Abstract
This paper gives a pathwise construction of Jackson-type queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanism are jointly stationary and ergodic in a sense that is made precise in the paper. The main tools for these results are the subadditive ergodic theorem, which is used to derive a strong law of large numbers, and basic theorems on monotone stochastic recursive sequences. The techniques which are proposed here apply to other and more general classes of discrete event systems, like Petri nets or GSMPs. The paper also provides new results on the Jackson-type networks with i.i.d. driving sequences which were studied in the past.
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The work of this author was supported in part by a grant from the European Commission DG XIII, under the BRA Qmips contract.
The work of this author was supported by a sabbatical grant from INRIA Sophia Antipolis.
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Baccelli, F., Foss, S. Ergodicity of Jackson-type queueing networks. Queueing Syst 17, 5–72 (1994). https://doi.org/10.1007/BF01158688
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DOI: https://doi.org/10.1007/BF01158688
Keywords
- Ordered directed graph
- Euler graphs
- Euler ordered directed graph
- switching sequence
- open Jackson-type queueing network
- point processes
- Euler network
- composition
- decomposition
- conservation rule
- departure and throughput processes
- first and second-order ergodic properties
- subadditive ergodic theorem
- solidarity property
- stochastic recursive sequences
- stationary solution
- coupling-convergence
- uniqueness of the stationary regime