Abstract
A new queueing model is proposed, namely the \(MM\,\sum_{k=1}^{K}{CPP_k}/GE/c/L\,G\)-Queue, also known as the Sigma queue. A steady state solution for this queue in a computable form is presented, based on an analytic methodology applying certain non-trivial transformations on the balance equations, which turn out to be of the QBD-M (Quasi Simultaneous-Multiple-Bounded Births and Deaths) type. As a consequence, existing methods such as the spectral expansion method or the matrix-geometric method can be used to obtain the steady state probabilities. The utility of this queue is demonstrated through the performance analysis of an optical burst switching (OBS) node. Numerical results show the impact of the burstiness of arrival process on the performance of the OBS multiplexer.
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References
3GPP Technical Report 25.214, version 7.0.0. Physical layer procedures (FDD). The 3GPP project (March 2006)
Chakka, R.: Performance and Reliability Modelling of Computing Systems Using Spectral Expansion. PhD thesis, University of Newcastle upon Tyne (Newcastle upon Tyne) (1995)
Chakka, R.: Spectral Expansion Solution for some Finite Capacity Queues. Annals of Operations Research 79, 27–44 (1998)
Kouvatsos, D.: Entropy Maximisation and Queueing Network Models. Annals of Operations Research 48, 63–126 (1994)
Do, T.V., Chakka, R., Harrison, P.G.: An integrated analytical model for computation and comparison of the throughputs of the UMTS/HSDPA user equipment categories. In: MSWiM 2007: Proceedings of the 10th ACM Symposium on Modeling, Analysis, and Simulation of Wireless and Mobile Systems, pp. 45–51. ACM, New York (2007)
Do, T.V., Do, N.H., Chakka, R.: Performance Evaluation of the High Speed Downlink Packet Access in Communications Networks Based on High Altitude Platforms. In: Al-Begain, K., Heindl, A., Telek, M. (eds.) ASMTA 2008. LNCS, vol. 5055, pp. 310–322. Springer, Heidelberg (2008)
El-Bawab, T.S., Shin, J.-D.: Optical Packet Switching in Core Networks: Between Vision and Reality. IEEE Communications Magazine, 60–65 (September 2002)
Fourneau, J.M., Gelenbe, E., Suros, R.: G-networks with Multiple Classes of Positive and Negative Customers. Theoretical Computer Science 155, 141–156 (1996)
Fretwell, R.J., Kouvatsos, D.D.: ATM traffic burst lengths are geometrically bounded. In: Proceedings of the 7th IFIP Workshop on Performance Modelling and Evaluation of ATM & IP Networks, Antwerp, Belgium. Chapman and Hall, Boca Raton (1999)
Haverkort, B., Ost, A.: Steady state analyses of infinite stochastic Petri nets: A comparison between the spectral expansion and the matrix geometric methods. In: Proceedings of the 7th International Workshop on Petri Nets and Performance Models, Saint Malo, France, pp. 335–346 (1997)
Meier-Hellstern, K.S.: The analysis of a queue arising in overflow models. IEEE Transactions on Communications 37, 367–372 (1989)
Mitrani, I., Chakka, R.: Spectral expansion solution for a class of Markov models: Application and comparison with the matrix-geometric method. Performance Evaluation 23, 241–260 (1995)
Naoumov, V., Krieger, U., Wagner, D.: Analysis of a Multi-server Delay-loss System with a General Markovian Arrival Process. In: Chakravarthy, S.R., Alfa, A.S. (eds.) Matrix-Analytical Methods in Stochastic Models, pp. 43–66. Marcel Dekker, New York (1997)
Neuts, M.F.: Matrix Geometric Soluctions in Stochastic Model. Johns Hopkins University Press, Baltimore (1981)
Norros, I.: A Storage Model with Self-similar Input. Queueing Systems and their Applications 16, 387–396 (1994)
Shah-Heydari, S., Le-Ngoc, T.: MMPP models for multimedia traffic. Telecommunication Systems 15(3-4), 273–293 (2000)
Stewart, W.J.: Introduction to Numerical Solution of Markov Chains. Princeton University Press, Princeton (1994)
Turner, J.S.: Terabit Burst Switching. Journal of High Speed Networks 8, 3–16 (1999)
Venkatesh, T., Sankar, A., Jayaraj, A., Siva Ram Murthy, C.: A Complete Framework to Support Controlled Burst Retransmission in Optical Burst Switching Networks. IEEE Journal on Selected Areas in Communications 26(S-3), 65–73 (2008)
Yao, S., Xue, F., Mukherjee, B., Ben Yoo, S.J., Dixit, S.: Electrical Ingress Buffering and Traffic Aggregation for Optical Packet Switching and Their Effect on TCP-Level Performance in Optical Mesh Networks. IEEE Communications Magazine, 66–72 (2002)
Zhang, Z., Liu, L., Yang, Y.: Slotted Optical Burst Switching (SOBS) networks. Computer Communications 30(18), 3471–3479 (2007)
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Chakka, R., Van Do, T. (2011). Some New Markovian Models for Traffic and Performance Evaluation of Telecommunication Networks. In: Kouvatsos, D.D. (eds) Network Performance Engineering. Lecture Notes in Computer Science, vol 5233. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02742-0_27
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DOI: https://doi.org/10.1007/978-3-642-02742-0_27
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