Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-12T19:47:39.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Heavy traffic analysis for continuous polling models

Part of: Stochastic analysis Stochastic processes Operations research and management science 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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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