Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T20:05:11.644Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Effect of Finite Buffer Truncation in a Two-Node Jackson Network

Part of: Markov processes Limit theorems Special processes

Published online by Cambridge University Press:  14 July 2016

Yutaka Sakuma  and
Masakiyo Miyazawa
Yutaka Sakuma*
Affiliation:
Tokyo University of Science
Masakiyo Miyazawa*
Affiliation:
Tokyo University of Science
*
∗Postal address: Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan.
∗Postal address: Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different cases for the limit. This generalizes some recent results obtained for the tandem Jackson network. Special cases and some numerical examples are also presented.

Keywords

Decay rateJackson networkfinite truncationstationary distribution

MSC classification

Secondary: 60K25: Queueing theory 60F10: Large deviations 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 199 - 222
Copyright
© Applied Probability Trust 2005 

References

Alsmeyer, G. (1994). On the Markov renewal theorem. Stoch. Process. Appl. 50, 37–56.CrossRefGoogle Scholar
Çinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Fujimoto, K., Takahashi, Y. and Makimoto, N. (1998). Asymptotic properties of stationary distributions in two-stage queueing systems. J. Operat. Res. Soc. Japan 41, 118–141.Google Scholar
Fujimoto, K., Takahashi, Y. and Makimoto, N. (2001). Geometric decay of the steady-state probabilities in a quasi-birth–death process with a countable number of phases. Stoch. Models 17, 1–24.Google Scholar
Höglund, T. (1991). The ruin problem for finite Markov chains. Ann. Prob. 19, 1298–1310.CrossRefGoogle Scholar
Jackson, J. R. (1957). Networks of waiting lines. Operat. Res. 5, 518–521.CrossRefGoogle Scholar
Kingman, J. F. C. (1961). A convexity property of positive matrices. Quart. J. Math. Oxford 12, 283–284.CrossRefGoogle Scholar
Kroese, D. P., Scheinhardt, W. R. W. and Taylor, P. G. (2004). Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process. Ann. Appl. Prob. 14, 2057–2089.CrossRefGoogle Scholar
McDonald, D. R. (1999). Asymptotics of first passage times for random walk in an orthant. Ann. Appl. Prob. 9, 110–145.CrossRefGoogle Scholar
Miyazawa, M. (2003). Conjectures on decay rates of tail probabilities in generalized Jackson and batch movement networks. J. Operat. Res. Soc. Japan 46, 74–98.Google Scholar
Miyazawa, M. (2004). The Markov renewal approach to M/G/1 type queues with countably many background states. Queueing Systems 46, 177–196.CrossRefGoogle Scholar
Miyazawa, M. and Zhao, Y. Q. (2004). The stationary tail asymptotics in the GI/G/1-type queue with countably many background states. Adv. Appl. Prob. 36, 1231–1251.CrossRefGoogle Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models (Johns Hopkins Ser. Math. Sci. 2). Johns Hopkins University Press, Baltimore, MD.Google Scholar
Seneta, E. (1981). Nonnegative Matrices and Markov Chains. Springer, New York.CrossRefGoogle Scholar
Shurenkov, V. M. (1984). On the theory of Markov renewal. Theory Prob. Appl. 29, 247–265.CrossRefGoogle Scholar