Queueing Systems

, Volume 43, Issue 1–2, pp 103–128 | Cite as

A Diffusion Approximation for a Markovian Queue with Reneging

  • Amy R. Ward
  • Peter W. Glynn

Abstract

Consider a single-server queue with a Poisson arrival process and exponential processing times in which each customer independently reneges after an exponentially distributed amount of time. We establish that this system can be approximated by either a reflected Ornstein–Uhlenbeck process or a reflected affine diffusion when the arrival rate exceeds or is close to the processing rate and the reneging rate is close to 0. We further compare the quality of the steady-state distribution approximations suggested by each diffusion.

Markovian queues reneging impatience deadlines reflected Ornstein–Uhlenbeck process reflected affine diffusion diffusion approximation steady-state 

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References

  1. [1]
    J. Abate and W. Whitt, Numerical inversion of Laplace transforms of probability distributions, ORSA J. Computing 7 (1995) 36–43.Google Scholar
  2. [2]
    C.J. Ancker and A. Gafarian, Queueing with impatient customers who leave at random, J. Industr. Engrg. 13 (1962) 84–90.Google Scholar
  3. [3]
    P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1999).Google Scholar
  4. [4]
    A. Birman and Y. Kogan, Asymptotic evaluation of closed queueing networks with many stations, Stochastic Models 8 (1992) 543–563.Google Scholar
  5. [5]
    S. Browne and W. Whitt, Piecewise-linear diffusion processes, in: Advances in Queueing: Theory, Methods, and Open Problems, ed. J. Dshalalow (CRC Press, Boca Raton, FL, 1995) pp. 463–480.Google Scholar
  6. [6]
    E. Coffman, A. Puhalskii, M. Reiman and P. Wright, Processor-shared buffers with reneging, Performance Evaluation 19 (1994) 25–46.Google Scholar
  7. [7]
    D. Duffie, J. Pan and K. Singleton, Transform analysis and asset pricing for affine jump-diffusions, Econometrica 6 (2000) 1343–1376.Google Scholar
  8. [8]
    P. Echeverria, A criterion for invariant measures of Markov processes, Z. Wahrsch. Verw. Gebiete 61 (1982) 1–16.Google Scholar
  9. [9]
    S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence (Wiley, New York, 1986).Google Scholar
  10. [10]
    P. Fleming, A. Stolyar and B. Simon, Heavy traffic limit for a mobile phone system loss model, in: Proc. of 2nd Internat. Conf. on Telecommunication Systems Mod. and Analysis, Nashville, TN, 1994.Google Scholar
  11. [11]
    O. Garnett, A. Mandelbaum and M. Reiman, Designing a call center with impatient customers, Preprint (2001).Google Scholar
  12. [12]
    P. Glynn, Strong approximations in queueing theory, in:Asymptotic Methods in Probability and Statistics, ed. B. Szyszkowitcz (Elsevier, Amsterdam, 1998) pp. 133–150.Google Scholar
  13. [13]
    J.M. Harrison, Brownian Motion and Stochastic Flow Systems (Wiley, New York, 1985).Google Scholar
  14. [14]
    D. Iglehart, Limit diffusion approximations for the many-server queue and the repairman problem, J. Appl. Probab. 2 (1965) 429–441.Google Scholar
  15. [15]
    D. Iglehart and W. Whitt, Multiple channel queues in heavy traffic I, Adv. in Appl. Probab. 2 (1970) 150–177.Google Scholar
  16. [16]
    P. Lions and A. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37 (1984) 511–537.Google Scholar
  17. [17]
    A. Mandelbaum and G. Pats, State-dependent queues: Approximations and applications, in: Stochastic Networks (Springer, Berlin, 1995) pp. 239–282.Google Scholar
  18. [18]
    D. Mitra and J.A. Morrison, Erlang capacity and uniform approximations for shared unbuffered resources, IEEE/ACM Trans. Networking 1 (1993) 664–667.Google Scholar
  19. [19]
    C. Palm, Etude des délais d'attente, Ericsson Technics 5 (1937) 37–56.Google Scholar
  20. [20]
    R. Srikant and W. Whitt, Simulation run lengths to estimate blocking probabilities, ACMTrans.Modeling Comput. Simulation 6 (1996) 7–52.Google Scholar
  21. [21]
    C.J. Stone, Limit theorems for birth and death processes and diffusion processes, Ph.D. thesis, Department of Statistics, Stanford University (1961).Google Scholar
  22. [22]
    R. Syski, Introduction to Congestion Theory in Telephone Systems (Oliver and Boyd, Edinborough, 1960).Google Scholar
  23. [23]
    A. Ward and P. Glynn, A diffusion approximation for a GI/G/1 queue with reneging, Working paper (2002).Google Scholar
  24. [24]
    A. Ward and P. Glynn, Properties of the reflected Ornstein–Uhlenbeck process,Working paper (2002).Google Scholar
  25. [25]
    W. Whitt, Heavy traffic limit theorems for queues: A survey, in: Lecture Notes in Economics and Mathematical Systems, Vol. 98 (Springer, Berlin, 1974) pp. 307–350.Google Scholar
  26. [26]
    W. Whitt, Improving service by informing customers about anticipated delays, Managm. Sci. 45(2) (1999) 192–207.Google Scholar
  27. [27]
    R.W. Wolff, Stochastic Modeling and the Theory of Queues (Prentice-Hall, Englewood Cliffs, NJ, 1989).Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Amy R. Ward
    • 1
  • Peter W. Glynn
    • 2
  1. 1.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of Management Science & EngineeringStanford UniversityStanfordUSA

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