Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T12:12:30.742Z Has data issue: false hasContentIssue false
  • English
  • Français

About the cumulative idle time in multiphase queues

Published online by Cambridge University Press:  15 October 2005

Saulius Minkevičius  and
Stasys Steišūnas
Saulius Minkevičius
Affiliation:
Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. stst@ktl.mii.lt
Stasys Steišūnas
Affiliation:
Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. stst@ktl.mii.lt
Get access

Abstract

The paper is designated to the analysis of queueing systems, arising in the network theory and communications theory (called multiphase queueing systems, tandem queues or series of queueing systems). Also we note that multiphase queueing systems can be useful for modelling practical multi-stage service systems in a variety of disciplines, especially on manufacturing (assembly lines), computer networking (packet switch structures), and in telecommunications (e.g. cellular mobile networks), etc. This research presents heavy traffic limit theorems for the cumulative idle time in multiphase queues. In this work, functional limit theorems are proved for the values of important probability characteristics of the queueing system (a cumulative idle time of a customer).

Keywords

Queueing systemsmultiphase queuesfunctional limit theoremsheavy traffica cumulative idle time of a customer.
Type
Research Article
Information
RAIRO - Operations Research , Volume 39 , Issue 2 , April 2005 , pp. 75 - 85
Copyright
© EDP Sciences, 2005

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

Adams, C., Gelenbe, E. and Vicard, J., An experimentally validated model of the paging drum. Acta Inform. 11 (1979) 103–117. CrossRef
Baccelli, F., Borovkov, A. and Mairesse, J., Asymptotic results on infinite tandem queueing networks. Probab. Theory Related Fields 118 (2000) 365–405. CrossRef
P. Billingsley, Convergence of Probability Measures. New York, Wiley (1968).
Gelenbe, E., On approximate computer system models. J. ACM 22 (1975) 261–269. CrossRef
Gelenbe, E., Probabilistic models of computer systems. II. Diffusion approximations, waiting times and batch arrivals. Acta Inform. 12 (1979) 285–303. CrossRef
E. Gelenbe, X. Mang and Y. Feng, Diffusion cell loss estimates for ATM with multiclass bursty traffic. Computer Systems – Science and Engineering. Special Issue: ATM Networks. 11 (1996) 325–333.
E. Gelenbe, X. Mang and R. Onvural, Diffusion based Call Admission Control in ATM. Performance Evaluation 27/28 (1996) 411–436, also in Proceedings of the IFIP WG 7.3/ACM-SIGMETRICS Performance'96 Conference.
E. Gelenbe and I. Mitrani, Analysis and Synthesis of Computer Systems. New York-London, Academic Press (1980).
Gelenbe, E. and Muntz, R.R., Probabilistic models of computer systems. I. Exact results. Acta Inform. 7 (1976) 35–60. CrossRef
G. Gelenbe and G. Pujolle, The behaviour of a single queue in a general queueing network. Acta Inform. 7 (1976/77) 123–136.
Iglehart, D., Weak convergence in queueing theory. Adv. Appl. Probab. 5 (1973) 570–594. CrossRef
D. Iglehart and W. Whitt, Multiple channel queues in heavy traffic. I. Adv. Appl. Probab. No. 2 (1970) 150–177.
F. Karpelevich and A. Kreinin, Heavy Traffic Limits for Multiphase Queues. American Mathematical Society, Providence, Rhode Island (1994).
Kella, O., Concavity and reflected Levy process. J. Appl. Probab. 29 (1992) 209–215.
Kobayashi, H., Application of the diffusion approximation to queueing networks: Parts I and II. J. ACM 21 (1974) 316–328, 459–469. CrossRef
H. Kobayashi and Q. Ren, A diffusion approximation analysis of an ATM statistical multiplexer with multiple state solutions: Part I: Equilibrium state solutions. Proc. ICC'93 (1993) 1047–1053.
X. Mang, ATM Network Performance Predictions and Control, Ph.D. Dissertation. Duke: Electrical and Computer Engineering Department of Duke University (1996).
X. Mang and E. Gelenbe, Call admission control in ATM using diffusion model. Proc. Globecom'96, London, UK (1996).
Milch, P. and Waggoner, M., A random walk approach to a shutdown queueing system. SIAM J. Appl. Math. 19 (1970) 103–115. CrossRef
Minkevičius, S., Weak convergence in multiphase queues. Lithuanian Math. J. 26 (1986) 717–722 (in Russian). CrossRef
S. Minkevičius, On the full idle time in multiphase queues. Lithuanian Math. J. (to appear).
Pike, M., Some numerical results for the queueing system D/Ek/1. J. R. Statist. Soc. Ser. B. 25 (1963) 477–488.
N. Prabhu, Stochastic Storage Processes. New York-Heidelberg-Berlin, Springer-Verlag (1968).
Puhalskii, A., Moderate deviations for queues in critical loading. Queue. Syst. Theor. Appl. 31 (1999) 359–392. CrossRef
Reiser, M., Kobayashi, H., Accuracy of the diffusion approximation for some queueing systems. IBM J. Res. Dev. 18 (1974) 110–124. CrossRef
Conditions, M. Ridel for stationarity in a single server queueing system. Zastos. Mat. 15 (1976) 17–24.
L. Takacs, Occupation time problems in the theory of queues, in Lecture Notes in Economics and Mathematical Systems. Berlin-Heidelberg-New York, Springer-Verlag 98 (1974) 91–131.
Whitt, W., Limits for cumulative input processes to queues. Probab. Engrg. Inform. Sci. 14 (2000) 123–150. CrossRef