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**Self-similar processes in communications networks.**
*(English)*
Zbl 0988.90003

From the introduction: The main objective of the present paper is to review and briefly discuss the known definitions and properties of second-order self-similar discrete-time processes, to supplement them with some more general conditions of self-similarity, to present a model for ATM cell traffic, and, finally, to find the conditions of model self-similarity.

Section II contains definitions of exactly and asymptotically second-order self-similar processes, which we adopt. The most essential second-order properties of these processes are presented. A novelty here is the presentation of some unknown proofs and properties, as well as the presentation of all these properties in one paper. A comparison of different definitions is done, with discussion and comments.

Section III gives a model of ATM cell traffic, the necessary and sufficient conditions for its exact self-similarity and a sufficient condition for its asymptotic self-similarity. The conditions are more general than others obtained earlier; they contain the known conditions as special cases. We reference earlier papers which are particularly relevant to the model and also discuss some other known models, which are linked, to our model.

The proots of our results are placed in Appendices A–D. In this presentation, we need to use the concepts of the Karamata slow- and regular-variation theory. The definitions of slowly and regularly varying functions and sequences are given in Appendix E. For other known results in the theory, we refer to N. H. Bingham, C. M. Goldie and J. L. Teugels [Regular Variation. Cambridge, New York: Cambridge Univ. Press (1987; Zbl 0617.26001)]. A brief presentation of our results was given in [N. Likhanov, B . Tsybakov and N. D. Georganas, “A model of self-similar communications-network traffic”, Proc. Int. Conf. “Distributed Computer Communication Networks” (DCCN’97) (Tel-Aviv, Israel, 1997), 212-217 (1997)].

Section II contains definitions of exactly and asymptotically second-order self-similar processes, which we adopt. The most essential second-order properties of these processes are presented. A novelty here is the presentation of some unknown proofs and properties, as well as the presentation of all these properties in one paper. A comparison of different definitions is done, with discussion and comments.

Section III gives a model of ATM cell traffic, the necessary and sufficient conditions for its exact self-similarity and a sufficient condition for its asymptotic self-similarity. The conditions are more general than others obtained earlier; they contain the known conditions as special cases. We reference earlier papers which are particularly relevant to the model and also discuss some other known models, which are linked, to our model.

The proots of our results are placed in Appendices A–D. In this presentation, we need to use the concepts of the Karamata slow- and regular-variation theory. The definitions of slowly and regularly varying functions and sequences are given in Appendix E. For other known results in the theory, we refer to N. H. Bingham, C. M. Goldie and J. L. Teugels [Regular Variation. Cambridge, New York: Cambridge Univ. Press (1987; Zbl 0617.26001)]. A brief presentation of our results was given in [N. Likhanov, B . Tsybakov and N. D. Georganas, “A model of self-similar communications-network traffic”, Proc. Int. Conf. “Distributed Computer Communication Networks” (DCCN’97) (Tel-Aviv, Israel, 1997), 212-217 (1997)].

### MSC:

90B18 | Communication networks in operations research |

94A05 | Communication theory |

90B20 | Traffic problems in operations research |