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On some time-non-homogeneous 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, Università di Salerno, 84100 Salerno, Italy.
∗Postal address: Dipartimento di Informatica e Applicazioni, 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

Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.

Keywords

M/M/1 MODELWIENER PROCESSORNSTEIN–UHLENBECK PROCESSFIRST-PASSAGE TIMEBUSY PERIOD
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 4 , December 1987 , pp. 974 - 994
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

Research carried out under CNR-JSPS Scientific Cooperation Programme, Contracts No. 83.00002.01 and No. 84.00227.01, and with MPI financial support.

References

[1] Anderssen, K. S. De Hoog, F. R. and Weiss, R. (1973) On the numerical solution of Brownian motion processes. J. Appl. Prob. 10, 409–418.CrossRefGoogle Scholar
[2] Conolly, B. W. (1975) Lecture Notes on Queueing Systems. Ellis Horwood, Chichester.Google Scholar
[3] Duda, A. (1983) Application of the transient diffusion approximation to adaptive routing. Res. Report no. 14, ISEM, Université de Paris-Sud.Google Scholar
[4] Durbin, J. (1971) Boundary crossing probabilities for Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Prob. 8, 431–453.Google Scholar
[5] Favella, L., Reineri, M. T., Ricciardi, L. M. and Sacerdote, L. (1982) First passage time problems and some related computational methods. Cybernetics and Systems 13, 95–128.Google Scholar
[6] Gaver, D. P. (1968) Diffusion approximations and models for certain congestion problems. J. Appl. Prob. 5, 607–623.CrossRefGoogle Scholar
[7] Gelenbe, E. and Mitrani, E. (1980) Analysis and Synthesis of Computer System. Academic Press, London.Google Scholar
[8] Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1985) On some diffusion approximations to queueing systems. Adv. Appl. Prob. 18, 991–1014.CrossRefGoogle Scholar
[9] Gradshteyn, I. S. and Ryzhik, I. M. (1980) Table of Integral Series and Products. Academic Press, New York.Google Scholar
[10] Kleinrock, L. (1976) Queueing Systems, Vol. II: Computer Applications. Wiley, New York.Google Scholar
[11] Kobayashi, H. (1974) Application of the diffusion approximation to the queueing networks: Part I. J. Assoc. Comput. Mach. 21, 316–328.Google Scholar
[12] Kobayashi, H. (1981) Diffusion approximations in queueing analysis. Research Report RC 8819, IBM Thomas J. Watson Res. Center, New York.Google Scholar
[13] Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985) A note on first-passage time and some related problems. J. Appl. Prob. 22, 346–359.Google Scholar
[14] 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
[15] 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
[16] Ricciardi, L. M. (1977) Diffusion Processes and Related Topics in Biology. Lecture Notes in Biomathematics, Springer-Verlag, Berlin.Google Scholar
[17] Ricciardi, L. M. and Sato, S. (1983) A note on the evaluation of the first passage time probability densities. J. Appl. Prob. 20, 197–201.Google Scholar
[18] Ricciardi, L. M., Sacerdote, L. and Sato, S. (1984) On an integral equation for first-passage-time probability densities. J. Appl. Prob. 21, 302–314.Google Scholar
[19] Williams, W. E. (1980) Partial Differential Equations. Clarendon Press, Oxford.Google Scholar
[20] Wong, E. (1971) Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York.Google Scholar