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Queueing Systems

, Volume 89, Issue 1–2, pp 81–125 | Cite as

Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment

  • A. Korhan Aras
  • Xinyun Chen
  • Yunan Liu
Article

Abstract

Extending Ward Whitt’s pioneering work “Fluid Models for Multiserver Queues with Abandonments, Operations Research, 54(1) 37–54, 2006,” this paper establishes a many-server heavy-traffic functional central limit theorem for the overloaded \(G{/}GI{/}n+GI\) queue with stationary arrivals, nonexponential service times, n identical servers, and nonexponential patience times. Process-level convergence to non-Markovian Gaussian limits is established as the number of servers goes to infinity for key performance processes such as the waiting times, queue length, abandonment and departure processes. Analytic formulas are developed to characterize the distributions of these Gaussian limits.

Keywords

Many-server queues Many-server heavy-traffic limits Nonexponential service times Efficiency-driven regime Customer abandonment Gaussian approximation Functional central limit theorem 

Mathematics Subject Classification

60B12 60B05 60F17 60K25 60M20 90B22 

Notes

Acknowledgements

We thank editors Amy Ward and Guodong Pang for inviting us to contribute to this special issue and anonymous referees for providing constructive comments. The third author would like to thank Ward Whitt for his support and guidance through the years, and for being a tremendous source of inspiration. The first and third authors acknowledge supports from NSF Grant CMMI 1362310.

References

  1. 1.
    Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, 766–801 (2001)CrossRefGoogle Scholar
  2. 2.
    Aras, A.K., Chen, X., Liu, Y.: Longer online appendix: many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment (2017). https://yunanliu.wordpress.ncsu.edu/files/2017/11/ArasLiuChenOLGGnGApp11282017.pdf
  3. 3.
    Aras, A.K., Liu, Y., Whitt, W.: Heavy-traffic limit for the initial content process. Stoch. Syst. 7, 95–142 (2017)CrossRefGoogle Scholar
  4. 4.
    Bassamboo, A., Randhawa, R.S.: On the accuracy of fluid models for capacity sizing in queueing systems with impatient customers. Oper. Res. 58, 1398–1413 (2010)CrossRefGoogle Scholar
  5. 5.
    Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, Berlin (2008)CrossRefGoogle Scholar
  6. 6.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley-Interscience, New York (1999)CrossRefGoogle Scholar
  7. 7.
    Cox, D.R.: Renewal Theory. Methuen, London (1962)Google Scholar
  8. 8.
    Dai, J., He, S., Tezcan, T.: Many server diffusion limits for \({G/Ph/n+GI}\) queues. Ann. Appl. Probab. 20, 1854–1890 (2010)CrossRefGoogle Scholar
  9. 9.
    Dai, J.G., He, S.: Customer abandonment in many-server queues. Math. Oper. Res. 35(2), 347–362 (2010)CrossRefGoogle Scholar
  10. 10.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)CrossRefGoogle Scholar
  11. 11.
    Gamarnik, D., Goldberg, D.: Steady-state \({GI/GI/n}\) queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23, 2382–2419 (2012)CrossRefGoogle Scholar
  12. 12.
    He, B., Liu, Y., Whitt, W.: Staffing a service system with non-Poisson nonstationary arrivals. Probab. Eng. Inf. Sci. 30, 593–621 (2016)CrossRefGoogle Scholar
  13. 13.
    He, S.: Diffusion approximation for efficiency-driven queues: a space–time scaling approach. Working paper, National University of SingaporeGoogle Scholar
  14. 14.
    Huang, J., Mandelbaum, A., Zhang, H., Zhang, J.: Refined models for efficiency-driven queues with applications to delay announcements and staffing. Working paper (2017)Google Scholar
  15. 15.
    Kang, W., Ramanan, K.: Fluid limits of many-server queues with reneging. Ann. Appl. Probab. 20, 2204–2260 (2010)CrossRefGoogle Scholar
  16. 16.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1988)CrossRefGoogle Scholar
  17. 17.
    Kaspi, H., Ramanan, K.: SPDE limits of many-server queue. Ann. Appl. Probab. 23, 145–229 (2013)CrossRefGoogle Scholar
  18. 18.
    Krichagina, E.V., Puhalskii, A.A.: A heavy-traffic analysis of a closed queueing system with a \({GI/\infty }\) service center. Queueing Syst. 25, 235–280 (1997)CrossRefGoogle Scholar
  19. 19.
    Lebovits, J.: Stochastic calculus with respect to Gaussian processes. Working paper (2017)Google Scholar
  20. 20.
    Liu, R., Kuhl, M.E., Liu, Y., Wilson, J.R.: Modeling and simulation of nonstationary non-Poisson processes. INFORMS J. Comput. (forthcoming)Google Scholar
  21. 21.
    Liu, Y.: Staffing to stabilize the tail probability of delay in service systems with time-varying demand. Oper. Res. (forthcoming)Google Scholar
  22. 22.
    Liu, Y., Whitt, W.: A network of time-varying many-server fluid queues with customer abandonment. Oper. Res. 59, 835–846 (2011)CrossRefGoogle Scholar
  23. 23.
    Liu, Y., Whitt, W.: The \(G_t/GI/s_t+GI\) many-server fluid queue. Queueing Syst. 71, 405–444 (2012)CrossRefGoogle Scholar
  24. 24.
    Liu, Y., Whitt, W.: A many-server fluid limit for the \(G_t/GI/s_t + GI\) queueing model experiencing periods of overloading. Oper. Res. Lett. 40, 307–312 (2012)CrossRefGoogle Scholar
  25. 25.
    Liu, Y., Whitt, W.: Algorithms for time-varying networks of many-server fluid queues. INFORMS J. Comput. 26, 59–73 (2013)CrossRefGoogle Scholar
  26. 26.
    Liu, Y., Whitt, W.: Many-server heavy-traffic limits for queues with time-varying parameters. Ann. Appl. Probab. 24, 378–421 (2014)CrossRefGoogle Scholar
  27. 27.
    Liu, Y., Whitt, W., Yu, Y.: Approximations for heavily-loaded \(G/GI/n+GI\) queues. Naval Res. Logist. 63, 187–217 (2016)CrossRefGoogle Scholar
  28. 28.
    Mandelbaum, A., Massey, W.A., Reiman, M.I.: Strong approximations for Markovian service networks. Queueing Syst. 30, 149–201 (1998)CrossRefGoogle Scholar
  29. 29.
    Mandelbaum, A., Momcilovic, P.: Queues with many servers and impatient customers. Math. Oper. Res. 37, 41–65 (2012)CrossRefGoogle Scholar
  30. 30.
    Mandelbaum, A., Zeltyn, S.: Staffing many-server queues with impatient customers: constraint satisfaction in call centers. Oper. Res. 57, 1189–1205 (2009)CrossRefGoogle Scholar
  31. 31.
    Pang, G., Whitt, W.: Two-parameter heavy-traffic limits for infinite-server queues. Queueing Syst. 65, 325–364 (2010)CrossRefGoogle Scholar
  32. 32.
    Pang, G., Whitt, W., Talreja, R.: Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4, 193–267 (2007)CrossRefGoogle Scholar
  33. 33.
    Reed, J.: The \({G/GI/N}\) queue in the Halfin–Whitt regime. Ann. Appl. Probab. 19, 2211–2269 (2009)CrossRefGoogle Scholar
  34. 34.
    Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales. Wiley, New York (1994)Google Scholar
  35. 35.
    Whitt, W.: Queues with superposition arrival process in heavy traffic. Stoch. Process. Appl. 21, 81–91 (1985)CrossRefGoogle Scholar
  36. 36.
    Whitt, W.: Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and there Application to Queues. Springer, Berlin (2002)Google Scholar
  37. 37.
    Whitt, W.: Efficiency-driven heavy-traffic approximations for many-server queues with abandonments. Manag. Sci. 50, 1449–1461 (2004)CrossRefGoogle Scholar
  38. 38.
    Whitt, W.: Fluid models for multiserver queues with abandonments. Oper. Res. 54, 37–54 (2006)CrossRefGoogle Scholar
  39. 39.
    Zhang, J.: Fluid models of many-server queues with abandonment. Queueing Syst. 73, 147–193 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.SAS InstituteCaryUSA
  2. 2.School of Economics and ManagementWuhan UniversityWuhanChina
  3. 3.Industrial and Systems Engineering DepartmentNorth Carolina State UniversityRaleighUSA

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