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Asymptotic analysis of single resource loss systems in heavy traffic, with applications to integrated networks

Part of: Computer system organization Special processes

Published online by Cambridge University Press:  01 July 2016

N. G. Bean ,
R. J. Gibbens  and
S. Zachary
N. G. Bean*
Affiliation:
University of Adelaide
R. J. Gibbens*
Affiliation:
University of Cambridge
S. Zachary*
Affiliation:
Heriot-Watt University
*
* Postal address: Teletraffic Research Centre, Department of Applied Mathematics, University of Adelaide, South Australia 5005, Australia (e-mail: nbean@maths.adelaide.edu.au).
** Postal address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge, CB2 ISB (e-mail: R.J.Gibbens@statslab.cam.ac.uk).
*** Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Riccarton, Edinburgh, EH144AS (e-mail: stan@ma.hw.ac.uk).
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Abstract

In this paper we consider the analysis of call blocking at a single resource with differing capacity requirements as well as differing arrival rates and holding times. We include in our analysis trunk reservation parameters which provide an important mechanism for tuning the relative call blockings to desired levels. We base our work on an asymptotic regime where the resource is in heavy traffic. We further derive, from our asymptotic analysis. methods for the analysis of finite systems. Empirical results suggest that these methods perform well for a wide class of examples.

Keywords

LOSS NETWORKSTIME-SCALE SEPARATIONTRUNK RESERVATIONBLOCKING PROBABILITIES

MSC classification

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Secondary: 60K15: Markov renewal processes, semi-Markov processes 60K30: Applications (congestion, allocation, storage, traffic, etc.) 68M20: Performance evaluation; queueing; scheduling

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 273 - 292
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Work carried out at the Statistical Laboratory, University of Cambridge and supported by the George Murray Scholarship and the Cambridge Commonwealth Trust.

Work supported by the SERC under Grant GR/F 94194 and a Royal Society University Research Fellowship.

Work supported by the SERC under Grant GR/F 32646.

References

[1] Bean, N. G., Gibbens, R. J. and Zachary, S. (1994). The performance of single resource loss systems in multiservice networks. In Proceedings of the 14th International Teletraffic Congress. ed. Labetoulle, J. and Roberts, J. W., pp. 13–21. Elsevier Science, Oxford.Google Scholar
[2] Bean, N. G., Gibbens, R. J. and Zachary, S. (1994) Dynamic and equilibrium behaviour of controlled loss networks. Submitted for publication.Google Scholar
[3] Burman, D. Y., Lehoczky, J. P. and Lim, Y. (1984). Insensitivity of blocking probabilities in a circuit-switched network. J. Appl. Prob. 21, 850–859.CrossRefGoogle Scholar
[4] Durrett, R. (1991) Probability: Theory and Examples. Wadsworth & Brooks/Cole, Pacific Grove, CA.Google Scholar
[5] Dziong, Z. and Roberts, J. W. (1987). Congestion probabilities in a circuit switched integrated services network. Performance Evaluation 7, 267–284.Google Scholar
[6] Gersht, A. and Lee, K. J. (1990) A bandwidth management strategy in ATM networks. Preprint, GTE Laboratories, 40 Sylvan Road, Waltham, MA 02254, USA.Google Scholar
[7] Gibbens, R. J. and Hunt, P. J. (1991) Effective bandwidths for the multitype UAS channel. Queueing Systems 9, 17–28.CrossRefGoogle Scholar
[8] Greenberg, A., Mitrani, I. and Stolyar, A. (1992) Unpublished.Google Scholar
[9] Heuser, H. G. (1982) Functional Analysis. Wiley, New York.Google Scholar
[10] Hui, J. Y. (1988) Resource allocation for broadband networks. IEEE J.S.A.C. 6, 1598–1608.Google Scholar
[11] Hunt, P. J. (1990) Limit theorems for stochastic loss networks. Ph.D. Dissertation, University of Cambridge.Google Scholar
[12] Hunt, P. J. and Kurtz, T. G. (1994) Large loss networks. Stoch. Proc. Appl. To appear.Google Scholar
[13] Johnsonbaugh, R. and Pfaffenberger, W. E. (1981) Foundations of Mathematical Analysis. Marcel Dekker, New York.Google Scholar
[14] Kaufman, J. S. (1981) Blocking in a shared resource environment. IEEE Trans. Comm. 29, 1474–1481.Google Scholar
[15] Kelly, F. P. (1986) Blocking probabilities in large circuit switched networks. Adv. Appl. Prob. 18, 473–505.Google Scholar
[16] Kelly, F. P. (1989) Fixed point models of loss networks. J. Austral. Math. Soc. B 31, 204–218.CrossRefGoogle Scholar
[17] Kelly, F. P. (1991) Effective bandwidths at multi-class queues. Queueing Systems 9, 5–16.CrossRefGoogle Scholar
[18] Kelly, F. P. (1991) Loss networks. Ann. Appl. Prob. 1, 319–378.Google Scholar
[19] Key, P. B. (1990) Optimal control and trunk reservation in loss networks. Prob. Eng. Inf. Sci. 4, 203–242.CrossRefGoogle Scholar
[20] Rudin, W. (1970) Real and Complex Analysis. McGraw-Hill, London.Google Scholar
[21] Tijms, H. C. and Van De Coevering, M. C. T. (1991) A simple numerical approach for infinite-state Markov chains. Prob. Eng. Inf. Sci. 5, 285–296.Google Scholar
[22] Tran-Gia, P. and Hübner, F. (1993) An analysis of trunk reservation and grade of service balancing mechanisms in multiservice broadband networks. In IFIP Workshop TC6, Modelling and Performance Evaluation of ATM Technology , La Martinique, French Caribbean, 25–27 January.Google Scholar
[23] Zachary, S. (1991) On blocking in loss networks. Adv. Appl. Prob. 23, 355–372.CrossRefGoogle Scholar