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On some diffusion approximations to queueing systems

Published online by Cambridge University Press:  01 July 2016

V. Giorno ,
A. G. Nobile  and
L. M. Ricciardi
V. Giorno*
Affiliation:
University of Salerno
A. G. Nobile*
Affiliation:
University of Salerno
L. M. Ricciardi*
Affiliation:
University of Naples
*
∗∗ Postal address: Dipartimento di Informatica e Applicazioni, Facoltà de Scienze, Università di Salerno, 84100 Salerno, Italy.
∗∗ Postal address: Dipartimento di Informatica e Applicazioni, Facoltà de Scienze, Università di Salerno, 84100 Salerno, Italy.
∗ Postal address: Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Mezzocannone 8, 80134 Napoli, Italy.
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Abstract

For a class of models of adaptive queueing systems an exact diffusion approximation is derived with the aim of obtaining information on the evolution of the systems. Our approximating diffusion process includes the Wiener and the Ornstein–Uhlenbeck processes with reflecting boundaries at 0. The goodness of the approximations is thoroughly discussed and the closed-form solutions obtained for the diffusion processes are compared with those holding for the queueing system in order to investigate the conditions under which reliable information can be obtained from the approximating continuous models. For the latter the transient behaviour is quantitatively analysed and the distribution of the busy period is determined in closed form.

Keywords

ADAPTIVE QUEUEING SYSTEMSM/M/1 MODELWIENER PROCESSORNSTEIN–UHLENBECK PROCESSFIRST-PASSAGE TIMEBUSY PERIOD

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 4 , December 1986 , pp. 991 - 1014
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Work performed under CNR-JSPS Scientific Cooperation Programme, Contracts no. 83.00032.01 and no. 84.00227.01, CNR Contract no. 85.00002.01 and under MPI financial support.

References

1. Chan, J. and Conolly, B. W. (1978) Comparative effectiveness of certain queueing systems with adaptive demand and service mechanisms. Comput. Operat. Res. 5, 187–196.CrossRefGoogle Scholar
2. Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam,.Google Scholar
3. Conolly, B. W. (1975) Lecture Notes on Queueing Systems. Ellis Horwood, Chichester.Google Scholar
4. Conolly, B. W. and Chan, J. (1977) Generalised birth and death queueing processes: recent results. Adv. Appl. Prob. 9, 125–140.Google Scholar
5. Cox, D. R. and Miller, H. D. (1972) The Theory of Stochastic Processes. Chapman and Hall, London.Google Scholar
6. Darling, O. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.CrossRefGoogle Scholar
7. Duda, A. (1983) Application of the transient diffusion approximation to adaptive routing. Res. Report no. 14, ISEM, Université de Paris-Sud.Google Scholar
8. Erdelyi, A., Magnus, W., Oberthettinger, F. and Tricomi, F. G. (1953) Higher Transcendental Functions , Vol. I. McGraw-Hill, New York.Google Scholar
9. Erdelyi, A., Magnus, W., Oberthettinger, F. and Tricomi, F. G. (1953) Higher Transcendental Functions , Vol. II. McGraw-Hill, New York.Google Scholar
10. Erdelyi, A., Magnus, W., Oberthettinger, F. and Tricomi, F. G. (1954) Tables of Integral Transforms , Vol. I. McGraw-Hill, New York.Google Scholar
11. Feller, W. (1951) Two singular diffusion processes. Ann. Math. 54, 173–182.CrossRefGoogle Scholar
12. Gaver, D. P. (1968) Diffusion approximations and models for certain congestion problems. J. Appl. Prob. 5, 607–623.Google Scholar
13. Gelenbe, E. and Mitrani, I. (1980) Analysis and Synthesis of Computer Systems. Academic Press, London.Google Scholar
14. Gradshteyn, I. S. and Ryzhik, I. M. (1980) Table of Integrals, Series and Products. Academic Press, New York.Google Scholar
15. Karlin, S. and Taylor, H. M. (1981) A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
16. Kleinrock, L. (1976) Queueing Systems, Vol. II: Computer Applications. Wiley, New York.Google Scholar
17. Kobayashi, H. (1974) Application of the diffusion approximation to the queueing networks: Part I. J. Assoc. Comput. Mach. 21, 316–328.Google Scholar
18. Kobayashi, H. (1981) Diffusion approximations in queueing analysis. Res. Report, RC 8819, IBM Thomas J. Watson Research Center, New York.Google Scholar
19. Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985) Exponential trends of Ornstein-Uhlenbeck first-passage-time densities. J. Appl. Prob. 22, 360–369.Google Scholar
20. Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985) Exponential trends of first passage time densities for a class of diffusion processes with steady-state distribution. J. Appl. Prob. 22, 611–618.Google Scholar
21. Tricomi, F. G. (1954) Funzioni Ipergeometriche Confluenti. Cremonese, Roma.Google Scholar
22. Van Hoorn, M. H. (1984) Algorithms and Approximations for Queueing Systems. CWI Tract, Amsterdam.Google Scholar