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Ergodicity of a Jackson network by batch arrivals

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

A. A. Borovkov  and
R. Schassberger
A. A. Borovkov*
Affiliation:
Russian Academy of Sciences
R. Schassberger*
Affiliation:
Technical University of Braunschweig
*
∗ Postal address: Institute of Mathematics, Russian Academy of Sciences, Novosibirsk, Russia 630090.
∗∗ Postal address: Institut für Mathematische Stochastik, Technical University of Braunschweig, Postfach 3329, D-38106 Braunschweig, Germany.
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Abstract

The Jackson network under study receives batch arrivals at i.i.d. intervals and features Markovian routing and exponentially distributed service times. The system is shown to be stable, in the sense of not being overloaded, if and only if, for each node, the total arrival rate of external and internal customers is less than the service rate. The method of proof is of more general interest.

Keywords

ERGODIC MARKOV CHAINQUEUEING NETWORK

MSC classification

Primary: 60J75: Jump processes
Secondary: 60K25: Queueing theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 31 , Issue 3 , September 1994 , pp. 847 - 853
Copyright
Copyright © Applied Probability Trust 1994 

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References

[1] Afanas'Eva, L. G. (1987) On the ergodicity of an open queuing network. Theory Prob. Appl. 32, 777–781.Google Scholar
[2] Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[3] Borovkov, A. A. (1986) Limit theorems for queueing networks I. Theory Prob. Appl. 31, 413–427.CrossRefGoogle Scholar
[4] Borovkov, A. A. and Schassberger, R. (1994) Ergodicity of a polling network. Stoch. Proc. Appl. 50, 253–262.CrossRefGoogle Scholar
[5] Fayolle, G., Malyshev, V. A., Menshikov, M. V. and Sidorenko, A. F. (1994) Lyapounov functions for Jackson network. Math. Operat. Res. 18, 916–927.CrossRefGoogle Scholar
[6] Goodman, J. B. and Massey, W. A. (1984) The non-ergodic Jackson network. J. Appl. Prob. 21, 860–869.CrossRefGoogle Scholar
[7] Jackson, J. R. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.CrossRefGoogle Scholar
[8] Rosberg, Z. (1980) A positive recurrence criterion associated with multidimensional queuing processes. J. Appl. Prob. 17, 790–801.CrossRefGoogle Scholar
[9] Sigman, K. (1990) The stability of open queuing networks. Stoch. Proc. Appl. 35, 11–25.CrossRefGoogle Scholar