar
  • Anderson, W. J., Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer-Verlag, 1991. MR1118840Google Scholar
  • Billingsley, P., Convergence of Probability Measures. Wiley-Interscience, 1999. MR1700749Google Scholar
  • Bramson, M., Stability of queueing networks. In École d’Été de, Probabilités de Saint-Flour XXXVI – 2006. Lecture Notes in Mathematics, vol. 1950. Springer, Berlin 2008. MR2445100Google Scholar
  • Brockwell, P. J., Hyndman, R. J., and Grunwald, G. K., Continuous time threshold autoregressive models. Statistica Sinica, 1(2):401–410, 1991. MR1130126Google Scholar
  • Browne, S. and Whitt, W., Piecewise-Linear Diffusion Processes. CRC Press, Boca Raton, FL 1995. MR1395170Google Scholar
  • Budhiraja, A. and Ghosh, A., Diffusion approximations for controlled stochastic network: An asymptotic bound for the value function. The Annals of Applied Probability, 16(4):1962–2006, 2006. MR2288710Google Scholar
  • Budhiraja, A. and Lee, C., Stationary distribution convergence for generalized jackson networks n heavy traffic. Mathematics of Operations Research, 34(1):45–56, 2009. MR2542988LinkGoogle Scholar
  • Chan, T. and Williams, D., An ‘excursion’ approach to an annealing problem. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 105, pages 169–176, 1989. MR0966154Google Scholar
  • Conolly, B. W., Parthasarathy, P. R., and Selvaraju, N., Double-ended queues with impatience. Computers & Operations Research, 29(14):2053–2072, 2002. MR1920589Google Scholar
  • Cont, R. and de Larrard, A., Order book dynamics in liquid markets: Limit theorems and diffusion approximations. Available online at http://ssrn.com/abstract=1757861, 2012.Google Scholar
  • Cont, R., Stoikov, S., and Talreja, R., A stochastic model for order book dynamics. Operations Research, 58(3):549–563, 2010. MR2680564LinkGoogle Scholar
  • Dai, J. G. and He, S., Customer abandonment in many-server queues. Math. Oper. Res., 35(2):347–362, 2010. MR2674724LinkGoogle Scholar
  • Dai, J. G., He, S., and Tezcan, T., Many-server diffusion limits for G/Ph/n + GI queues. Ann. Appl. Probab., 20(5):1854–1890, 2010. MR2724423Google Scholar
  • Dai, J. G., Dieker, A. B., and Gao, X., Validity of heavy-traffic steady-state approximations in many-server queues with abandonment. arXiv: 1306.5346, 2013. MR3238006Google Scholar
  • Degirmenci, I. T., Asymptotic analysis and performance-based design of large scale service and inventory systems. Ph.D. Dissertation, Department of Business Administration, Duke University, 2010.Google Scholar
  • Dieker, A. B. and Gao, X., Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov functions. Ann. Appl. Probab., 23(4):1291–1317, 2013. MR3098433Google Scholar
  • Gamarnik, D. and Zeevi, A., Validity of heavy traffic steady-state approximation in generalized Jackson networks. Annals of Applied Probability, 16(1):56–90, 2006. MR2209336Google Scholar
  • Garnet, O., Mandelbaum, A., and Reiman, M., Designing a call center with impatient customers. Manufacturing & Service Operations Management, 4(3):208–227, 2002.Google Scholar
  • Karatzas, I. and Shreve, S., Brownian Motion and Stochastic Calculus. Springer, second edition, 1991. MR1121940Google Scholar
  • Karlin, S. and Taylor, H. M., A Second Course in Stochastic Processes. Academic Press, 1981. MR0611513Google Scholar
  • Kashyap, B. R. K., The double-ended queue with bulk service and limited waiting space. Operations Research, 14(5):822–834, 1966. MR0215386LinkGoogle Scholar
  • Kim, W. K., Yoon, K. P., Mendoza, G., and Sedaghat, M., Simulation model for extended double-ended queueing. Computers & Industrial Engineering, 59(2):209–219, 2010.Google Scholar
  • Kulkarni, V. G., Modeling and Analysis of Stochastic Systems. Chapman & Hall/CRC, 1996. MR2643433Google Scholar
  • Lee, C. and Weerasinghe, A., Convergence of a queueing system in heavy traffic with general patience-time distributions. Stochastic Processes and Their Applications, 121(11):2507–2552, 2011. MR2832412Google Scholar
  • Liu, X., Diffusion approximations for double-ended queues with general distributed patience times. In preparation.Google Scholar
  • Mandelbaum, A. and Momčilović, P., Queues with many servers and impatient customers. Mathematics of Operations Research, 37(1):41–65, 2012. MR2891146LinkGoogle Scholar
  • Meyn, S. P. and Down, D., Stability of generalized Jackson networks. Annals of Applied Probability, 4(1):124–148, 1994. MR1258176Google Scholar
  • Perry, D. and Stadje, W., Perishable inventory systems with impatient demands. Mathematical Methods of Operations Research, 50(1):77–90, 1999. MR1711118Google Scholar
  • Peszat, S. and Zabczyk, J., Strong Feller property and irreducibility for diffusions on Hilbert spaces. The Annuals of Probability, 23(1):157–172, 1995. MR1330765Google Scholar
  • Prabhakar, B., Bambos, N., and Mountford, T. S., The synchronization of Poisson processes and queueing networks with service and synchronization nodes. Advances in Applied Probability, 32(3):824–843, 2000. MR1788097Google Scholar
  • Protter, P., Stochastic Integration and Differential Equations. Springer, 2008.Google Scholar
  • Reed, J. and Tezcan, T., Hazard rate scaling of the abandonment distribution for the GI/M/n + GI queue in heavy traffic. Oper. Res., 60(4):981–995, 2012. MR2979435LinkGoogle Scholar
  • Reed, J. and Ward, A. R., A diffusion approximation for a generalized Jackson network with reneging. In Proceedings of the 42nd Annual Allerton Conference on Communication, Control, and Computing, 2004.Google Scholar
  • Reed, J. E. and Ward, A. R., Approximating the GI/GI/1 + GI queue with a nonlinear drift diffusion: Hazard rate scaling in heavy traffic. Mathematics of Operations Research, 33(3):606–644, 2008. MR2442644LinkGoogle Scholar
  • Stramer, O., Brockwell, P. J., and Tweedie, R. L., Continuous-time threshold AR(1) processes. Advances in Applied Probability, 28(3):728–746, 1996. MR1404307Google Scholar
  • Stramer, O. and Tweedie, R.L., Existence and stability of weak solutions to stochastic differential equations with non-smooth coefficients. Statistica Sinica, 7(3):577–593, 1997. MR1467449Google Scholar
  • Tong, H. and Yeung, I., Threshold autoregressive modeling in continuous time. Statistica Sinica, 1(2):411–430, 1991. MR1130127Google Scholar
  • van Leeuwaarden, J. S. H. and Knessl, C., Spectral gap of the erlang a model in the halfin-whitt regime. Stochastic Systems, 2(1):149–207, 2012.LinkGoogle Scholar
  • Ward, A. R. and Glynn, P. W., A diffusion approximation for a Markovian queue with reneging. Queueing Syst. Theory Appl., 43(1/2):103–128, 2003. MR1957808Google Scholar
  • Ward, A. R. and Glynn, P. W., A diffusion approximation for a GI/GI/1 queue with balking or reneging. Queueing Syst. Theory Appl., 50(4):371–400, 2005. MR2172907Google Scholar
  • Whitt, W., Heavy-traffic limits for the G/H*2/N/M queue. Math. Oper. Res., 30(1):1–27, 2005. MR2125135LinkGoogle Scholar
  • Zeltyn, S. and Mandelbaum, A., Call centers with impatient customers: Many-server asymptotics of the M/M/n + G queue. Queueing Systems, 51(3-4):361–402, 2005. MR2189598Google Scholar
  • Zenios, S. A., Modeling the transplant waiting list: A queueing model with reneging. Queueing systems, 31(3):239–251, 1999. MR1706048Google Scholar
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