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Heavy traffic analysis for continuous polling models

Part of: Stochastic analysis Operations research and management science Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Dirk P. Kroese
Dirk P. Kroese*
Affiliation:
University of Twente
*
∗Postal address: Faculty of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands.
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Abstract

We consider a continuous polling system in heavy traffic. Using the relationship between such systems and age-dependent branching processes, we show that the steady-state number of waiting customers in heavy traffic has approximately a gamma distribution. Moreover, given their total number, the configuration of these customers is approximately deterministic.

Keywords

QUEUEING THEORYCONTINUOUS POLLING SYSTEMHEAVY TRAFFICQUEUE LENGTHAGE-DEPENDENT BRANCHING PROCESSEXPONENTIAL APPROXIMATIONGAMMA APPROXIMATIONRANDOM MEASUREGENERATING FUNCTION

MSC classification

Primary: 60K25: Queueing theory 90B22: Queues and service 60G55: Point processes
Secondary: 60G57: Random measures 60H05: Stochastic integrals
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 3 , September 1997 , pp. 720 - 732
Copyright
Copyright © Applied Probability Trust 1997 

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References

[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
[2] Boxma, O. J. and Takagi, H., (eds) (1992) Special issue on polling systems. Queueing Systems 11.CrossRefGoogle Scholar
[3] Coffman, E. G. Jr. and Gilbert, E. N. (1986) A continuous polling system with constant service times. IEEE Trans. Inform. Theory 32, 584–591.CrossRefGoogle Scholar
[4] Coffman, E. G. Jr., Puhalskii, A. A. and Reiman, M. I. (1995) Polling systems with zero switchover times: a heavy-traffic averaging principle. Ann. Appl. Prob. 5, 681–719.CrossRefGoogle Scholar
[5] Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[6] Feller, W. (1970) An Introduction to Probability Theory and Its Applications. Vol. II. 2nd edn. Wiley, New York.Google Scholar
[7] Fuhrmann, S. W. and Cooper, R. B. (1985) Application of decomposition principle in M/G/1 vacation model to two continuum cyclic queueing models – especially token-ring LANs. AT&T Tech. J. 64, 1091–1098.Google Scholar
[8] Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
[9] Kingman, J. F. C. (1962) On queues in heavy traffic. J. R. Statist. Soc. B 24, 383–392.Google Scholar
[10] Kingman, J. F. C. (1982) Queue disciplines in heavy traffic. Math. Operat. Res. 7, 262–271.CrossRefGoogle Scholar
[11] Kroese, D. P. and Schmidt, V. (1992) A continuous polling system with general service times. Ann. Appl. Prob. 2, 906–927.CrossRefGoogle Scholar
[12] Kroese, D. P. and Schmidt, V. (1994) Single-server queues with spatially distributed arrivals. Queueing Systems. 17, 317–345.CrossRefGoogle Scholar
[13] Resing, J. A. C. (1993) Polling systems and multiple branching processes. Queueing Systems 13, 409–426.CrossRefGoogle Scholar
[14] Szczotka, W. (1990) Exponential approximation of waiting time and queue size for queues in heavy traffic. Adv. Appl. Prob. 22, 230–240.CrossRefGoogle Scholar
[15] Takagi, H. (1986) Analysis of Polling Systems. MIT Press, Cambridge, MA.Google Scholar
[16] Van Der Mei, R. (1995) Polling systems and the power series algorithm. PhD thesis. Tilburg University.Google Scholar
[17] Whitt, W. (1974) Heavy trafffic limit theory for queues: a survey. In Mathematical Methods in Queueing Theory. (Lecture Notes in Economics and Mathematical Systems 98.) ed. Clarke, A. B. Springer, Berlin, pp. 307–350.CrossRefGoogle Scholar