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A Diffusion Approximation for a GI/GI/1 Queue with Balking or Reneging


Consider a single-server queue with a renewal arrival process and generally distributed processing times in which each customer independently reneges if service has not begun within a generally distributed amount of time. We establish that both the workload and queue-length processes in this system can be approximated by a regulated Ornstein-Uhlenbeck (ROU) process when the arrival rate is close to the processing rate and reneging times are large. We further show that a ROU process also approximates the queue-length process, under the same parameter assumptions, in a balking model. Our balking model assumes the queue-length is observable to arriving customers, and that each customer balks if his or her conditional expected waiting time is too large.

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  1. : 2000, Extend: Professional simulation Tools. Imagine That, 6830 Via Del Oro, Suite 230, San Jose, CA 95119, version 5 edition.

  2. : 2002, Cisco: Behind the hype. Business Week.

  3. C.J. Ancker and A.V. Gafarian, Queueing with impatient customers who leaveat random, Journal of Industrial Engineering 13 (1962) 84–90.

    Google Scholar 

  4. F. Baccelli, P. Boyer and G. Hebuterne, Single-server queues with impatient customers, Adv. Appl. Prob. 16 (1984) 887–905.

    Google Scholar 

  5. L. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn and L. Zhao, Statistical analysis of a Telephone Call Center: A queueing-science perspective, Working Paper, (2002).

  6. S. Browne and W. Whitt, Piecewise-linear diffusion processes, in: Advances in Queueing: Theory, Methods, and Open Problems, ed. J. Dshalalow (CRC Press, 1995) pp. 463–480.

  7. H. Chen, Generalized regulated Mapping: Fluid and diffusion limits, Notes prepared for Avi Mandelbaum, 1990.

  8. H. Chen, and D.D. Yao, Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization (Springer-Verlag, New York, 2001).

    Google Scholar 

  9. E. Coffman, A. Puhalskii, M. Reiman and P. Wright, Processor-shared buffers with reneging, Performance Evaluation 19 (1994) 25–46.

    Article  Google Scholar 

  10. B. Doytchinov, J. Lehoczky and S. Shreve, real-time queues in heavy traffic with earliest-deadline-first queue discipline, Annals of Applied Probability 11 (2001) 332–378.

    Article  Google Scholar 

  11. D. Gamarnik and A. Zeevi, Validity of heavy traffic steady-state approximations in open queueing networks, Working Paper, 2004.

  12. O. Garnett, A. Mandelbaum and M. Reiman, Designing a call center with impatient customers, Manufacturing and Service Operations Management 4 (2002) 208–227.

    Article  Google Scholar 

  13. P. Hall and C.C. Heyde, Martingale Limit Theory and its Application (Academic Press, Inc., Boston, 1980).

    Google Scholar 

  14. J.M. Harrison, Brownian Motion and Stochastic Flow Systems (John Wiley & Sons, New York, 1985).

    Google Scholar 

  15. D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic, I and II, Adv. Appl. Prob. 2 (1970) 150–177 and 355–364.

    Google Scholar 

  16. R. Lillo and M. Martin, Stability in queues with impatient customers, Stochastic Models 17 (2001).

  17. A. Mandelbaum and S. Zeltyn, The impact of customers’ patience on delay and abandonment: Some empirically-driven experiments with the M/M/N + G queue, OR Spectrum 26(3) (2004) 377–411. Special Issue on Call Centers.

    Article  Google Scholar 

  18. C. Palm, Etude des delais d’attente, Ericson Technics 5 (1937) 37–56.

    Google Scholar 

  19. E.L. Plambeck, S. Kumar and J.M. Harrison, A multiclass queue in heavy traffic with throughput time constraints: Asymptotically optimal dynamic controls, Queueing Systems 39 (2001) 23–54.

    Article  Google Scholar 

  20. J. Reed and A.R. Ward, A diffusion approximation for a generalized Jackson network with reneging, in: Proceedings of the 42nd Annual Allerton Conference on Communication, Control, and Computing, 2004.

  21. M.I. Reiman, Some diffusion approximations with state space collapse, in: Lecture Notes in Control and Information Sciences, eds. F. Baccelli and G. Fayolle (Springer, 1984) vol. 60 pp. 209–240.

  22. R.E. Stanford, Reneging phenomena in single channel queues, Mathematics of Operations Research 4 (1979) 162–178.

    Google Scholar 

  23. A.R. Ward and P.W. Glynn, A diffusion approximation for a Markovian queue with reneging, Queueing Systems 43 (2003a) 103–128.

    Article  Google Scholar 

  24. A.R. Ward and P.W. Glynn, Properties of the reflected ornstein-uhlenbeck process, Queueing Systems 44 (2003b) 109–123.

    Article  Google Scholar 

  25. W. Whitt, Improving service by informing customers about anticipated delays, Management Science 45(2) (1999) 192–207.

    Google Scholar 

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Correspondence to Amy R. Ward.

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Ward, A.R., Glynn, P.W. A Diffusion Approximation for a GI/GI/1 Queue with Balking or Reneging. Queueing Syst 50, 371–400 (2005).

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  • deadlines
  • reneging
  • balking
  • impatience
  • GI/GI/1-GI queue
  • Ornstein-Uhlenbeck process
  • regulated diffusion
  • reflected diffusion