Abstract
In this paper we deal with transient analysis of networks of queues. These systems most often have enormous state space and the exact computation of their transient behavior is not possible. We propose to apply an approximate technique based on assumptions on the structure of the transient probabilities. In particular, we assume that the transient probabilities of the model can be decomposed into a quasi product form. This assumption simplifies the dependency structure of the model and leads to a relatively small set of ordinary differential equations (ODE) that can be used to compute an approximation of the transient probabilities. We provide the derivation of this set of ODEs and illustrate the accuracy of the approach on numerical examples.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Angius, A., Horváth, A.: Product form approximation of transient probabilities in stochastic reaction networks. Electronic Notes on Theoretical Computer Science 277, 3–14 (2011)
Angius, A., Horváth, A., Wolf, V.: Quasi product form approximation for markov models of reaction networks. In: Priami, C., Petre, I., de Vink, E. (eds.) Transactions on Computational Systems Biology XIV. LNCS, vol. 7625, pp. 26–52. Springer, Heidelberg (2012)
Anselmi, J., Casale, G., Cremonesi, P.: Approximate solution of multiclass queuing networks with region constraints. In: Proc. of 15th Int. Symp. on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems (MASCOTS 2007), pp. 225–230 (2007)
Baskett, F., Chandy, K.M., Muntz, R.R., Palacios, G.: Open, closed, and mixed networks of queues with different classes of customers. Journal of the ACM 22(2), 248–260 (1975)
Bazan, P., German, R.: Approximate transient analysis of large stochastic models with WinPEPSY-QNS. Computer Networks 53, 1289–1301 (2009)
Boucherie, R.J.: Product-form in queueing networks. PhD thesis, Vrije Universiteit, Amsterdam (1992)
Boucherie, R.J., Taylor, P.G.: Transient product form distributions in queueing networks. Discrete Event Dynamic Systems: Theory and Applications 3, 375–396 (1993)
Casale, G.: Approximating passage time distributions in queueing models by Bayesian expansion. Perform. Eval. 67(11), 1076–1091 (2010)
Chandy, K.M., Herzog, U., Woo, L.: Parametric analysis of queueing networks. IBM Journal of Research and Development 19(1), 36–42 (1975)
Chen, H., Mandelbaum, A.: Discrete flow networks: Bottleneck analysis and fluid approximations. Mathematics of Operations Research 16(2), 408–446 (1991)
Gordon, W.J., Newell, G.F.: Cyclic queueing networks with exponential servers. Operations Research 15(2), 254–265 (1967)
Harrison, J.M., Lemoine, A.J.: A note on networks of infinite-server queues. J. Appl. Probab. 18(2), 561–567 (1981)
Harrison, P.G.: Transient behaviour of queueing networks. Journal of Applied Probability 18(2), 482–490 (1981)
Horváth, A., Horváth, G., Telek, M.: A traffic based decomposition of two-class queueing networks with priority service. Computer Networks 53, 1235–1248 (2009)
Horváth, A., Horváth, G., Telek, M.: A joint moments based analysis of networks of MAP/MAP/1 queues. Performance Evaluation 67, 759–778 (2010)
Jackson, J.R.: Jobshop-like queueing systems. Management Science 10(1), 131–142 (1963)
Kuehn, P.: Approximate analysis of general queuing networks by decomposition. IEEE Transactions on Communications 27(1), 113–126 (1979)
Massey, W.A., Whitt, W.: Networks of infinite-server queues with nonstationary Poisson input. Queueing Systems 13, 183–250 (1993)
Matis, T.I., Feldman, R.M.: Transient analysis of state-dependent queueing networks via cumulant functions. Journal of Applied Probability 38(4), 841–859 (2001)
Melamed, B.: Characterizations of Poisson traffic streams in jackson queueing networks. Advances in Applied Probability 11(2), 422–438 (1979)
Sauer, C.H.: Approximate solution of queueing networks with simultaneous resource possession. IBM Journal of Research and Development 25(6), 894–903 (1981)
Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press (1995)
Whitt, W.: The queueing network analyzer. Bell System Technical Journal 62(9), 2779–2815 (1983)
Whitt, W.: Untold horrors of the waiting room. What the equilibrium distribution will never tell about the queue-length process. Management Science 29(4), 395–408 (1983)
Whitt, W.: Decomposition approximations for time-dependent Markovian queueing networks. Operations Research Letters 24, 97–103 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Angius, A., Horváth, A., Wolf, V. (2013). Approximate Transient Analysis of Queuing Networks by Quasi Product Forms. In: Dudin, A., De Turck, K. (eds) Analytical and Stochastic Modeling Techniques and Applications. ASMTA 2013. Lecture Notes in Computer Science, vol 7984. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39408-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-39408-9_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39407-2
Online ISBN: 978-3-642-39408-9
eBook Packages: Computer ScienceComputer Science (R0)