{"title":"System Overflow\/Blocking Transients For Queues with Batch Arrivals Using a Family of Polynomials Resembling Chebyshev Polynomials","authors":"Vitalice K. Oduol, C. Ardil","volume":67,"journal":"International Journal of Electrical and Computer Engineering","pagesStart":731,"pagesEnd":738,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/3687","abstract":"<p>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.<\/p>\r\n","references":"[1] H. P. Schwefel, L. Lipsky, M. Jobmann, \"On the Necessity of Transient\r\nPerformance Analysis in Telecommunication Networks,\" 17th\r\nInternational Teletraffic Congress (ITC17), Salvador da Bahia, Brazil,\r\nSeptember 24-28 2001\r\n[2] B. van Holt, C. Blondia, \"Approximated Transient Queue Length and\r\nWaiting Time Distribution via Steady State Analysis\", Stochastic\r\nModels 21, pp.725-744, 2005\r\n[3] T. Hofkens, K. Spacy, C. Blondia, \"Transient Analysis of the DBMAP\/\r\nG\/1 Queue with an Applications to the Dimensioning of Video\r\nPlayout Buffer for VBR Traffic\", Proceedings of Networking, Athens\r\nGreece, 2004\r\n[4] D. M. Lucantoni, G. L. Choudhury, W. Witt, \"The Transient BMAP\/PH\/1\r\nQueue\", Stochastic Models 10, pp.461-478, 1994\r\n[5] W. B\u00f6hm. S. G. Mohanty, \"Transient Analysis of Queues with\r\nHeterogeneous Arrivals\" , Queuing Systems, Vol.18, pp.27-45, 1994\r\n[6] V. K. Oduol, \"Transient Analysis of a Single-Server Queue with Batch\r\nArrivals Using Modeling and Functions Akin to the Modified Bessel\r\nFunctions\" International Journal of Applied Science, Engineering and\r\nTechnology, Vol. 5, No.1, pp.34-39, April 2009\r\n[7] V. K. Oduol, C. Ardil, \"Transient Analysis of a Single-Server Queue with\r\nFixed-Size Batch Arrivals\",International Journal of Electronics,Communications and Computer Engineering, Vol. 1, No.1, pp.55-50,\r\nMay 2009\r\n[8] G. L. Choudhury, D. M. Lucantoni, W. Witt, \"Multidimensional\r\nTransform Inversion with Application to the Transient M\/G\/1 Queue\",\r\nAnnals of Applied Probability, 4, 1994, pp.719-740.\r\n[9] J. Abate, G. L. Choudhury, W. Whitt, \"An Introduction to Numerical\r\nTransform Inversion and its Application to Probability Models\" In: W.\r\nGrassman, (ed.) Computational Probability, pp. 257-323. Kluwer,\r\nBoston , 1999.\r\n[10] G. N. Higginbottom, Performance Evaluation of Communication\r\nNetworks, Artech House 1998\r\n[11] I. S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products,\r\nAlan Jeffrey and Daniel Zwillinger (eds.) Seventh edition, Academic\r\nPress, Feb. 2007\r\n[12] M. Abramowitz, I. A. Stegun (eds.), \"Chapter 22\", Handbook of\r\nMathematical Functions with Formulas, Graphs, and Mathematical\r\nTables, New York, Dover, 1965.\r\n[13] P. K. Suetin,. \"Chebyshev polynomials\", in Hazewinkel, Michiel,\r\nEncyclopaedia of Mathematics, Kluwer Academic Publishers, (2001)\r\n[14] Wikipedia \"Chebyshev polynomials\" Available at\r\nhttp:\/\/en.wikipedia.org\/wiki\/Chebyshev_polynomials","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 67, 2012"}