# On mean-field (GI/GI/1) queueing model: existence and uniqueness

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## Abstract

A mean-field extension of the queueing system (GI/GI/1) is considered. The process is constructed as a Markov solution of a martingale problem. Uniqueness in distribution is also established under a slightly different set of assumptions on intensities in comparison with those required for existence.

## Keywords

GI/GI/1 Mean-field Existence Weak uniqueness Skorokhod lemma## Mathematics Subject Classification

60-02 60K25 90B22## Notes

### Acknowledgements

The techniques used in this paper were stimulated by the methods developed in a long-term joint work on formally quite different McKean–Vlasov SDE equations with Yu. Mishura, as well as in fruitful discussions of the author on the same subject with D. Šiska, and L. Szpruch. S. Pirogov, A. Rybko, and G. Zverkina helped to find some (quite a few) technicalities to be corrected in the earlier versions of the text. The author is sincerely thankful to all these colleagues, as well as to the anonymous referee. The deepest gratitude is to Professor Alexander Dmitrievich Solovyev (06.09.1927—06.04.2001) who was the author’s supervisor at BSc and MSc programmes at University.

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