Abstract
Queueing theory is a useful tool in design of computer networks and their performance evaluation. The literature concerning this subject is abundant. However, it is in general limited to the analysis of steady states. It means that flows of customers considered in models are constant and obtained solutions do not depend on time. It is in glaring contrast with the flows observed in real networks where the perpetual changes of traffic intensities are due to the nature of users, sending variable quantities of data, cf. multimedia traffic, and also due to the performance of traffic control algorithms which are trying to avoid congestion in networks, e.g. the algorithm of congestion window used in TCP protocol which is adapting the rate of the sent traffic to the observed losses or transmission delays. We discuss here the means used to analyse transient states in queueing models. In computer applications a mathematical model is useful only when it furnishes quantitative results. Therefore practical issues related to numerical side of models are of importance and are here discussed. We present three approaches—Markov models solved numerically, fluid flow approximation and diffusion approximation. A particular importance is given to the latter as the author has here over 20 year experience in development and application of this method. He is also convinced of the qualities of this approach—its flexibility to treat various variants of queueing models. Traffic intensity observed in computer networks have a complex stochastic nature that influences the network performances. We discuss also this side of implemented queueing models.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A.K. Erlang, The theory of probabilities and telephone conversations. Nyt Tidsskr. Mat. B 20, 33–39 (1909)
A.K. Erlang, Solutions of some problems in the theory of probabilities of significance in automatic telephone exchnges. Electroteknikeren 13, 5–13 (1917)
T.O. Engset, Die Wahrscheinlichkeitsrechnung zur Bestimmung der Whlerzahl in automatischen Fernsprechmtern, Elektrotechnische Zeitschrift, Heft 31 (1918)
D.G. Kendall, Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann. Math. Stat. 24(3), 338 (1953)
L. Kleinrock, Queueing Systems, volume I: Theory, volume II: Computer Applications (Wiley, New York, 1975/1976)
M. Stasiak, M. Gabowski, A. Wisniewski, P. Zwierzykowski, Modelling and Dimensioning of Mobile Networks, from GSM to LTE, (Wiley, 2011)
A.L. Sherr, An analysis of time-shared computer system, Ph.D. Thesis, Project MAC, MIT Press, Cambridge, 1967
A.Y. Khinchin, On the average stopping time of machines (in Russian). Mat. Sb. 40, 119–123 (1933)
E.D. Lazowska, J. Zahorjan, G.S. Graham, K.C. Sevcik, Computer System Analysis Using Queueing Network Models (Prentice-Hall Inc, New Jersey, 1984)
H. Kobayashi, Modeling and Analysis: An Introduction to System Performance Evalution Methodology, Quantitative System Performance (Addison Wesley, Reading, 1978.)
E. Gelenbe, I. Mitrani Analysis and synthesis of computer systems (Academic Press, London, 1980)
W. Willinger, W.E. Leland, M.S. Taqqu, On the self-similar nature of ethernet traffic. IEEE/ACM Trans. Netw. 2, 1–15 (1994)
E. Gelenbe, On approximate computer systems models. J. ACM 22(2), 261–269 (1975)
T. Czachórski, A method to solve diffusion equation with instantaneous return processes acting as boundary conditions. Bulletin of Polish Academy of Sciences. Tech. Sci. 41(4), 417–451 (1993)
F. Baskett, M. Chandy, R. Muntz, J. Palacios, Open, closed and mixed networks of queues with different classes of customers. J. ACM 22(2), 248–260 (1975)
P. Reinecke, T. Krauß, K. Wolter, Hyperstar: phase-type fitting made easy. in 9th International Conference on the Quantitative Evaluation of Systems (QEST) 2012. 201202 Tool Presentation (September 2012)
D.C. Champernowne, An elementary method of solution of the queueing problem with a single server and constant parameters. J. R. Stat. Soc. B18, 125–128 (1956)
L. Takâcs, Introduction to the Theory of Queues (Oxford University Press, Oxford, 1960)
A.M.K. Tarabia, Transient analysis of M/M/1/N queue—an alternative approach. Tamkang J. Sci. Eng. 3(4), 263–266 (2000)
T.C.T. Kotiah, Approximate transient analysis of some queueing systems. Oper. Res. 26(2), 334–346 (1978)
S.K. Jones, R.K. Cavin, D.A. Johnston, An efficient computational procedure for the evaluation of the \(M/M/1\) transient state occupancy probabilities. IEEE Trans. Commun. COM–28(12), 2019–2020 (1980)
B. Mandelbrot, J.V. Ness, Fractional brownian motions, fractional noises and applications. SIAM Review, vol. 10 (1968)
D.R. Cox, Long-Range Dependance: A Review, Statistics: An Appraisal (Lowa State University Press, Lowa, 1984)
I. Norros, On the use of fractional Brownian motion in the theory of connectionless networks. IEEE J. Sel. Areas Commun. 13(6), 953–962 (1995)
T. Mikosch, S. Resnick, H. Rootzen, A. Stegeman, Is network traffic approximated by stable levy motion or fractional Brownian motion? Anal. Appl. Probab. 12(1), 23–68 (2002)
A. Erramilli, R.P. Singh, P. Pruthi, An application of determinic chaotic maps to model packet traffic. Queueing Syst. 20(1–2), 171–206 (1995)
J.R. Gallardo, D. Makrakis, L. Orozco-Barbosa, Use a \(\alpha \)-stable self-similar stochastic processes for modeling traffic in broadband networks. Perform. Eval. 40(1–3), 71–98 (2000)
F. Harmantzis, D. Hatzinakos, Heavy network traffic modeling and simulation using stable FARIMA processes. in 19th International Teletraffic Congress (Beijing, 2005)
N. Laskin, I. Lambadatis, F.C. Harmantzis, M. Devetsikiotis, Fractional levy motion and its application to network traffic modeling. Comput. Netw. 40(3), 363–375 (2002)
G. Casale, Building accurate workload models using Markovian arrival processes, SIGMETRICS’11, (San Jose, U.S.A., 2011), pp. 7–11
A.T. Andersen, B.F. Nielsen, A markovian approach for modeling packet traffic with long-range dependence. IEEE J. Sel. Areas Commun. 16(5), 719–732 (1998)
W. Fischer, K. Meier-Hellstern, The Markov-modulated Poisson process (MMPP) cookbook. Perform. Eval. 18(2), 149–171 (1993)
J. Domanska, A. Domanski, T. Czachorski, Internet Traffic Source Based on Hidden Markov Model, NEW2AN 2011 (St. Petersburg, Russia, 2011)
D. Potier, New user’s introduction to QNAP2, Rapport Technique no. 40, INRIA, Rocquencourt (1984)
W. Stewart, Introduction to the Numerical Solution of Markov Chains (Princeton University Press, Chichester, 1994)
PRISM—probabilistic model checker, www.prismmodelchecker.org/
C. Moler, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 30–49 (2003)
P. Pecka, S. Deorowicz, M. Nowak, Efficient representation of transition matrix in the markov process modeling of computer networks, in Man-Machine Interactions 2, Advances in Intelligent and Soft Computing no. 103, ed. by T. Czachórski, et al. (Springer, 2011), pp. 457–464
C. Scientifique, B. Philippe, R.B. Sidje, Transient solutions of Markov processes by krylov subspaces. 2nd International Workshop on the Numerical Solution of Markov Chains (1989)
R.B. Sidje, K. Burrage, S. McNamara, Inexact uniformization method for computing transient distributions of Markov chains. SIAM J. Sci. Comput. 29(6), 2562–2580 (2007)
R.B. Sidje, W.J. Stewart, A numerical study of large sparse matrix exponentials arising in Markov chains. Comput. Stat. Data Anal. 29, 345–368 (1999)
R.B. Sidje, Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24(1), 130–156 (1998)
Numerical computation for Markov chains on GPU: building chains and bounds, algorithms and applications. Project POLONIUM 2012–2013, bilateral cooperation PRISM-Université de Versailles and IITiS PAN, Polish Academy of Sciences
E. Gelenbe, G. Pujolle, The behaviour of a single queue in a general queueing network. Acta Inform. 7(Fasc. 2), 123–136 (1976)
R.P. Cox, H.D. Miller, The Theory of Stochastic Processes (Chapman and Hall, London, 1965)
H. Stehfest, Algorithm 368: numeric inversion of laplace transform. Commun. ACM 13(1), 47–49 (1970)
P.J. Burke, The output of a queueing system. Oper. Res. 4(6), 699–704 (1956)
T. Czachorski, J.-M. Fourneau, T. Nycz, F. Pekergin, Diffusion approximation model of multiserver stations with losses. Electron. Notes Theor. Comput. Sci. 232, 125–143 (2009)
T. Czachrski, K. Grochla, T. Nycz, F. Pekergin, Diffusion approximation models for transient states and their application to priority queues. IARIA J. Int. J. Adv. Netw. Serv. 2(3), 205–217 (2009)
K. Hollot, Y. Liu, V. Misra, D. Towsley, W.B. Gong, Fluid methods for modeling large heterogeneous networks. Technical report AFRL-IF-RS-TR-2005-282 (2005)
Y. Liu, F. Lo Presti, V. Misra, Y. Gu, Fluid Models and Solutions for Large-Scale IP Networks, ACM/SigMetrics (2003)
V. Misra, W. Gong, D. Towsley, A fluid-based Analysis of a network of AQM routers supporting TCP flows with an application to RED. in Proceedings of the Conference on Applications, Technologies, Architectures and Protocols for Computer Communication (SIGCOMM 2000), pp. 151–160 (2000)
A. Domański, J. Domańska, T. Czachórski, Comparison of CHOKe and gCHOKe Active Queues Management Algorithms with the use of Fluid Flow Approximation, Communications in Computer and Information Science, vol 370 (Springer, Berlin, 2013)
A. Domański, J. Domańska, T. Czachórski, Comparison of AQM Control Systems with the Use of Fluid Flow Approximation, Communications in Computer and Information Science, vol 291 (Springer, Heidelberg, 2012)
J. Domańska, A. Domański, T. Czachórski, Fluid Flow Analysis of RED Algorithm with Modified Weighted Moving Average, Communications in Computer and Information Science, vol 356 (Springer, Berlin, 2013)
T. Czachórski, M. Nycz, T. Nycz, F. Pekergin, Analytical and numerical means to model transient states in computer networks’, in 20th International Conference, CN 2013, (Lwowek Slaski, Poland, June 17–21, 2013). Springer Proceedings Series: Communications in Computer and Information Science, Vol. 370, pp. 426–435, ISBN: 978-3-642-38864-4
T. Nycz, M. Nycz, T. Czachórski, A numerical comparison of diffusion and fluid-flow approximations used in modelling transient states of TCP/IP networks, in Proceedings of Computer Networks, ed. by A. Kwiecie, P. Gaj, P. Stera (Springer, Berlin, 2014)
OMNET++ Community Site, www.omnetpp.org
Acknowledgments
This work was supported by Polish project NCN nr 4796/B/T02/2011/40 “Models for transmissions dynamics, congestion control and quality of service in Internet”.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Czachórski, T. (2015). Queueing Models for Performance Evaluation of Computer Networks—Transient State Analysis. In: Mityushev, V., Ruzhansky, M. (eds) Analytic Methods in Interdisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol 116. Springer, Cham. https://doi.org/10.1007/978-3-319-12148-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-12148-2_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12147-5
Online ISBN: 978-3-319-12148-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)