**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30172

##### System Overflow/Blocking Transients For Queues with Batch Arrivals Using a Family of Polynomials Resembling Chebyshev Polynomials

**Authors:**
Vitalice K. Oduol,
C. Ardil

**Abstract:**

The paper shows that in the analysis of a queuing system with fixed-size batch arrivals, there emerges a set of polynomials which are a generalization of Chebyshev polynomials of the second kind. The paper uses these polynomials in assessing the transient behaviour of the overflow (equivalently call blocking) probability in the system. A key figure to note is the proportion of the overflow (or blocking) probability resident in the transient component, which is shown in the results to be more significant at the beginning of the transient and naturally decays to zero in the limit of large t. The results also show that the significance of transients is more pronounced in cases of lighter loads, but lasts longer for heavier loads.

**Keywords:**
batch arrivals,
blocking probability,
generalizedChebyshev polynomials,
overflow probability,
queue transientanalysis

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1330773

**References:**

[1] 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

[2] B. van Holt, C. Blondia, "Approximated Transient Queue Length and Waiting Time Distribution via Steady State Analysis", Stochastic Models 21, pp.725-744, 2005

[3] 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

[4] D. M. Lucantoni, G. L. Choudhury, W. Witt, "The Transient BMAP/PH/1 Queue", Stochastic Models 10, pp.461-478, 1994

[5] W. Böhm. S. G. Mohanty, "Transient Analysis of Queues with Heterogeneous Arrivals" , Queuing Systems, Vol.18, pp.27-45, 1994

[6] 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, pp.34-39, April 2009

[7] V. K. Oduol, C. Ardil, "Transient Analysis of a Single-Server Queue with Fixed-Size Batch Arrivals",International Journal of Electronics,Communications and Computer Engineering, Vol. 1, No.1, pp.55-50, May 2009

[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] G. N. Higginbottom, Performance Evaluation of Communication Networks, Artech House 1998

[11] I. S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Alan Jeffrey and Daniel Zwillinger (eds.) Seventh edition, Academic Press, Feb. 2007

[12] M. Abramowitz, I. A. Stegun (eds.), "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, Dover, 1965.

[13] P. K. Suetin,. "Chebyshev polynomials", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, (2001)

[14] Wikipedia "Chebyshev polynomials" Available at http://en.wikipedia.org/wiki/Chebyshev_polynomials