A Diffusion Approximation for a Markovian Queue with Reneging
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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.
- J. Abate and W. Whitt, Numerical inversion of Laplace transforms of probability distributions, ORSA J. Computing 7 (1995) 36–43.
- C.J. Ancker and A. Gafarian, Queueing with impatient customers who leave at random, J. Industr. Engrg. 13 (1962) 84–90.
- P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1999).
- A. Birman and Y. Kogan, Asymptotic evaluation of closed queueing networks with many stations, Stochastic Models 8 (1992) 543–563.
- 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.
- E. Coffman, A. Puhalskii, M. Reiman and P. Wright, Processor-shared buffers with reneging, Performance Evaluation 19 (1994) 25–46.
- D. Duffie, J. Pan and K. Singleton, Transform analysis and asset pricing for affine jump-diffusions, Econometrica 6 (2000) 1343–1376.
- P. Echeverria, A criterion for invariant measures of Markov processes, Z. Wahrsch. Verw. Gebiete 61 (1982) 1–16.
- S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence (Wiley, New York, 1986).
- 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.
- O. Garnett, A. Mandelbaum and M. Reiman, Designing a call center with impatient customers, Preprint (2001).
- P. Glynn, Strong approximations in queueing theory, in:Asymptotic Methods in Probability and Statistics, ed. B. Szyszkowitcz (Elsevier, Amsterdam, 1998) pp. 133–150.
- J.M. Harrison, Brownian Motion and Stochastic Flow Systems (Wiley, New York, 1985).
- D. Iglehart, Limit diffusion approximations for the many-server queue and the repairman problem, J. Appl. Probab. 2 (1965) 429–441.
- D. Iglehart and W. Whitt, Multiple channel queues in heavy traffic I, Adv. in Appl. Probab. 2 (1970) 150–177.
- P. Lions and A. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37 (1984) 511–537.
- A. Mandelbaum and G. Pats, State-dependent queues: Approximations and applications, in: Stochastic Networks (Springer, Berlin, 1995) pp. 239–282.
- D. Mitra and J.A. Morrison, Erlang capacity and uniform approximations for shared unbuffered resources, IEEE/ACM Trans. Networking 1 (1993) 664–667.
- C. Palm, Etude des délais d'attente, Ericsson Technics 5 (1937) 37–56.
- R. Srikant and W. Whitt, Simulation run lengths to estimate blocking probabilities, ACMTrans.Modeling Comput. Simulation 6 (1996) 7–52.
- C.J. Stone, Limit theorems for birth and death processes and diffusion processes, Ph.D. thesis, Department of Statistics, Stanford University (1961).
- R. Syski, Introduction to Congestion Theory in Telephone Systems (Oliver and Boyd, Edinborough, 1960).
- A. Ward and P. Glynn, A diffusion approximation for a GI/G/1 queue with reneging, Working paper (2002).
- A. Ward and P. Glynn, Properties of the reflected Ornstein–Uhlenbeck process,Working paper (2002).
- 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.
- W. Whitt, Improving service by informing customers about anticipated delays, Managm. Sci. 45(2) (1999) 192–207.
- R.W. Wolff, Stochastic Modeling and the Theory of Queues (Prentice-Hall, Englewood Cliffs, NJ, 1989).
- A Diffusion Approximation for a Markovian Queue with Reneging
Volume 43, Issue 1-2 , pp 103-128
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- Markovian queues
- reflected Ornstein–Uhlenbeck process
- reflected affine diffusion
- diffusion approximation
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