Transient Analysis of a Single-Server Queue with Fixed-Size Batch Arrivals

Authors: Vitalice K. Oduol, C. Ardil

Abstract:

The transient analysis of a queuing system with fixed-size batch Poisson arrivals and a single server with exponential service times is presented. The focus of the paper is on the use of the functions that arise in the analysis of the transient behaviour of the queuing system. These functions are shown to be a generalization of the modified Bessel functions of the first kind, with the batch size B as the generalizing parameter. Results for the case of single-packet arrivals are obtained first. The similarities between the two families of functions are then used to obtain results for the general case of batch arrival queue with a batch size larger than one.

Keywords: batch arrivals, generalized Bessel functions, queuetransient analysis, time-varying probabilities.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1062256

References:

[1] G. N. Higginbottom, Performance Evaluation of Communication Networks, Artech House 1998
[2] H. P. Schwefel, L. Lipsky, M. Jobmann, "On the Necessity of Transient Performance Analysis in Telecommunication Networks," 17th International Teletraffic Congress (ITC17), Salvador da Bahia, Brazil, September 24-28 2001
[3] B. van Holt, C. Blondia, "Approximated Transient Queue Length and Waiting Time Distribution via Steady State Analysis", Stochastic Models 21, pp.725-744, 2005
[4] T. Hofkens, K. Spacy, C. Blondia, "Transient Analysis of the DBMAP/ G/1 Queue with an Applications to the Dimensioning of Video Playout Buffer for VBR Traffic", Proceedings of Networking, Athens Greece, 2004
[5] D. M. Lucantoni, G. L. Choudhury, W. Witt, "The Transient BMAP/PH/1 Queue", Stochastic Models 10, pp.461-478, 1994
[6] W. Böhm. S. G. Mohanty, "Transient Analysis of Queues with Heterogeneous Arrivals" , Queuing Systems, Vol.18, pp.27-45, 1994
[7] V. K. Oduol, "Transient Analysis of a Single-Server Queue with Batch Arrivals Using Modeling and Functions akin to the Modified Bessel Functions", International Journal of Applied Science, Engineering and Technology, Vol.5. No.1,2009 pp.34-39.
[8] G. L. Choudhury, D. M. Lucantoni, W. Witt, "Multidimensional Transform Inversion with Application to the Transient M/G/1 Queue", Annals of Applied Probability, 4, 1994, pp.719-740.
[9] J. Abate, G. L. Choudhury, W. Whitt, "An Introduction to Numerical Transform Inversion and its Application to Probability Models" In: W. Grassman, (ed.) Computational Probability, pp. 257-323. Kluwer, Boston , 1999.
[10] I. S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Alan Jeffrey and Daniel Zwillinger (eds.) Seventh edition, Academic Press (Feb 2007)
[11] L. Kleinrock, R. Gail, Queuing Systems: Problems and Solutions, John Wiley & Sons, 1996.
[12] R. G. Hohlfeld, J. I. F. King, T. W. Drueding, G. v. H. Sandri, "Solution of convolution inegral equations by the method of differential inversion", SIAM Journal on Applied Mathematics, Vol. 53 , No.1 (February 1993), Pages: 154 - 167
[13] A. S. Vasudeva Murthy, "A note on the differential inversion method of Hohlfeld et al.",SIAM Journal on Applied Mathematics, Vol. 55 , No.3 (June 1995), pp. 719 - 722
[14] P. L. Bharatiya , "The Inversion of a Convolution Transform Whose Kernel is a Bessel Function", The American Mathematical Monthly, Vol. 72, No. 4. (Apr., 1965), pp. 393-397